prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the effect of the proximity of superconductivity on the Kondo effect?

Read the following text and answer briefly.

Context 1: \section{Introduction}
\label{sec:Intro}

The exchange interactions control the magnetic order and properties of a vast number of materials
\cite{White2006Dec}
and lead to many fascinating phenomena, such as various types of the Kondo effect 
\cite{Kondo,NozieresBlandin,Pustilnik_Glazman} 
Double quantum dots (DQDs), and in general multi-impurity systems, constitute
a convenient and controllable playground,
where nearly as much different exchange mechanisms compete with each other to
shape the ground state of the system 
\emph{Local exchange} between the spin of a quantum dot (QD)
and the spin of conduction band electrons gives rise to the
Kondo effect \cite{Kondo,Hewson_book}  
\emph{Direct exchange} arriving with an additional side-coupled QD may destroy it or lead to the 
two-stage Kondo screening \cite{Pustilnik_Glazman,Cornaglia,Granger,ZitkoBonca,ZitkoPRB2010,Ferreira} 
In a geometry where the two QDs contact the same lead, conduction band electrons 
mediate the \emph{RKKY exchange} \cite{RK,K,Y}  The RKKY interaction competes
with the Kondo effect and leads to the quantum phase transition of a still debated nature
\cite{Doniach,Jones,Affleck,Bork,Neel,KondoRKKYexp,Hans,Hans2,Fabian} 
Moreover, in DQDs coupled in series also \emph{superexchange} can alter the Kondo physics significantly
\cite{Zitko_2QDEx,Sela} 

Recently, hybrid quantum devices, in which the interplay between various magnetic correlations
with superconductivity (SC) plays an important role, have become an important direction of research
\cite{hybridQDs,SCspintronics}  In particular, chains of magnetic atoms on SC surface have proven 
to contain self-organized Majorana quasi-particles and exotic spin textures
\cite{Braunecker,Klinovaja,Vazifeh,Yazdani},
while hybrid DQD structures have been used to split the Cooper pairs coherently into two entangled 
electrons propagating to separated normal leads \cite{CPS1,CPS2,CPS4,CPS5,CPS9} 
The latter is possible due to non-local (\emph{crossed}) Andreev reflections (CARs),
in which each electron of a Cooper pair tunnels into different QD, and
subsequently to attached lead  Such processes give rise to an exchange mechanism \cite{Yao},
that we henceforth refer to as \emph{the CAR exchange}, which can greatly modify
the low-temperature transport behavior of correlated hybrid nanostructures 

The CAR exchange may be seen as RKKY-like interaction between
two nearby impurities on SC surface \cite{Yao} 
The effect can be understood as a consequence
of spin-dependent hybridization of the Yu-Shiba-Rusinov (YSR)
states \cite{Yu,Shiba,Rusinov} in SC contact,
caused both by the overlap of their wave functions
and their coupling to Cooper-pair condensate 
This process is the most effective when the YSR states 
are close to the middle of the SC gap, {\it e g } in the YSR-screened phase \cite{YSRscreening} 
The mechanism presented here is essentially the same,
yet in the considered regime can be understood
perturbatively without referring to YSR states,
as a consequence of the non-local pairing induced by SC electrode  
In particular, the presence of YSR bound states close to the Fermi level 
is not necessary for significant consequences for the Kondo physics, 
as long as some inter-dot pairing is present  


The proximity of SC induces pairing in QDs \cite{RozhkovArovas,Buitelaar} 
and tends to suppress the Kondo effect if the superconducting energy gap $2\Delta$ 
becomes larger than the relevant Kondo temperature $T_K$ 
\cite{Buitelaar2002Dec,adatomsSC,Kondo_vs_SC1,Kondo_vs_SC2,Zitko_Kondo-Andreev,Zitko_S-QD-N,IW_Sau,YSRscreening} 
Moreover, the strength of SC pairing can greatly affect the Kondo physics in the sub-gap transport regime:
For QDs attached to SC and normal contacts, it can enhance the Kondo effect
\cite{DomanskiIW,KWIW,part1}, while
for DQD-based Cooper pair splitters, it tends to suppress both the $\mathrm{SU}(2)$ and $\mathrm{SU}(4)$ Kondo effects \cite{IW_Kacper} 
Our main result is that the non-local pairing induced by superconducting 
proximity effect, which gives rise to CAR exchange, can be the sole cause of the Kondo screening 
Moreover, relatively small values of coupling to SC, $\GS{}\ll U$, are sufficient for the effect to occur 
This is in contrast to the DQD system considered in Ref ~\cite{part1},
where only one of the quantum dots is proximized, such that 
CAR exchange cannot arise,
and the Kondo physics becomes qualitatively
affected only for $\GS{}\sim U/2$ %


\begin{figure}[bt]
\centering
\includegraphics[width=1\linewidth]{Fig1 png}
\caption{
		 (a) Schematic of the considered system  Left/right (L/R) lead
		 is coupled to the first quantum dot (QD1), while superconductor
		 is attached to both QD1 and QD2 
		 (b)-(d) illustrate an example of direct spin exchange:
		 spin-up electron from the initial state (b) hops to the other QD (c) and spin-down electron 
		 hops back (d)  Note, that the final state is in fact the same singlet state, 
		 only with opposite sign 
		 (e)-(g) show an example of process contributing to crossed Andreev reflection (CAR) exchange 
		 A Cooper pair from SC approaches DQD (e) and two singlets of the same charge 
		 are formed (f), before the Cooper pair is re-emitted (g) 
		 (h)-(j) present an example of RKKY process: an electron scattered off
		 one QD (h) mediates the spin exchange towards the other (i), before it is finally scattered
		 off there, too (j) 
		 }
\label{fig:system}
\end{figure}


In this paper we discuss the CAR-induced Kondo screening in a setup comprising T-shaped DQD
with normal and superconducting contacts, see \fig{system}(a) 
We note that despite quite generic character of CAR exchange,
and its presence in systems containing at least two localized electrons
coupled close to each other to the same SC bath,
to best of our knowledge CAR-induced screening
has hardly been identified in previous studies
\cite{CPS1,CPS2,CPS4,CPS5,CPS9,IW_Kacper,IW_Sau,Zitko_Josephson,Zitko_S2QD,Martinek2017} 
In the system proposed here [\fig{system}(a)], its presence is evident 
Moreover, CAR exchange magnitude can be directly related to the relevant energy scales, such as the Kondo 
temperature, which provides a fingerprint for quantitative experimental verification of our predictions  

The paper is organized as follows  In \Sec{model} we describe the considered system 
and present the model we use to study it  In \Sec{scales} the relevant energy scales are estimated
to make the discussion of main results concerning CAR-induced Kondo effect in \Sec{main} more clear  
Finally, the influence of effects neglected in \Sec{main} are presented in the following sections,
including CAR exchange interplay with RKKY interaction (\Sec{RKKY}), particle-hole asymmetry (\Sec{asym}),
couplings asymmetry (\Sec{x}) and reduced efficiency of CAR coupling (\Sec{coef})  In summary,
the effects discussed in \Sec{main} remain qualitatively valid in all these cases 
The paper is concluded in \Sec{conclusions} 


\section{Model}
\label{sec:model}

The schematic of the considered system is depicted in \fig{system}(a) 
It contains two QDs attached to a common SC lead 
Only one of them (QD1) is directly attached to the left (L) and right (R) normal leads,
while the other dot (QD2) remains coupled only through QD1 
The SC is modeled by the BCS Hamiltonian, 
$H_{\mathrm{S}}=\sum_{\mathbf{k}\sigma}\xi_{\mathbf{k}}a_{\mathbf{k}\sigma}^{\dag}a_{\mathbf{k}\sigma}-\Delta\sum_{\mathbf{k}}(a^\dag_{\mathbf{k}\uparrow}a_{-\mathbf{k}\downarrow}^{\dag}+a_{-\mathbf{k}\downarrow}a_{\mathbf{k}\uparrow})$,
with energy dispersion $\xi_{\mathbf{k}}$, energy gap $2\Delta>0$ and $a_{\mathbf{k}\sigma}$ annihilation operator 
of electron possessing spin $\sigma$ and momentum $\mathbf{k}$  The coupling between
SC and QDs is described by the hopping Hamiltonian 
$H_{\mathrm{TS}}=\sum_{i\mathbf{k}\sigma}v_{\mathrm{S}i}(d^\dagger_{i\sigma}a^{}_{\mathbf{k}\sigma}+h c )$,
with $d^\dagger_{i\sigma}$ creating a spin-$\sigma$ electron at QD$i$  The matrix element 
$v_{\mathrm{S}i}$ and the normalized density of states of SC in normal state, $\rho_{\rm S}$, 
contribute to the coupling of QD$i$ to SC electrode as $\GS{i} = \pi \rho_{\rm S} |v_{{\rm S}i}|^2$  
We focus on the sub-gap regime, therefore, we integrate out SC degrees of freedom lying outside the energy gap \cite{RozhkovArovas} 
This gives rise to the following effective Hamiltonian,
$H_{\mathrm{eff}}=H_{\mathrm{SDQD}}+H_{\rm L}+H_{\rm R}+H_{\rm T}$, 
where 
\begin{eqnarray}
H_{\rm SDQD} 	& = & 
				\sum_{i\sigma} \varepsilon_{i} n_{i\sigma} 
				+\sum_{i} U n_{i\uparrow} n_{i\downarrow} 
				+U' (n_1-1)(n_2-1) 
				\nonumber\\
				&+&\sum_\sigma t(d^\dagger_{1\sigma}d^{}_{2\sigma} + h c ) 
				+J \vec{S}_1\vec{S}_2
				\nonumber\\
				&+&\sum_{i} \!\!\left[ \Gamma_{{\rm S}i} (d^\dagger_{i\uparrow} d^\dagger_{i\downarrow} \!+\! h c )
				+\Gamma_{\rm SX} (d^\dagger_{i\uparrow} d^\dagger_{\bar{i}\downarrow} \!+\! h c ) \right]
	\label{H_DQD} 
\end{eqnarray}
is the Hamiltonian of the SC-proximized DQD
\cite{IW_Kacper,Walldorf2018Feb}, with QD$i$ energy level $\varepsilon_i$,
inter-site (intra-site) Coulomb interactions $U'$ ($U$),
inter-dot hopping $t$, and CAR coupling $\GS{\rm X}$ 
$n_{i\sigma}=d^\dagger_{i\sigma}d^{}_{i\sigma}$ denotes the electron number operator 
at QD$i$, $n_i=n_\uparrow+n_\downarrow$, and $\bar{i}\equiv 3-i$  
Our model is strictly valid in the regime where $\Delta$ is the largest 
energy scale  Nevertheless, all discussed phenomena are
present in a full model for energies smaller than SC gap 
Moreover, by eliminating other consequences of the presence of SC lead,
our model pinpoints the fact that the non-local pairing is 
sufficient for the occurrence of the CAR exchange 
The presence of out-gap states shall result mainly in additional broadening of DQD energy levels,
changing the relevant Kondo temperatures 
We note that the procedure of integrating out out-gap states neglects the 
RKKY interaction mediated by SC lead and other possible indirect exchange mechanisms%
 \footnote{
 Note, that by RKKY interaction we mean only such an effective exchange, 
 which arises due to multiple scattering of a single electron or hole, see \fig{system}(h)-(j) 
 Other mechanisms leading to the total indirect exchange are considered separately 
 In particular, in the large gap limit, exchange described in Ref ~\cite{Yao} is in fact reduced to
 the CAR exchange, and additional antiferromagnetic contribution would arise for finite gap 
 }  
To compensate for this,
we explicitly include the Heisenberg term $ J \vec{S}_1\vec{S}_2$ in
$H_{\rm SDQD}$, with $\vec{S}_i$ denoting the spin operator of QD$i$
and a Heisenberg coupling $J$ substituting the genuine RKKY exchange 

The normal leads are treated as reservoirs of noninteracting electrons,
$H_{r}=\sum_{\mathbf{k}\sigma}\varepsilon_{r\mathbf{k}}c^\dagger_{r\mathbf{k}\sigma}c^{}_{r\mathbf{k}\sigma}$,
where $c^{}_{r\mathbf{k}\sigma}$ annihilates an electron of spin 
$\sigma$ and momentum $\mathbf{k}$ in lead $r$ ($r={\rm L,R}$) with the corresponding energy $\varepsilon_{r\mathbf{k}\sigma}$ 
The tunneling Hamiltonian reads,
$H_{\rm T} = \sum_{r\mathbf{k}\sigma} v_{r} (d^\dagger_{1\sigma}c^{}_{r\mathbf{k}\sigma} + h c )$,
giving rise to coupling between lead $r$ and QD$i$ of strength $\Gamma_r = \pi \rho_r |v_r|^2$,
with $\rho_r$ the normalized density of states of lead $r$ and $v_r$ the 
local hopping matrix element, assumed momentum-independent 
We consider a wide-band limit, assuming constant $\Gamma_r=\Gamma/2$
within the cutoff $\pm D = \pm 2U$ around the Fermi level  

For thorough analysis of the CAR exchange mechanism and its consequences
for transport, we determine the linear conductance between the two normal leads from
\begin{equation}
G = \frac{2e^2}{h} \pi \Gamma \int \left[ -\frac{\partial f_T}{\partial\omega} \right] \mathcal{A}(\omega) {\rm d} \omega ,
\label{G}
\end{equation}
where $f_T$ is the Fermi function at temperature $T$,
while $\mathcal{A}(\omega)$ denotes the normalized local spectral density 
of QD1 \cite{fn1} 
Henceforth, unless we state otherwise, we assume a maximal CAR coupling, 
$\GS{\rm X} = \sqrt{\GS{1}\GS{2}}$ \cite{IW_Kacper,Walldorf2018Feb},
$\GS{1}=\GS{2}=\GS{}$ and consider DQD tuned to the particle-hole symmetry point, 
$\varepsilon_1=\varepsilon_2=-U/2$  However, these assumptions are not crucial for the results presented
here, as discussed in Secs ~\ref{sec:asym}-\ref{sec:coef} 

\section{Estimation of relevant energy scales}
\label{sec:scales}

Since we analyze a relatively complex system, let us build up the understanding of its behavior starting
from the case of a QD between two normal-metallic leads, which can be obtained in our 
model by setting $t=\GS{}=J=U'=0$  Then, the conductance as a function of temperature, $G(T)$, grows
below the Kondo temperature $T_K$ and reaches maximum for $T\to 0$, $G(T\!=\!0)=G_{\rm max}$ 
At particle-hole symmetry point, the unitary transmission is achieved, $G_{\rm max}= G_0 = 2e^2/h$;
see short-dashed line in \fig{G-T}(a) 
An experimentally relevant definition of $T_K$ is that at $T=T_K$ 
$G(T)=G_{\rm max}/2$  $T_K$ is exponentially small in 
the local exchange $J_0 = 8\Gamma / (\pi \rho U)$, and is approximated by
$T_K \approx D \exp[-1/(\rho J_0)]$ \cite{Hewson_book} 

The presence of a second side-coupled QD, $t,U'>0$, significantly enriches the physics of the system 
by introducing direct exchange between QDs, see \fig{system}(b-d) 
In general, effective inter-dot exchange can be defined as energy difference between 
the triplet and singlet states of isolated DQD, 
$J^{\mathrm{eff}} = E_{S=1} - E_{\rm GS}$  Unless $U$ becomes very large, superexchange can be neglected
\cite{Zitko_2QDEx} and $J^{\mathrm{eff}}$ is determined by \emph{direct exchange}, $J^{\mathrm{eff}}\approx 4t^2/(U-U')>0$ 
When the hopping $t$ is tuned small \cite{CPS1}, one can expect $J^{\mathrm{eff}}\lesssim T_K$, which 
implies the two-stage Kondo screening \cite{Pustilnik_Glazman,Cornaglia} 
Then, for $T \ll T_K$, the local spectral density of QD1 serves as a band of width $\sim T_K$ for QD2 
The spin of an electron occupying QD2 
experiences the Kondo screening below the associated Kondo temperature
\begin{equation}
T^* = a T_K \exp(- b T_K / J_{\rm eff})
\label{Tstar}
\end{equation}
with $a$ and $b$ constants of order of unity \cite{Pustilnik_Glazman,Cornaglia} 
This is reflected in conductance, which drops to $0$ with lowering $T$, maintaining characteristic 
Fermi-liquid 
$G\sim T^2$ dependence \cite{Cornaglia}; see the curves indicated with squares 
in \fig{G-T}(a)  Similarly to $T_K$, experimentally relevant definition of $T^*$ is that 
$G(T\!=\!T^*) = G_{\rm max}/2$  Even at the particle-hole 
symmetry point $G_{\rm max} < G_0$, because the single-QD strong-coupling fixed point 
is unstable in the presence of QD2 and $G(T)$ does not achieve $G_0$ exactly,
before it starts to decrease 


The proximity of SC gives rise to two further exchange mechanisms that
determine the system's behavior  First of all, the (conventional)
\emph{RKKY interaction} appears, $J \sim \GS{}^2$ \cite{RK,K,Y}  
Moreover, the \emph{CAR exchange} emerges as a consequence of finite $\GS{}$ \cite{Yao}  
It can be understood on the basis 
of perturbation theory as follows  DQD in the inter-dot singlet state may absorb
and re-emit a Cooper pair approaching from SC; see \fig{system}(e)-(g)  As a second-order
process, it reduces the energy of the singlet, which is the ground state of isolated DQD 
A similar process is not possible in the triplet state due to spin conservation 
Therefore, the singlet-triplet energy splitting $J^{\mathrm{eff}}$ is increased (or generated for $t=J=0$)  
More precisely, the leading ($2$nd-order in $t$ and $\GS{}$) terms
in the total exchange are 
\begin{equation}
J^{\mathrm{eff}} 	\approx 	J + \frac{4t^2}{U-U'+\frac{3}{4}J} + \frac{4\GS{}^2}{U+U'+\frac{3}{4}J} 
\label{Jeff}
\end{equation}
Using this estimation, one can predict $T^*$ for finite $\GS{}$, $t$ and $J$ with \eq{Tstar} 
Apparently, from three contributions corresponding to:
(i) RKKY interaction, (ii) direct exchange and (iii) CAR exchange, only the first may bear a negative (ferromagnetic) sign 
The two other contributions always have an anti-ferromagnetic nature 
More accurate expression for $J^{\mathrm{eff}}$ is derived in Appendix~\ref{sec:downfolding}
[see \eq{A_J}] by the Hamiltonian down-folding procedure  The relevant terms differ 
by factors important only for large $\GS{}/U$  
Finally, it seems worth stressing that normal leads are not necessary for CAR exchange to occur 
At least one of them is inevitable for the Kondo screening though, and two symmetrically coupled 
normal leads allow for measurement of the normal conductance 


It is also noteworthy that inter-dot Coulomb interactions
decrease the energy of intermediate states contributing to direct exchange 
[\fig{system}(c)], while increasing the energy of intermediate
states causing the CAR exchange [\fig{system}(f)] 
This results in different dependence of corresponding terms in \eq{Jeff} on $U'$ 
As can be seen in \figs{G-T}(b) and \ref{fig:G-T}(c), it has a significant effect 
on the actual values of $T^*$ 

\begin{figure}
\includegraphics[width=1\linewidth]{Fig2 pdf}
\caption{(a) Linear conductance $G$ as function of $T$ calculated for 
		 $\varepsilon_1=\varepsilon_2=-U/2$, $\Gamma=U/5$, $U'=U/10$ and different situations, 
		 as indicated  The quantity $\xi\equiv\sqrt{\GS{}^2+t^2}$ is fixed 
		 for different curves drawn with the same dashing style 
		 Note the logarithmic scale on both axes 
		 %
		 (b) Points show $T^*/T_K$ calculated by NRG from curves in subfigure (a)  
		 Lines present the fit to \eq{Tstar} with $J^{\mathrm{eff}}$ obtained from \eq{Jeff} 
		 %
		 (c) The same as (b), only for $U'=0$ 
		 %
		 (d) and (e) show the residual conductance $G_{\mathrm{min}} \equiv G(T \!=\! 0)$ as a function of
		 $\GS{}$ for $t=0$ (denoted "CAR") and $t=\GS{}$ (denoted "Both")  
		 Dotted line is a guide for eyes  $U'=U/10$ in (b) and (d) and $U'=0$ in (c) and (e) 
		}
\label{fig:G-T}
\end{figure}

\section{CAR exchange and Kondo effect}
\label{sec:main}

To verify \eqs{Tstar}-(\ref{Jeff}) we calculate $G$ using
accurate full density matrix numerical renormalization group (NRG) technique \cite{WilsonNRG,Weichselbaum,FlexibleDMNRG,fn2} 
We compare $U'=0$ case with experimentally relevant value $U'=U/10$ \cite{Keller2013Dec} 
While for two close adatoms on SC surface RKKY interactions may lead to prominent consequences
\cite{Klinovaja}, the conventional ({\it i e } non-CAR) contribution should 
vanish rapidly when the inter-impurity distance $r$ exceeds a few lattice constants \cite{RKKYrange,SC_RKKY}  
Meanwhile, the CAR exchange may remain significant for $r$ of the order
of coherence length of the SC contact \cite{Yao}  Therefore, we first neglect the conventional RKKY coupling and analyze its consequences in Sec ~\ref{sec:RKKY} 

The main results are presented in \fig{G-T}(a), showing the temperature dependence of $G$
for different circumstances  
For reference, results for $\GS{}=0$ are shown, exhibiting 
the two-stage Kondo effect caused by \emph{direct} exchange mechanism 
As can be seen in \figs{G-T}(b) and \ref{fig:G-T}(c), an excellent agreement of $T^*$ found from NRG calculations and \eq{Tstar} 
is obtained with $a=0 42$ and $b=1 51$, the same for both $U'=0$ and $U'=U/10$  Note, 
however, that $J^{\mathrm{eff}}$ is different in these cases, cf  \eq{Jeff},
and $U'$ leads to increase of $T^*$ 

Furthermore, for $t=0$ and $\GS{}>0$ the two-stage Kondo effect caused solely by the \emph{CAR
exchange} is present; see \fig{G-T}(a) 
Experimentally, this situation
corresponds to a distance between the two QDs smaller than the superconducting coherence length,
but large enough for the exponentially suppressed direct hopping to be negligible 
While intuitively one could expect pairing to compete with any kind of magnetic ordering,
the Kondo screening induced by CAR exchange is a beautiful example of a superconductivity
in fact leading to magnetic order, namely the formation of the Kondo singlet 
This CAR-exchange-mediated Kondo screening is our main finding 
For such screening, \eq{Tstar} is still fulfilled with very similar 
parameters, $a=0 37$ ($a=0 35$) and $b=1 51$ ($b=1 50$) for $U'=0$ ($U'=U/10$),
correspondingly; see \figs{G-T}(b-c) 
Moreover, as follows from \eq{Jeff}, $U'$ reduces CAR exchange, and therefore diminishes $T^*$ 
For the same values of $J^{\mathrm{eff}}$, the dependence of $G(T)$ for $t=0$ and $\GS{}>0$ is hardly different 
from the one for $\GS{}=0$ and $t>0$ for $T\geq T^*$ (results not shown) 
However, $G(T)$ saturates at residual value $G_{\mathrm{min}}$ as $T\to 0$ only for finite
$\GS{}$, which at particle-hole symmetry makes $G_{\mathrm{min}}$
the hallmark of SC proximity and the corresponding CAR exchange processes 
From numerical results, one can estimate it as
\begin{equation}
G_{\mathrm{min}} = \frac{e^2}{h} \cdot c \, \frac{\GS{}^2}{U^2} 
	\qquad {\scriptstyle (\GS{1}=\GS{2}=\GS{})} ,
\label{Gmin}
\end{equation}
with $c\approx 2 25$, barely depending on $U'$ and getting smaller for $t>0$  
This is illustrated in \figs{G-T}(d-e), where the dotted line corresponds to \eq{Gmin} with $c=2 25$  

Lastly, in \fig{G-T}(a) we also present the curves obtained for $t=\GS{}$ chosen such, 
that the quantity $\xi=\sqrt{t^2+\GS{}^2}$ remains the same 
in all the cases 
This is to illustrate what happens when \emph{both} (direct and CAR) exchange interactions are
present  \fig{G-T}(c) clearly shows that $T^*$ remains practically unaltered for $U'=0$ 
The comparison with \fig{G-T}(b) proves that in this case it practically does not depend 
on $U'$  The enhancement of direct exchange is compensated by the decrease of the CAR one  
On the contrary, $G_{\mathrm{min}}$ decreases for larger $t$ below the estimation given by Eq ~(\ref{Gmin}), 
as can be seen in \figs{G-T}(d-e)  

While analyzing the results concerning $G_{\mathrm{min}}(\GS{})$ plotted in \figs{G-T}(d-e) 
one needs to keep in mind that $G_{\mathrm{min}}$ is obtained at deeply cryogenic conditions  To illustrate
this better, $G(\GS{})$ obtained for $t=0$ and $T=10^{-6}U$ is plotted with solid line 
in \fig{3}  Clearly, for weak $\GS{}$ the system exhibits rather conventional (single-stage)
Kondo effect with $G=G_{\mathrm{max}}\approx 2e^2/h$, while QD2 is effectively decoupled ($G_{\mathrm{max}}<2e^2/h$
in the proximity of SC lead \cite{KWIW})  Only for larger values of $\GS{}$
the CAR exchange is strong enough, such that $T^*>T$ and the dependence $G(\GS{})$ continuously 
approaches the $T=0$ limit estimated by \eq{Gmin} and presented in \figs{G-T}(d-e) 

\section{CAR-RKKY competition}
\label{sec:RKKY}

\begin{figure}
\includegraphics[width=0 98\linewidth]{Fig3 pdf}
\caption{Linear conductance $G$ vs  $\GS{}$ calculated
		 for $t=0$, $\Gamma=U/5$, $U'=U/10$, finite $T=10^{-6}U$
		 and different values of RKKY coupling $J$, as indicated  
		 Inset shows QD1 spectral function $\mathcal{A}(\omega)$ as a function of energy $\omega$
		 for points on $J=-0 1U$ curve, indicated with corresponding symbols 
		}
\label{fig:3}
\end{figure}

Let us now discuss the effects introduced by the conventional RKKY interaction 
We choose $t=0$ for the sake of simplicity and
analyze a wide range of $\GS{}$, starting from the case of anti-ferromagnetic 
RKKY interaction ($J>0$)  Large $J>0$ leads to the formation of a molecular singlet in the 
nanostructure  This suppresses the conductance, unless $\GS{}$ becomes of the order of $U/2$, 
when the excited states of DQD are all close to the ground state  This is illustrated 
by double-dotted line in \fig{3} 
Smaller value of $J>0$ causes less dramatic consequences, namely it just increases $J^{\mathrm{eff}}$ according
to \eq{Jeff}, leading to enhancement of $T^*$, cf  \eq{Tstar}  This is presented with
dot-dashed line in \fig{3} 

The situation changes qualitatively for ferromagnetic RKKY coupling, $J<0$ 
Then, RKKY exchange and CAR exchange have opposite signs and compete with each other 
Depending on their magnitudes and temperature, one
of the following scenarios may happen 

For $J^{\mathrm{eff}} > 0$, {\it i e } large enough $\GS{}$, and $T<T^*$, the system is in the 
singlet state due to the two-stage Kondo screening of DQD spins  $G(T\!=\!0)$ is reduced 
to $G_{\mathrm{min}}$, which tends to increase for large negative $J$; see dashed lines in \fig{3}  
In the inset to \fig{3}, the spectral density of QD1 representative for this regime is plotted 
as curve indicated by triangle  It corresponds to a point on the $J=-0 1U$ curve in the main 
plot, also indicated by triangle  The dip in $\mathcal{A}(\omega)$ has width of order of $T^*$ 

For finite $T$, there is always a range of sufficiently small $|J^{\mathrm{eff}}|$, where QD2 becomes effectively
decoupled, and, provided $T<T_K$, $G$ reaches $G_{\mathrm{max}}$ due to conventional Kondo effect 
at QD1  This is the case for sufficiently small $\GS{}$ for $J=0$ or $J=-0 01U$, and in the narrow
range of $\GS{}$ around the point indicated by a circle in \fig{3} for $J=-0 1U$ (for $J=0 01U$, 
the considered $T$ is close to $T^*$ and $G$ does not reach $G_{\rm max}$)  The conventional Kondo effect manifests itself with 
a characteristic peak in $\mathcal{A}(\omega)$, as illustrated in the inset in \fig{3} with line denoted by circle 

Finally, large enough $J^{\mathrm{eff}} < 0$ and low $T$, give rise to an effective ferromagnetic coupling of DQDs 
spins into triplet state  Consequently, the underscreened Kondo effect occurs 
\cite{Mattis,NozieresBlandin} for weak $\GS{}$ and, {\it e g }, $J=-0 1U$; 
see the point indicated by square in \fig{3} 
This leads to $G=G_{\mathrm{max}}$ and a peak in $\mathcal{A}(\omega)$, whose shape is significantly different from the
Kondo peak, cf  the curve denoted by square in the inset in \fig{3}  



\section{Effects of detuning from the particle-hole symmetry point}
\label{sec:asym}

\begin{figure}
\includegraphics[width=0 98\linewidth]{Fig4 pdf}
\caption{
         (a) Linear conductance between the normal leads $G$ as a function of temperature $T$
         for parameters corresponding to \fig{G-T}(a) with $\xi=U/10$, and additional curves for finite 
         detuning from particle-hole symmetry point, $\delta_1=-\delta_2$, 
         and two values of $\xi=\sqrt{t^2+\GS{}^2}$, as indicated in the figure 
         (b) $G_{\mathrm{min}} \equiv G(T \!=\! 0)$ as a function of QD1 detuning $\delta_1$ for different
         exchange mechanisms, $\xi=U/10$ and $\delta_2=\pm\delta_1$ (as indicated) 
		}
\label{fig:asym}
\end{figure}

At PHS $G_{\mathrm{min}}=G(T \!=\! 0)=0$ in the absence of superconducting lead, making $G_{\mathrm{min}} > 0$ a hallmark
of SC-induced two-stage Kondo effect  However, outside of PHS point $G_{\mathrm{min}} > 0$ even in the case of 
the two-stage Kondo effect caused by the direct exchange  
Exact PHS conditions are hardly possible in real systems, and the fine-tuning of the QD energy
levels to PHS point is limited to some finite accuracy 
Therefore, there may appear a question, if the results obtained at PHS are of any importance for the
realistic setups  As we show below --- they are,
in a reasonable range of detunings $\delta_i=\varepsilon_i +U/2$ 

In \fig{asym}(a) we present the $G(T)$ dependence in and outside the PHS, corresponding to 
parameters of \fig{G-T}(a)  
Clearly, for considered small values of $\delta_1=\delta_2=\delta$, 
$G_{\mathrm{min}}<10^{-3}e^2/h$ for direct exchange only, while $G_{\mathrm{min}}$ in the presence of a superconductor is 
significantly increased and close to the PHS value  Furthermore, for $|\delta_1| \sim |\delta_2| 
\sim \delta$, the residual conductance caused by the lack of PHS, $G_{\mathrm{min}} \approx e^2/h \cdot (\delta/U)^2$,
which is a rapidly decreasing function in the vicinity of PHS point, as illustrated in \fig{asym}(b)
with lines denoted by a square  Evidently, in the regime $|\delta_i| < 0 01U$ the residual conductance
caused by SC is orders of magnitude larger, leading to the plateau in $G_{\mathrm{min}}(\delta_1)$ dependence,
visible in \fig{asym}(b) 
Taking into account that the realistic values of $U$ in the semiconductor quantum dots are rather 
large, this condition seems to be realizable by fine-tuning of QD gate voltages 

Lastly, let us point out that while in the presence of only one exchange mechanism, \emph{CAR} or
\emph{direct}, $G_{\mathrm{min}}(\delta_1)$ dependencies depicted in \fig{asym}(b) are symmetrical with respect
to sign change of $\delta_1$, for \emph{both} exchange mechanisms the dependence is non-symmetric  

\section{Effects of asymmetry of couplings to superconductor}
\label{sec:x}

\begin{figure}
\includegraphics[width=0 98\linewidth]{Fig5 pdf}
\caption{
		 (a) Linear conductance between the normal leads, $G$, as a function of temperature, $T$,
		 for parameters corresponding to \fig{G-T}(a) with $\xi=U/10$, for different values 
		 of asymmetry coefficient $x$ [see \eq{xGS}], in the presence of \emph{CAR} exchange only 
		 %
		 (b) The second-stage Kondo temperature $T^*$ normalized by $T_K$ as a function of $x$, 
		 calculated with the aid of NRG (points) and a fit to \eq{Tstar} (lines) 
		 with $J^{\mathrm{eff}}$ from \eq{Jeff} 
		 %
		 (c) The zero-temperature conductance $G_{\mathrm{min}}$ as a function of QD1 coupling to SC lead, $\GS{1}$,
		 compiled from data obtained at different circumstances (as indicated in the legend)
		 for different $x$  Dotted line corresponds to \eq{Gmin2} with $c=2 25$ 
		}
\label{fig:x}
\end{figure}

Similarly to PHS, the ideal symmetry in the coupling between respective QDs and SC lead is hardly possible
in experimental reality  As shown below, it does not introduce any qualitatively new features 
On the other hand, it decreases the second stage Kondo temperature, which is already small, therefore,
quantitative estimation of this decrease may be important for potential experimental approaches 
To analyze the effects of $\GS{1}\neq\GS{2}$, we introduce the asymmetry parameter $x$ and extend
the definition of $\GS{}$,
\beq
x = \frac{\GS{1}-\GS{2}}{\GS{1}+\GS{2}}, \quad \GS{} = \frac{\GS{1}+\GS{2}}{2} 
\label{xGS}
 \end{equation} 
Note, that even for a fixed $\GS{}$, the actual CAR coupling $\GS{\rm X}=\GS{}\sqrt{1-x^2}$ decreases
with increasing $|x|$, which is a main mechanism leading to a decrease of $T^*$ outside the $x=0$ point
visible in \figs{x}(a) and (b)  To illustrate this, the curves corresponding to \emph{both} exchange
mechanisms were calculated using $x$-dependent $t=\GS{\rm X}$ instead of $t=\xi/\sqrt{2}$  
Therefore, $\xi$ was generalized for $x\neq 0$ by setting $\xi=\sqrt{t^2(1-x^2)^{-1}+\GS{}^2}$ 
Clearly, in \fig{x}(b) the curves for different exchange mechanisms are very similar and differ mainly 
by a constant factor, resulting from different influence of $U'$; see \Sec{scales}  
The magnitude of $T^*$ changes is quite large, exceeding an order of magnitude for $x=\pm 0 5$ 
and $\xi=U/20$  Moreover, $T^* \to 0$ for $x\to\pm 1$  Consequently, for strongly asymmetric
devices one cannot hope to observe the second stage of Kondo screening 

A careful observer can note that the $T^*(x)$ dependency is not symmetrical; note for example different 
$T^*$ for $x=\pm 0 5$ in \fig{x}(a)  This is caused by the dependence of the first stage Kondo temperature
$T_K$ on $\GS{1}$ \cite{part1,DomanskiIW},
\beq
\widetilde{T}_K(\GS{1}) = T_K \cdot \exp\!\left( \frac{\pi}{2} \frac{\GS{1}^2}{\Gamma U}\right) 
 \end{equation} 
Here, $T_K$ is, as earlier, defined in the absence of SC, while $\widetilde{T}_K$ is a function 
of $\GS{1}$, such that $G(\widetilde{T}_K) = G_{\rm max}(\GS{1})/2$ in the absence of QD2  
As $\widetilde{T}_K$ grows for increasing $\GS{1}$ (or $x$), $T^*$ decreases according to \eq{Tstar}  
Its $\GS{}$ dependence can be accounted for by small changes in the coefficients $a$ and $b$ in \eq{Tstar}, 
as long as $x$ is kept constant  

To close the discussion of $T^*(x)$ dependence let us point out, that in \eq{A_J} 
there appears a correction to \eq{Jeff} for $x\neq 0$  However, it is very small due to additional
factor $\GS{}^2/U^2$ in the leading order  Its influence on curves plotted in \fig{x}(b) is hardly visible 

In turn, let us examine the $x$ dependence of the $T=0$ conductance $G_{\mathrm{min}}$  As can be seen 
in \fig{x}(a), it monotonically increases with $x$, as it crosses $x=0$ point  In fact, \eq{Gmin}
can be generalized to
\beq
G_{\mathrm{min}} = \frac{e^2}{h} \cdot c \, \frac{\GS{1}^2}{U^2} ,
\label{Gmin2}
 \end{equation} 
with $c\approx 2 25$ (indicated by a dotted line in \fig{x}(c))  Note that $G_{\mathrm{min}}$ is proportional to 
$\GS{1}^2=(x+1)^2 \GS{}^2$, instead of simply $\GS{}$, cf  \eq{Gmin}  The values of $G_{\mathrm{min}}$ obtained
from all analyzed $G(T)$ dependencies for different $x$ have been compiled in \fig{x}(c) 
It is evident, that \eq{Gmin2} is approximately fulfilled for all the considered cases 

Finally, it seems noteworthy that the normal-lead coupling asymmetry, 
$\Gamma_{\rm L}\neq \Gamma_{\rm R}$, is irrelevant for the results except for a constant factor
diminishing the conductance $G$ \cite{KWIWJB-asym} 



\section{The role of CAR efficiency}
\label{sec:coef}

\begin{figure}[tb]
\includegraphics[width=0 98\linewidth]{Fig6 pdf}
\caption{Linear conductance between the normal leads
		 $G$ as a function of coupling to SC lead, $\GS{}$, for indicated values of RKKY exchange $J$
		 and the efficiency of CAR processes reduced by factor (a) $\mathcal{C}=0 9$ and (b) $\mathcal{C}=0 5$ 
		 Other parameters as in \fig{3} 
		 Insets: QD1 local spectral density $\mathcal{A}(\omega)$ as a function of energy $\omega$
		 for points on $J=-0 1U$ curve, indicated with corresponding symbols 
		} 
\label{fig:C}
\end{figure}

Up to this point we assumed $\GS{\rm X} = \sqrt{\GS{1}\GS{2}}$, which is valid when the two 
quantum dots are much closer to each other than the coherence length in the superconductor 
This does not have to be the case in real setups, yet relaxing this assumption does not 
introduce qualitative changes  Nevertheless, the model cannot be extended to inter-dot 
distances much larger than the coherence length, where $\GS{\rm X}\to 0$ 

To quantitatively analyze the consequences of less effective Andreev coupling we define the 
CAR efficiency as $\mathcal{C} \equiv \GS{\rm X} / \sqrt{\GS{1}\GS{2}}$ and analyze $\mathcal{C} < 1$
in the wide range of $\GS{1}=\GS{2}=\GS{}$ and other parameters corresponding to \fig{3}  
The results are presented in \fig{C} 

Clearly, decreasing $\mathcal{C}$ from $\mathcal{C}=1$ causes diminishing of $\GS{\rm X}$, and consequently of CAR 
exchange  For a change as small as $\mathcal{C}=0 9$, the consequences reduce to some shift of the 
conventional Kondo regime, compare \fig{C}(a) with \fig{3}  Stronger suppression of CAR may, 
however, increase the SC coupling necessary to observe the second stage of Kondo screening caused
by CAR outside the experimentally achievable range, see \fig{C}(b)  Moreover, the reduced $T^*$
leads to narrowing of the related local spectral density dip, while the
increased critical $\GS{}$ necessary for the observation of the second stage of screening leads to the
shallowing of the dip  This is visible especially in the inset in \fig{C}(b) 


\section{Conclusions}
\label{sec:conclusions}

The CAR exchange mechanism is present in any system comprising at least
two QDs or magnetic impurities coupled to the same superconducting contact
in a way allowing for crossed Andreev reflections 
In the considered setup, comprised of two quantum dots in a T-shaped geometry 
with respect to normal leads and proximized by superconductor,
it leads to the two-stage Kondo
screening even in the absence of other exchange mechanisms 
This CAR induced exchange screening is characterized by a residual 
low-temperature conductance at particle-hole symmetric case 
We have also shown that the competition between CAR exchange and RKKY
interaction may result in completely different Kondo screening scenarios 

The presented results bring further insight into the low-temperature
behavior of hybrid coupled quantum dot systems, which hopefully could be verified
with the present-day experimental techniques 
Moreover, non-local pairing is present also in bulk systems such as non-$s$-wave superconductors 
The question if an analogue of discussed CAR exchange may play a role there
seems intriguing in the context of tendencies of many strongly correlated materials
to possess superconducting and anti-ferromagnetic phases 


\begin{acknowledgments}
This work was supported by the National Science Centre in Poland through project no 
2015/19/N/ST3/01030 
We thank J  Barna\'{s} and T  Maier for valuable discussions 
\end{acknowledgments}





Context 3: Paper Info

Title: Conflict Optimization for Binary CSP Applied to Minimum Partition into Plane Subgraphs and Graph Coloring
Publish Date: 25 Mar 2023
Author List: Loïc Crombez (from LIMOS, Université Clermont Auvergne), Guilherme Da Fonseca (from LIS, Aix-Marseille Université), Florian Fontan (from Independent Researcher), Yan Gerard (from LIMOS, Université Clermont Auvergne), Aldo Gonzalez-Lorenzo (from LIS, Aix-Marseille Université), Pascal Lafourcade (from LIMOS, Université Clermont Auvergne), Luc Libralesso (from LIMOS, Université Clermont Auvergne), Benjamin Momège (from Independent Researcher), Jack Spalding-Jamieson (from David R  Cheriton School of Computer Science, University of Waterloo), Brandon Zhang (from Independent Researcher), Da Zheng (from Department of Computer Science, University of Illinois at Urbana-Champaign)

Figure

Figure 1: A partition of the input graph of the CG:SHOP2022 instance vispecn2518 into 57 plane graphs It is the smallest instance of the challenge with 2518 segments On top left, you see all 57 colors together On top right, you see a clique of size 57, hence the solution is optimal Each of the 57 colors is then presented in small figures 
Figure 2: Number of colors over time for the instance vispecn13806 using different values p The algorithm uses σ = 0 15, easy vertices, q max = 59022, but does not use the BDFS nor any clique 
Figure 3: Number of colors over time with different values of q max obtained on the instance vispecn13806 Parameters are σ = 0 15, p = 1 2, no clique knowledge, and no BDFS 
Figure 4: Number of colors over time with and without clique knowledge and BDFS obtained on the instance vispecn13806 Parameters are σ = 0 15, p = 1 2, and q max = 1500000 
Figure 5: Number of colors over time for the instance vispecn13806 for different values of σ In both figures the algorithm uses p = 1 2, easy vertices, q max = 59022, but does not use the BDFS nor any clique For σ ≥ 0 25, no solution better than 248 colors is found 
Figure 6: Number of colors over time (in hours) for the instance vispecn13806 
Several CG:SHOP 2022 results We compare the size of the largest known clique to the smallest coloring found by each team on a selection of 14 CG:SHOP 2022 instances 
[20][21][22][23][24][25] with state-of-the-art graph coloring algorithms The conflict optimizer underperforms except on the geometric graphs r* and dsjr* CE39-0007), SEVERITAS (ANR-20-CE39-0005) and by the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP[20][21][22][23][24][25] The work of Luc Libralesso is supported by the French ANR PRC grant DECRYPT (ANR-18-CE39-0007) 

abstract

CG:SHOP is an annual geometric optimization challenge and the 2022 edition proposed the problem of coloring a certain geometric graph defined by line segments  Surprisingly, the top three teams used the same technique, called conflict optimization  This technique has been introduced in the 2021 edition of the challenge, to solve a coordinated motion planning problem 
In this paper, we present the technique in the more general framework of binary constraint satisfaction problems (binary CSP)  Then, the top three teams describe their different implementations of the same underlying strategy  We evaluate the performance of those implementations to vertex color not only geometric graphs, but also other types of graphs 

Introduction

The CG:SHOP challenge (Computational Geometry: Solving Hard Optimization Problems) is an annual geometric optimization competition, whose first edition took place in 2019  The 2022 edition proposed a problem called minimum partition into plane subgraphs  The input is a graph G embedded in the plane with edges drawn as straight line segments, and the goal is to partition the set of edges into a small number of plane graphs (Fig  )  
This goal can be formulated as a vertex coloring problem on a graph G defined as follows  The vertices of G are the segments defining the edges of G, and the edges of G correspond to pairs of crossing segments (segments that intersect only at a common endpoint are not considered crossing)  The three top-ranking teams (Lasa, Gitastrophe, and Shadoks) on the CG:SHOP 2022 challenge all used a common approach called conflict optimization while the fourth team used a SAT-Boosted Tabu Search  
Conflict optimization is a technique used by Shadoks to obtain the first place in the CG:SHOP 2021 challenge for low-makespan coordinated motion planning , and the main ideas of the technique lent themselves well to the 2022 challenge  Next, we describe the conflict optimizer as a metaheuristic to solve constraint satisfaction problems (CSP)  
We start by describing a CSP  A CSP is a triple of • variables X = (x 1 ,       , x n ), Each of the 57 colors is then presented in small figures  • domains D = (D 1 ,       , D n ), and • constraints R  Each variable x i must be assigned a value in the corresponding domain D i such that all constraints are satisfied 
In general, the constraints may forbid arbitrary subsets of values  We restrict our attention to a particular type of constraints (binary CSP ), which only involve pairs of assignments  A partial evaluation is an assignment of a subset of the variables, called evaluated, with the remaining variables called non-evaluated 
All constraints involving a non-evaluated variable are satisfied by default  We only consider assignments and partial assignments that satisfy all constraints  The conflict optimizer iteratively modifies a partial evaluation with the goal of emptying the set S of non-evaluated variables, at which point it stops 
At each step, a variable x i is removed from S  If there exists a value x ∈ D i that satisfies all constraints, then we assign the value x to the variable x i   Otherwise, we proceed as follows  For each possible value x ∈ D i , we consider the set K(i, x) of variables (other than x i ) that are part of constraints violated by the assignment x i = x 
We assign to x i the value x that minimizes where w(j) is a weight function to be described later  The variables x j ∈ K(i, x) become non-evaluated and added to S  The weight function should be such that w(j) increases each time x j is added to S, in order to avoid loops that keep moving the same variables back and forth from S  Let q(j) be the number of times x j became non-evaluated 
A possible weight function is w(j) = q(j)  More generally, we can have w(j) = q(j) p for some exponent p (typically between 1 and 2)  Of course, several details of the conflict optimizer are left open  For example, which element to choose from S, whether some random noise should be added to w, and the decision to restart the procedure from scratch after a certain time 
The CSP as is, does not apply to optimization problems  However, we can, impose a maximum value k of the objective function in order to obtain a CSP  The conflict optimizer was introduced in a low makespan coordinated motion planning setting  In that setting, the variables are the robots, the domains are their paths (of length at most k) and the constraints forbid collisions between two paths 
In the graph coloring setting, the domains are the k colors of the vertices and the constraints forbid adjacent vertices from having the same color  The conflict optimizer can be adapted to non-binary CSP, but in that case multiple variables may be unassigned for a single violated constraint  The strategy has some resemblance to the similarly named min-conflicts algorithm , but notable differences are that a partial evaluation is kept instead of an invalid evaluation and the weight function that changes over time 
While the conflict optimization strategy is simple, there are different ways to apply it to the graph coloring problem  The goal of the paper is to present how the top three teams applied it or complemented it with additional strategies  We compare the relative benefits of each variant on the instances given in the CG:SHOP 2022 challenge 
We also compare them to baselines on some instances issued from graph coloring benchmarks  The paper is organized as follows  Section 2 presents the details of the conflict optimization strategy applied to graph coloring  In the three sections that follow, the three teams Lasa, Gitastrophe, and Shadoks present the different parameters and modified strategies that they used to make the algorithm more efficient for the CG:SHOP 2022 challenge 
The last section is devoted to the experimental results 

Literature Review

The study of graph coloring goes back to the 4-color problem (1852) and it has been intensively studied since the 1970s (see for surveys)  Many heuristics have been proposed , as well as exact algorithms   We briefly present two classes of algorithms: greedy algorithms and exact algorithms  Greedy algorithms 
These algorithms are used to find good quality initial solutions in a short amount of time  The classic greedy heuristic considers the vertices in arbitrary order and colors each vertex with the smallest non-conflicting color  The two most famous modern greedy heuristics are DSATUR and Recursive Largest First (RLF )  
At each step (until all vertices are colored), DSATUR selects the vertex v that has the largest number of different colors in its neighbourhood  Ties are broken by selecting a vertex with maximum degree  The vertex v is colored with the smallest non-conflicting color  RLF searches for a large independent set I, assigns the vertices I the same color, removes I from G , and repeats until all vertices are colored 
Exact algorithms  Some exact methods use a branch-and-bound strategy, for example extending the DSATUR heuristic by allowing it to backtrack   Another type of exact method (branch-and-cut-and-price) decomposes the vertex coloring problem into an iterative resolution of two sub-problems   The "master problem" maintains a small set of valid colors using a set-covering formulation 
The "pricing problem" finds a new valid coloring that is promising by solving a maximum weight independent set problem  Exact algorithms are usually able to find the optimal coloring for graphs with a few hundred vertices  However, even the smallest CG:SHOP 2022 competition instances involve at least a few thousands vertices 

Conflict Optimization for Graph Coloring

Henceforth, we will only refer to the intersection conflict graph G induced by the instance  Vertices will refer to the vertices V (G ), and edges will refer to the edges E(G )  Our goal is to partition the vertices using a minimum set of k color classes C = {C 1 ,       , C k }, where no two vertices in the same color class C i are incident to a common edge 

Conflict Optimization

TABUCOL inspired neighbourhood One classical approach for the vertex coloring involves allowing solutions with conflicting vertices (two adjacent vertices with the same color)  It was introduced in 1987 and called TABUCOL  It starts with an initial solution, removes a color (usually the one with the least number of vertices), and assigns uncolored vertices with a new color among the remaining ones 
This is likely to lead to some conflicts (i e  two adjacent vertices sharing a same color)  The local search scheme selects a conflicting vertex, and tries to swap its color, choosing the new coloring that minimises the number of conflicts  If it reaches a state with no conflict, it provides a solution with one color less than the initial solution 
The process is repeated until the stopping criterion is met



ground_truth: It tends to suppress the Kondo effect.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the average magnetic moment per column in Ge$_{1-x}$Mn$_{x}$ films?

Read the following text and answer briefly.

Context 2: \section{introduction}
In the past few years, the synthesis of ferromagnetic semiconductors has become a major challenge for spintronics  Actually, growing a magnetic and semiconducting material could lead to promising advances like spin injection into non magnetic semiconductors, or electrical manipulation of carrier induced magnetism in magnetic semiconductors \cite{ohno00,Bouk02}  Up to now, major efforts have focused on diluted magnetic semiconductors (DMS) in which the host semiconducting matrix is randomly substituted by transition metal (TM) ions such as Mn, Cr, Ni, Fe or Co \cite{Diet02}  However Curie temperatures ($T_{C}$) in DMS remain rather low and TM concentrations must be drastically raised in order to increase $T_{C}$ up to room temperature  That usually leads to phase separation and the formation of secondary phases  It was recently shown that phase separation induced by spinodal decomposition could lead to a significant increase of $T_{C}$ \cite{Diet06,Fuku06}  For semiconductors showing $T_{C}$ higher than room temperature one can foresee the fabrication of nanodevices such as memory nanodots, or nanochannels for spin injection  Therefore, the precise control of inhomogeneities appears as a new challenge which may open a way to industrial applications of ferromagnetism in semiconductors 

The increasing interest in group-IV magnetic semiconductors can also be explained by their potential compatibility with the existing silicon technology  In 2002, carrier mediated ferromagnetism was reported in  MBE grown Ge$_{0 94}$Mn$_{0 06}$ films by Park \textit{et al } \cite{Park02}  The maximum critical temperature was 116 K  Recently many publications indicate a significant increase of $T_{C}$ in Ge$_{1-x}$Mn$_{x}$ material depending on growth conditions \cite{Pint05,Li05,tsui03}  Cho \textit{et al } reported a Curie temperature as high as 285 K \cite{Cho02}  
Taking into account the strong tendency of Mn ions to form intermetallic compounds in germanium, a detailed investigation of the nanoscale structure is required  Up to now, only a few studies have focused on the nanoscale composition in Ge$_{1-x}$Mn$_{x}$ films  Local chemical inhomogeneities have been recently reported by Kang \textit{et al } \cite{Kang05} who evidenced a micrometer scale segregation of manganese in large Mn rich stripes  Ge$_3$Mn$_5$ as well as Ge$_8$Mn$_{11}$ clusters embedded in a germanium matrix have been reported by many authors  However, Curie temperatures never exceed 300 K \cite{Bihl06,Morr06,Pass06,Ahle06}  Ge$_3$Mn$_5$ clusters exhibit a Curie temperature of 296 K \cite{Mass90}  This phase frequently observed in Ge$_{1-x}$Mn$_{x}$ films is the most stable (Ge,Mn) alloy  The other stable compound Ge$_8$Mn$_{11}$ has also been observed in nanocrystallites surrounded with pure germanium \cite{Park01}  Ge$_8$Mn$_{11}$ and Ge$_3$Mn$_5$ phases are ferromagnetic but their metallic character considerably complicates their potential use as spin injectors 
Recently, some new Mn-rich nanostructures have been evidenced in Ge$_{1-x}$Mn$_{x}$ layers  Sugahara \textit{et al } \cite{Sugh05} reported the formation of high Mn content (between 10 \% and 20 \% of Mn) amorphous Ge$_{1-x}$Mn$_x$ precipitates  in a Mn-free germanium matrix  Mn-rich coherent cubic clusters were observed by Ahlers \textit{et al } \cite{Ahle06} which exhibit a Curie temperatures below 200 K  Finally, high-$T_{C}$ ($>$ 400 K) Mn-rich nanocolumns have been evidenced \cite{Jame06} which could lead to silicon compatible room temperature operational devices \\
In the present paper, we investigate the structural and magnetic properties of Ge$_{1-x}$Mn$_x$ thin films for low growth temperatures ($<$ 200$^{\circ}$C) and low Mn concentrations (between 1 \% and 11 \%)  By combining TEM, x-Ray diffraction and SQUID magnetometry, we could identify different magnetic phases  We show that depending on growth conditions, we obtain either Mn-rich nanocolumns or Ge$_{3}$Mn$_{5}$ clusters embedded in a germanium matrix  We discuss the structural and magnetic properties of these nanostructures as a function of manganese concentration and growth temperature  We also discuss the magnetic anisotropy of nanocolumns and  
Ge$_3$Mn$_5$ clusters  

\section{Sample growth}

Growth was performed using solid sources molecular beam epitaxy (MBE) by co-depositing Ge and Mn evaporated from standard Knudsen effusion cells   Deposition rate was low ($\approx$ 0 2 \AA s$^{-1}$)  Germanium substrates were epi-ready Ge(001) wafers with a residual n-type doping and resistivity of 10$^{15}$ cm$^{-3}$ and 5 $\Omega cm$ respectively  After thermal desorption of the surface oxide, a 40 nm thick Ge buffer layer was grown at 250$^{\circ}$C, resulting in a 2 $\times$ 1 surface reconstruction as observed by reflection high energy electron diffraction (RHEED) (see Fig  1a)  Next, 80 nm thick Ge$_{1-x}$Mn$_{x}$ films were subsequently grown  at low substrate temperature (from 80$^{\circ}$C to 200$^{\circ}$C)  Mn content has been determined by x-ray fluorescence measurements performed on thick samples ($\approx$ 1 $\mu m$ thick) and complementary Rutherford Back Scattering (RBS) on thin Ge$_{1-x}$Mn$_{x}$ films grown on silicon  Mn concentrations range from 1 \% to 11\% Mn 

For Ge$_{1-x}$Mn$_{x}$ films grown at substrate temperatures below 180$^{\circ}$C, after the first monolayer (ML) deposition, the  2 $\times$ 1 surface reconstruction almost totally disappears  After depositing few MLs, a slightly diffuse 1 $\times$ 1 streaky RHEED pattern and a very weak 2 $\times$ 1 reconstruction (Fig  1b) indicate a predominantly two-dimensional growth  For growth temperatures above 180$^{\circ}$C additional spots appear in the RHEED pattern during the Ge$_{1-x}$Mn$_{x}$ growth (Fig  1c)  These spots may correspond to the formation of very small secondary phase crystallites  The nature of these crystallites will be discussed below 

Transmission electron microscopy (TEM) observations were performed using a JEOL 4000EX microscope with an acceleration voltage of 400 kV  Energy filtered transmission electron microscopy (EFTEM) was done using a JEOL 3010 microscope equipped with a Gatan Image Filter   Sample preparation was carried out by standard mechanical polishing and argon ion milling for cross-section investigations and plane views were prepared by wet etching with H$_3$PO$_4$-H$_2$O$_2$ solution \cite{Kaga82} 

\begin{figure}[htb]
    \center
    \includegraphics[width= 29\linewidth]{ /fig1a eps}
    \includegraphics[width= 29\linewidth]{ /fig1b eps}
    \includegraphics[width= 29\linewidth]{ /fig1c eps}
    \caption{RHEED patterns recorded during the growth of Ge$_{1-x}$Mn$_{x}$ films: (a) 2 $\times$ 1 surface reconstruction of the germanium buffer layer  (b) 1 $\times$ 1 streaky RHEED pattern obtained at low growth temperatures ($T_g<$180$^{\circ}$C)  (c) RHEED pattern of a sample grown at $T_g=$180$^{\circ}$C  The additional spots reveal the presence of Ge$_3$Mn$_5$ clusters at the surface of the film }
\label{fig1}
\end{figure}

\section{Structural properties \label{structural}}

\begin{figure}[htb]
    \center
	\includegraphics[width= 49\linewidth]{ /fig2a eps}
	\includegraphics[width= 49\linewidth]{ /fig2b eps}
	 \includegraphics[width= 49\linewidth]{ /fig2c eps}
	 \includegraphics[width= 49\linewidth]{ /fig2d eps}
    \caption{Transmission electron micrographs of a Ge$_{1-x}$Mn$_{x}$ film grown at 130$^{\circ}$C and containing 6 \% of manganese  (a) cross-section along the [110] axis : we clearly see the presence of nanocolumns elongated along the growth axis  (b) High resolution image of the interface between the Ge$_{1-x}$Mn$_{x}$ film and the Ge buffer layer  The Ge$_{1-x}$Mn$_{x}$ film exhibits the same diamond structure as pure germanium  No defect can be seen which could be caused by the presence of nanocolumns  (c) Plane view micrograph performed on the same sample confirms the columnar structure and gives the density and size distribution of nanocolumns  (d) Mn chemical map obtained by energy filtered transmission electron microcopy (EFTEM)  The background was carefully substracted from pre-edge images  Bright areas correspond to Mn-rich regions }
\label{fig2}
\end{figure}

In samples grown at 130$^{\circ}$C and containing 6 \% Mn, we can observe vertical elongated nanostructures \textit{i e } nanocolumns as shown in Fig  2a  Nanocolumns extend through the whole Ge$_{1-x}$Mn$_{x}$ film thickness  From the high resolution TEM image shown in Fig  2b, we deduce their average diameter around 3 nm  Moreover in Fig  2b, the interface between the Ge buffer layer and the Ge$_{1-x}$Mn$_{x}$ film is flat and no defect propagates from the interface into the film  The Ge$_{1-x}$Mn$_{x}$ film is a perfect single crystal in epitaxial relationship with the substrate  In Fig  2c is shown a plane view micrograph of the same sample confirming the presence of nanocolumns in the film  From this image, we can deduce the size and density of nanocolumns  The nanocolumns density is 13000 $\rm{\mu m}^{-2}$ with a mean diameter of 3 nm which is coherent with cross-section measurements  In order to estimate the chemical composition of these nanocolumns, we further performed chemical mapping using EFTEM  In Fig  2d we show a cross sectional Mn chemical map of the Ge$_{1-x}$Mn$_{x}$ film  This map shows that the formation of nanocolumns is a consequence of Mn segregation  Nanocolumns are Mn rich and the surrounding matrix is Mn poor  However, it is impossible to deduce the Mn concentration in Ge$_{1-x}$Mn$_{x}$ nanocolumns from this cross section  Indeed, in cross section observations, the columns diameter is much smaller than the probed film thickness and the signal comes from the superposititon of the Ge matrix and Mn-rich nanocolumns  In order to quantify Mn concentration inside the nanocolumns and inside the Ge matrix, EELS measurements (not shown here) have been performed in a plane view geometry \cite{Jame06}  These observations revealed that the matrix Mn content is below 1 \% (detection limit of our instrument)  Measuring the surface occupied by the matrix and the nanocolumns in plane view TEM images, and considering the average Mn concentration in the sample (6 \%), we can estimate the Mn concentration in the nanocolumns  The Mn concentration measured by EELS being between 0\% and 1\%, we can conclude that the Mn content in the nanocolumns is between 30 \% and 38 \% \\
For samples grown between 80$^\circ$C and 150$^\circ$C cross section and plane view TEM observations reveal the presence of Mn rich nanocolumns surrounded with a Mn poor Ge matrix  In order to investigate the influence of Mn concentration on the structural properties of Ge$_{1-x}$Mn$_{x}$ films, ten samples have been grown at 100$^\circ$C and at 150$^\circ$C with Mn concentrations of 1 3 \%, 2 3 \%, 4 \%, 7 \% and 11 3 \%  Their structural properties have been investigated by plane view TEM observations  

\begin{figure}[htb]
    \center
    \includegraphics[width= 98\linewidth]{ /fig3a eps}
	\includegraphics[width= 45\linewidth]{ /fig3b eps}
		\includegraphics[width= 45\linewidth]{ /fig3c eps}
    \caption{Nanocolumns size and density as a function of growth conditions  Samples considered have been grown at 100$^{\circ}$C and 150$^{\circ}$C respectively  (a) Mn concentration dependence of the size distribution  (b) columns density as a function of Mn concentration  (c) Volume fraction of the nanocolumns as a function of Mn concentration }
 \label{fig3}
\end{figure}

For samples grown at 100$^\circ$C with Mn concentrations below 5 \% the nanocolumns mean diameter is 1 8$\pm$0 2 nm  The evolution of columns density as a fonction of Mn concentration is reported in figure 3b  By increasing the Mn concentration from 1 3 \% to 4 \% we observe a significant increase of the columns density from 13000 to 30000 $\rm{\mu m}^{-2}$  For Mn concentrations higher than 5 \% the density seems to reach a plateau corresponding to 35000 $\rm{\mu m}^{-2}$ and their diameter slightly increases from 1 8 nm at 4 \% to 2 8 nm at 11 3 \%  By plotting the volume fraction occupied by the columns in the film as a function of Mn concentration, we observe a linear dependence for Mn contents below 5 \%  The non-linear behavior above 5 \% may indicate that the mechanism of Mn incorporation is different in this concentration range, leading to an increase of Mn concentration in the columns or in the matrix  For samples grown at 100$^\circ$C, nanocolumns are always fully coherent with the surrounding matrix (Fig  4a)  

Increasing the Mn content in the samples grown at 150$^\circ$C from 1 3 \% to 11 3 \% leads to a decrease of the columns density (fig 3b)  Moreover, their average diameter increases significantly and size distributions become very broad (see Fig  3a)  For the highest Mn concentration (11 3 \%) we observe the coexistence of very small columns with a diameter of 2 5 nm and very large columns with a diameter of 9 nm  In samples grown at 150$^\circ$C containing 11 3 \% of Mn, the crystalline structure of nanocolumns is also highly modified  In plane view TEM micrographs, one can see columns exhibiting several different crystalline structures  We still observe some columns which are fully coherent with the Ge matrix like in the samples grown at lower temperature  Nevertheless, observations performed on these samples grown at 150$^\circ$C and with 11 3\% Mn reveal some uniaxially \cite{Jame06} or fully relaxed columns exhibiting a misfit of 4 \% between the matrix and the columns and leading to misfit dislocations at the interface between the column and the matrix (see fig  4b)  Thus we can conclude that coherent columns are probably in strong compression and the surrounding matrix in tension  On the same samples (T$_g$=150$^{\circ}$C, 11 3\% Mn), we also observe a large number of highly disordered nanocolumns leading to an amorphous like TEM contrast(fig  4c) 

\begin{figure}[htb]
    \center
   \includegraphics[width= 31\linewidth]{ /fig4a eps}
	\includegraphics[width= 31\linewidth]{ /fig4b eps}
	\includegraphics[width= 31\linewidth]{ /fig4c eps}
    \caption{Plane view high resolution transmission electron micrographs of different types of nanocolumns : (a) typical structure of a column grown at 100$^{\circ}$C  The crystal structure is exactly the same as germanium   (b) Partially relaxed nanocolumn  One can see dislocations at the interface between the columns and the matrix leading to stress relaxation  (c) Amorphous nanocolumn  These columns are typical in samples grown at 150$^{\circ}$C with high Mn contents }
 \label{fig4}
\end{figure}

In conclusion, we have evidenced a complex mechanism of Mn incorporation in Mn doped Ge films grown at low temperature  In particular Mn incorporation is highly inhomogeneous  For very low growth temperatures (below 120$^\circ$C) the diffusion of Mn atoms leads to the formation of Mn rich, vertical nanocolumns  Their density mostly depends on Mn concentration and their mean diameter is about 2 nm  These results can be compared with the theoretical predictions of Fukushima \textit{et al } \cite{Fuku06}: they proposed a model of spinodal decomposition in (Ga,Mn)N and (Zn,Cr)Te based on layer by layer growth conditions and a strong pair attraction between Mn atoms which leads to the formation of nanocolumns  This model may also properly describe the formation of Mn rich nanocolumns in our samples  Layer by layer growth conditions can be deduced from RHEED pattern evolution during growth  For all the samples grown at low temperature, RHEED observations clearly indicate two-dimensional growth  Moreover, Ge/Ge$_{1-x}$Mn$_{x}$/Ge heterostructures have been grown and observed by TEM (see Fig  5)  Ge$_{1-x}$Mn$_{x}$/Ge (as well as Ge/Ge$_{1-x}$Mn$_{x}$) interfaces are very flat and sharp thus confirming a two-dimensional, layer by layer growth mode  Therefore we can assume that the formation of Mn rich nanocolumns is a consequence of 2D-spinodal decomposition 

\begin{figure}[htb]
    \center
	\includegraphics[width= 7\linewidth]{ /fig5 eps}
    \caption{Cross section high resolution micrograph of a Ge/Ge$_{1-x}$Mn$_{x}$/Ge/Ge$_{1-x}$Mn$_{x}$/Ge heterostructure  This sample has been grown at 130 $^{\circ}$C with 6\% Mn  Ge$_{1-x}$Mn$_{x}$ layers are 15 nm thick and Ge spacers 5 nm thick  We clearly see the sharpness of both Ge$_{1-x}$Mn$_{x}$/Ge and Ge/Ge$_{1-x}$Mn$_{x}$ interfaces  Mn segregation leading to the columns formation already takes place in very thin Ge$_{1-x}$Mn$_{x}$ films }
\label{fig5}
\end{figure}

For growth temperatures higher than 160$^\circ$C, cross section TEM and EFTEM observations (not shown here) reveal the coexistence of two Mn-rich phases: nanocolumns and Ge$_{3}$Mn$_{5}$ nanoclusters embedded in the germanium matrix  A typical high resolution TEM image is shown in figure 6  
Ge$_{3}$Mn$_{5}$ clusters are not visible in RHEED patterns for temperatures below 180$^\circ$C  To investigate the nature of these clusters, we performed x-ray diffraction in $\theta-2\theta$ mode  Diffraction scans were acquired on a high resolution diffractometer using the copper K$_\alpha$ radiation and on the GMT station of the BM32 beamline at the European Synchrotron Radiation Facility (ESRF)  Three samples grown at different temperatures and/or annealed at high temperature were investigated  The two first samples are Ge$_{1-x}$Mn$_{x}$ films grown at 130$^\circ$C and 170$^\circ$C respectively  The third one has been grown at 130$^\circ$C and post-growth annealed at 650$^\circ$C  By analysing x-ray diffraction spectra, we can evidence two different crystalline structures  For the sample grown at 130$^\circ$C, the $\theta-2\theta$ scan only reveals the (004) Bragg peak of the germanium crystal, confirming the good epitaxial relationship between the layer and the substrate, and the absence of secondary phases in the film in spite of a high dynamics of the order of 10$^7$  For both samples grown at 170$^\circ$C and annealed at 650$^\circ$C, $\theta-2\theta$ spectra are identical  In addition to the (004) peak of germanium, we observe three additional weak peaks  The first one corresponds to the (002) germanium forbidden peak which probably comes from a small distortion of the germanium crystal, and the two other peaks are respectively attributed to the (002) and (004) Bragg peaks of a secondary phase  The $c$ lattice parameter of Ge$_3$Mn$_5$ hexagonal crystal is 5 053 \AA \ \cite{Fort90} which is in very good agreement with the values obtained from diffraction data for both (002) and (004) lines assuming that the $c$ axis of Ge$_3$Mn$_5$ is along the [001] direction of the Ge substrate 

\begin{figure}[htb]
    \center
	\includegraphics[width= 7\linewidth]{ /fig6 eps}
	\caption{Cross section high resolution transmission electron micrograph of a sample grown at 170$^{\circ}$C  We observe the coexistence of two different Mn-rich phases: Ge$_{1-x}$Mn$_{x}$ nanocolumns and Ge$_3$Mn$_5$ clusters }
\label{fig6}
\end{figure}

In summary, in a wide range of growth temperatures and Mn concentrations, we have evidenced a two-dimensional spinodal decomposition leading to the formation of Mn-rich nanocolumns in Ge$_{1-x}$Mn$_{x}$ films  This decomposition is probably the consequence of: $(i)$ a strong pair attraction between Mn atoms, $(ii)$ a strong surface diffusion of Mn atoms in germanium even at low growth temperatures and $(iii)$ layer by layer growth conditions  We have also investigated the influence of growth parameters on the spinodal decomposition: at low growth temperatures (100$^{\circ}$C), increasing the Mn content leads to higher columns densities while at higher growth temperatures (150$^{\circ}$C), the columns density remains nearly constant whereas their size increases drastically  By plotting the nanocolumns density as a function of Mn content, we have shown that the mechanism of Mn incorporation in Ge changes above 5 \% of Mn  Finally, using TEM observations and x-ray diffraction, we have shown that Ge$_3$Mn$_5$ nanoclusters start to form at growth temperatures higher than 160$^\circ$C 

\section{Magnetic properties \label{magnetic}}

We have thoroughly investigated the magnetic properties of thin Ge$_{1-x}$Mn$_{x}$ films for different growth temperatures and Mn concentrations  In this section, we focus on Mn concentrations between 2 \% and 11 \%  We could clearly identify four different magnetic phases in Ge$_{1-x}$Mn$_{x}$ films : diluted Mn atoms in the germanium matrix, low $T_{C}$ nanocolumns ($T_{C}$ $\leq$ 170 K), high $T_{C}$ nanocolumns ($T_{C}$ $\geq$ 400 K) and Ge$_{3}$Mn$_{5}$ clusters ($T_{C}$ $\thickapprox$ 300 K)  The relative weight of each phase clearly depends on the growth temperature and to a lesser extend on Mn concentration  For low growth temperature ($<$ 120$^{\circ}$C), we show that nanocolumns are actually made of four uncorrelated superparamagnetic nanostructures  Increasing T$_{g}$ above 120$^{\circ}$C, we first obtain continuous columns exhibiting low $T_{C}$ ($<$ 170 K) and high $T_{C}$ ($>$ 400 K) for $T_{g}\approx$130$^{\circ}$C  The larger columns become ferromagnetic \textit{i e } $T_{B}>T_{C}$  Meanwhile Ge$_{3}$Mn$_{5}$ clusters start to form  Finally for higher $T_{g}$, the magnetic contribution from Ge$_{3}$Mn$_{5}$ clusters keeps increasing while the nanocolumns signal progressively disappears 

\begin{figure}[htb]
\center
   \includegraphics[width= 6\linewidth]{ /fig7a eps}
   \includegraphics[width= 3\linewidth]{ /fig7b eps}
\caption{(a) Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{0 93}$Mn$_{0 07}$ samples for different growth temperatures  The magnetic field is applied in the film plane  The inset shows the temperature dependence of a sample grown at 130$^{\circ}$C and annealed at 650$^{\circ}$C for 15 minutes  After annealing, the magnetic signal mostly arises from Ge$_{3}$Mn$_{5}$ clusters  (b) ZFC-FC measurements performed on Ge$_{0 93}$Mn$_{0 07}$ samples for different growth temperatures  The in-plane applied field is 0 015 T  The ZFC peak at low temperature ($\leq$150 K) can be attributed to the superparamagnetic nanocolumns  This peak widens and shifts towards high blocking temperatures when increasing growth temperature  The second peak above 150 K in the ZFC curve which increases with increasing growth temperature is attributed to superparamagnetic Ge$_{3}$Mn$_{5}$ clusters  The increasing ZFC-FC irreversibility at $\approx$ 300 K is due to the increasing contribution from large ferromagnetic Ge$_{3}$Mn$_{5}$ clusters  The nanocolumns signal completely vanishes after annealing at 650$^{\circ}$C for 15 minutes }
\label{fig7}
\end{figure}

In Fig  7a, the saturation magnetization at 2 Tesla in $\mu_{B}$/Mn of Ge$_{1-x}$Mn$_{x}$ films with 7 \% of Mn is plotted as a function of temperature for different growth temperatures ranging from $T_{g}$=90$^{\circ}$C up to 160$^{\circ}$C  The inset shows the temperature dependence of the magnetization at 2 Tesla after annealing at 650$^{\circ}$C during 15 minutes  Figure 7b displays the corresponding Zero Field Cooled - Field Cooled (ZFC-FC) curves recorded at 0 015 Tesla  In the ZFC-FC procedure, the sample is first cooled down to 5 K in zero magnetic field and the susceptibility is subsequently recorded at 0 015 Tesla while increasing the temperature up to 400 K (ZFC curve)  Then, the susceptibility is recorded under the same magnetic field while decreasing the temperature down to 5 K (FC curve)  Three different regimes can be clearly distinguished  \\
For $T_{g}\leq$120$^{\circ}$C, the temperature dependence of the saturation magnetization remains nearly the same while increasing growth temperature  The overall magnetic signal vanishing above 200 K is attributed to the nanocolumns whereas the increasing signal below 50 K originates from diluted Mn atoms in the surrounding matrix  The Mn concentration dependence of the saturation magnetization is displayed in figure 8  For the lowest Mn concentration (4 \%), the contribution from diluted Mn atoms is very high and drops sharply for higher Mn concentrations (7 \%, 9 \% and 11 3 \%)  Therefore the fraction of Mn atoms in the diluted matrix decreases with Mn concentration probably because Mn atoms are more and more incorporated in the nanocolumns  In parallel, the Curie temperature of nanocolumns increases with the Mn concentration reaching 170 K for 11 3 \% of Mn  This behavior may be related to different Mn compositions and to the increasing diameter of nanocolumns (from 1 8 nm to 2 8 nm) as discussed in section \ref{structural} 

\begin{figure}[htb]
\center
   \includegraphics[width= 7\linewidth]{ /fig8 eps}
    \caption{Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{1-x}$Mn$_{x}$ films grown at 100$^{\circ}$C plotted for different Mn concentrations: 4 1 \%; 7 \%; 8 9 \% and 11 3 \% }
\label{fig8}
\end{figure}

ZFC-FC measurements show that the nanocolumns are superparamagnetic  The magnetic signal from the diluted Mn atoms in the matrix is too weak to be detected in susceptibility measurements at low temperature  In samples containing 4 \% of Mn, ZFC and FC curves superimpose down to low temperatures  As we do not observe hysteresis loops at low temperature, we believe that at this Mn concentration nanocolumns are  superparamagnetic in the whole temperature range and the blocking temperature cannot be measured  For higher Mn contents, the ZFC curve exhibits a very narrow peak with a maximum at the blocking temperature of 15 K whatever the Mn concentration and growth temperature (see Fig  7b)  Therefore the anisotropy barrier distribution is narrow and assuming that nanocolumns have the same magnetic anisotropy, this is a consequence of the very narrow size distribution of the nanocolumns as observed by TEM  To probe the anisotropy barrier distribution, we have performed ZFC-FC measurements but instead of warming the sample up to 400 K, we stopped at a lower temperature $T_{0}$  

\begin{figure}[htb]
\center
   \includegraphics[width= 6\linewidth]{ /fig9 eps}
\caption{Schematic drawing of the anisotropy barrier distribution n($E_{B}$) of superparamagnetic nanostructures  If magnetic anisotropy does not depend on the particle size, this distribution exactly reflects their magnetic size distribution  In this drawing the blocking temperature ($T_{B}$) corresponds to the distribution maximum  At a given temperature $T_{0}$ such that 25$k_{B}T_{0}$ falls into the anisotropy barrier distribution, the largest nanostructures with an anisotropy energy larger than 25$k_{B}T_{0}$ are blocked whereas the others are superparamagnetic }
\label{fig9}
\end{figure}

If this temperature falls into the anisotropy barrier distribution as depicted in Fig  9, the FC curve deviates from the ZFC curve  Indeed the smallest nanostructures have become superparamagnetic at $T_{0}$ and when decreasing again the temperature, their magnetization freezes along a direction close to the magnetic field and the FC susceptibility is higher than the ZFC susceptibility  Therefore any irreversibility in this procedure points at the presence of superparamagnetic nanostructures  The results are given in Fig  10a  ZFC and FC curves clearly superimpose up to $T_{0}$=250 K thus the nanocolumns are superparamagnetic up to their Curie temperature and no Ge$_{3}$Mn$_{5}$ clusters could be detected  Moreover for low $T_{0}$ values, a peak appears at low temperature in FC curves which evidences strong antiferromagnetic interactions between the nanocolumns \cite{Chan00} 

\begin{figure}[htb]
\center
    \includegraphics[width= 35\linewidth]{ /fig10a eps}
    \includegraphics[width= 63\linewidth]{ /fig10b eps}
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 115$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 30 K, 50 K, 100 K, 150 K, 200 K and 250 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0 015 T) }
\label{fig10}
\end{figure}

In order to derive the magnetic size and anisotropy of the Mn-rich nanocolumns embedded in the Ge matrix, we have fitted the inverse normalized in-plane (resp  out-of-plane) susceptibility: $\chi_{\parallel}^{-1}$ (resp  $\chi_{\perp}^{-1}$)  The corresponding experimental ZFC-FC curves are reported in Fig  10b  Since susceptibility measurements are performed at low field (0 015 T), the matrix magnetic signal remains negligible  In order to normalize susceptibility data, we need to divide the magnetic moment by the saturated magnetic moment recorded at 5 T  However the matrix magnetic signal becomes very strong at 5 T and low temperature so that we need to subtract it from the saturated magnetic moment using a simple Curie function  From Fig  10b, we can conclude that nanocolumns are isotropic  Therefore to fit experimental data we use the following expression well suited for isotropic systems or cubic anisotropy: $\chi_{\parallel}^{-1}= \chi_{\perp}^{-1}\approx 3k_{B}T/M(T)+\mu_{0}H_{eff}(T)$  $k_{B}$ is the Boltzmann constant, $M=M_{s}v$ is the magnetic moment of a single-domain nanostructure (macrospin approximation) where $M_{s}$ is its magnetization and $v$ its volume  The in-plane magnetic field is applied along $[110]$ or $[-110]$ crystal axes  Since the nanostructures Curie temperature does not exceed 170 K, the temperature dependence of the saturation magnetization is also accounted for by writting $M(T)$  Antiferromagnetic interactions between nanostructures are also considered by adding an effective field estimated in the mean field approximation \cite{Fruc02}: $\mu_{0}H_{eff}(T)$ 
The only fitting parameters are the maximum magnetic moment (\textit{i e } at low temperature) per nanostructure: $M$ (in Bohr magnetons $\mu_{B}$) and the maximum interaction field (\textit{i e } at low temperature): $\mu_{0}H_{eff}$ 

\begin{figure}[htb]
\center
   \includegraphics[width= 7\linewidth]{ /fig11 eps}
\caption{Temperature dependence of the inverse in-plane (open circles) and out-of-plane (open squares) normalized susceptibilities of a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 115$^{\circ}$C  Fits were performed assuming isotropic nanostructures or cubic anisotropy  Dashed line is for in-plane susceptibility and solid line for out-of-plane susceptibility }
\label{fig11}
\end{figure}

In Fig  11, the best fits lead to $M\approx$1250 $\mu_{B}$ and $\mu_{0}H_{eff}\approx$102 mT for in-plane susceptibility and $M\approx$1600 $\mu_{B}$ and $\mu_{0}H_{eff}\approx$98 mT for out-of-plane susceptibility  It gives an average magnetic moment of 1425 $\mu_{B}$ per column and an effective interaction field of 100 mT  Using this magnetic moment and its temperature dependence, magnetization curves could be fitted using a Langevin function and $M(H/T)$ curves superimpose for $T<$100 K  However, from the saturated magnetic moment of the columns and their density (35000 $\rm{\mu m}^{-2}$), we find almost 6000 $\mu_{B}$ per column  Therefore, for low growth temperatures, we need to assume that nanocolumns are actually made of almost four independent elongated magnetic nanostructures  The effective field for antiferromagnetic interactions between nanostructures estimated from the susceptibility fits is at least one order of magnitude larger than what is expected from pure magnetostatic coupling  This difference may be due to either an additional antiferromagnetic coupling through the matrix which origin remains unexplained or to the mean field approximation which is no more valid in this strong coupling regime  As for magnetic anisotropy, the nanostructures behave as isotropic magnetic systems or exhibit a cubic magnetic anisotropy  First we can confirm that nanostructures are not amorphous otherwise shape anisotropy would dominate leading to out-of-plane anisotropy  We can also rule out a random distribution of magnetic easy axes since the nanostructures are clearly crystallized in the diamond structure and would exhibit at least a cubic anisotropy (except if the random distribution of Mn atoms within the nanostructures can yield random easy axes)  Since the nanostructures are in strong in-plane compression (their lattice parameter is larger than the matrix one), the cubic symmetry of the diamond structure is broken and magnetic cubic anisotropy is thus unlikely  We rather believe that out-of-plane shape anisotropy is nearly compensated by in-plane magnetoelastic anisotropy due to compression leading to a \textit{pseudo} cubic anisotropy  From the blocking temperature (15 K) and the magnetic volume of the nanostructures , we can derive their magnetic anisotropy constant using $Kv=25k_{B}T_{B}$: K$\approx$10 kJ m$^{-3}$ which is of the same order of magnitude as shape anisotropy 

\begin{figure}[htb]
\center
    \includegraphics[width= 35\linewidth]{ /fig12a eps}
    \includegraphics[width= 63\linewidth]{ /fig12b eps} 
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 93}$Mn$_{0 07}$ sample grown at 122$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 50 K, 100 K, 150 K, 200 K and 250 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields  (0 015 T) }
\label{fig12}
\end{figure}

For growth temperatures $T_{g}\geq$120$^{\circ}$C and Mn concentrations $\geq$ 7 \%, samples exhibit a magnetic signal above 200 K corresponding to Ge$_{3}$Mn$_{5}$ clusters (see Fig  7a)  As we can see, SQUID measurements are much more sensitive to the presence of Ge$_{3}$Mn$_{5}$ clusters, even at low concentration, than TEM and x-ray diffraction used in section \ref{structural}  We also observe a sharp transition in the ZFC curve (see Fig  7b, Fig  12a and 12b): the peak becomes very large and is shifted towards high blocking temperatures (the signal is maximum at $T=$23 K)  This can be easily understood as a magnetic percolation of the four independent nanostructures obtained at low growth temperatures into a single magnetic nanocolumn  Therefore the magnetic volume increases sharply as well as blocking temperatures  At the same time, the size distribution widens as observed in TEM  In Fig  12a, we have performed ZFC-FC measurements at different $T_{0}$ temperatures  The ZFC-FC irreversibility is observed up to the Curie temperature of $\approx$120 K meaning that a fraction of nanocolumns is ferromagnetic (\textit{i e } $T_{B}\geq T_{C}$) 
In Fig  12b, in-plane and out-of-plane ZFC curves nearly superimpose for $T\leq$150 K due to the isotropic magnetic behavior of the nanocolumns: in-plane magnetoelastic anisotropy is still compensating out-of-plane shape anisotropy  Moreover the magnetic signal above 150 K corresponding to Ge$_{3}$Mn$_{5}$ clusters that start to form in this growth temperature range is strongly anisotropic  This perpendicular anisotropy confirms the epitaxial relation: (0002) Ge$_{3}$Mn$_{5}$ $\parallel$ (002) Ge discussed in Ref \cite{Bihl06}  The magnetic easy axis of the clusters lies along the hexagonal $c$-axis which is perpendicular to the film plane 

\begin{figure}[ht]
\center
   \includegraphics[width= 35\linewidth]{ /fig13a eps}
   \includegraphics[width= 63\linewidth]{ /fig13b eps} 
\caption{(a) ZFC-FC measurements performed on a Ge$_{0 887}$Mn$_{0 113}$ sample grown at 145$^{\circ}$C  The in-plane applied field is 0 015 T  Magnetization was recorded up to different T$_{0}$ temperatures: 50 K, 100 K, 150 K, 200 K, 250 K and 300 K  Curves are shifted up for more clarity  (b) ZFC-FC curves for in-plane and out-of-plane applied fields (0 015 T) }
\label{fig13}
\end{figure}

For growth temperatures $T_{g}\geq$145$^{\circ}$C the cluster magnetic signal dominates (Fig  13b)  Superparamagnetic nanostructures are investigated performing ZFC-FC measurements at different $T_{0}$ temperatures (Fig  13a)  The first ZFC peak at low temperature \textit{i e } $\leq$ 150 K is attributed to  low-$T_{C}$ nanocolumns ($T_{C}\approx$130 K)  This peak is wider than for lower growth temperatures and its maximum is further shifted up to 30 K  These results are in agreement with TEM observations: increasing $T_{g}$ leads to larger nanocolumns (\textit{i e } higher blocking temperatures) and wider size distributions  ZFC-FC irreversibility is observed up to the Curie temperature due to the presence of ferromagnetic columns  The second peak above 180 K in the ZFC curve is attributed to Ge$_{3}$Mn$_{5}$ clusters and the corresponding ZFC-FC irreversibility persisting up to 300 K means that some clusters are ferromagnetic  We clearly evidence the out-of-plane anisotropy of Ge$_{3}$Mn$_{5}$ clusters and the isotropic magnetic behavior of nanocolumns (Fig  13b)  In this growth temperature range, we have also investigated the Mn concentration dependence of magnetic properties  

\begin{figure}[ht]
\center
    \includegraphics[width= 49\linewidth]{ /fig14a eps}
    \includegraphics[width= 49\linewidth]{ /fig14b eps} 
\caption{Temperature dependence of the saturation magnetization (in $\mu_{B}$/Mn) of Ge$_{1-x}$Mn$_{x}$ films grown at 150$^{\circ}$C plotted for different Mn concentrations: 2 3 \%; 4 \%; 7 \%; 9 \%; 11 3 \%  (b) ZFC-FC measurements performed on Ge$_{1-x}$Mn$_{x}$ films grown at 150$^{\circ}$C  The in-plane applied field is 0 025 T for 2 3 \% and 4 \% and 0 015 T for 8 \% and 11 3 \%  }
\label{fig14}
\end{figure}

In Fig  14a, for low Mn concentrations (2 3 \% and 4 \%) the contribution from diluted Mn atoms in the germanium matrix to the saturation magnetization is very high and nearly vanishes for higher Mn concentrations (7 \%, 9 \% and 13 \%) as observed for low growth temperatures  Above 7 \%, the magnetic signal mainly comes from nanocolumns and Ge$_{3}$Mn$_{5}$ clusters  We can derive more information from ZFC-FC measurements (Fig  14b)  Indeed, for 2 3 \% of Mn, ZFC and FC curves nearly superimpose down to low temperature meaning that nanocolumns are superparamagnetic in the whole temperature range  Moreover the weak irreversibility arising at 300 K  means that some Ge$_{3}$Mn$_{5}$ clusters have already formed in the samples even at very low Mn concentrations  For 4 \% of Mn, we can observe a peak with a maximum at the blocking temperature (12 K) in the ZFC curve  We can also derive the Curie temperature of nanocolumns: $\approx$45 K  The irresversibility arising at 300 K still comes from Ge$_{3}$Mn$_{5}$ clusters  Increasing the Mn concentration above 7 \% leads to: higher blocking temperatures (20 K and 30 K) due to larger nanocolumns and wider ZFC peaks due to wider size distributions in agreement with TEM observations (see Fig  3a)  Curie temperatures also increase (110 K and 130 K) as well as the contribution from Ge$_{3}$Mn$_{5}$ clusters \\
Finally when increasing $T_{g}$ above 160$^{\circ}$C, the nanocolumns magnetic signal vanishes and only Ge$_{3}$Mn$_{5}$ clusters and diluted Mn atoms coexist  The overall magnetic signal becomes comparable to the one measured on annealed samples in which only Ge$_{3}$Mn$_{5}$ clusters are observed by TEM (see Fig  7a) \\
The magnetic properties of high-$T_{C}$ nanocolumns obtained for $T_{g}$ close to 130$^{\circ}$C are discussed in detail in Ref \cite{Jame06} \\
In conclusion, at low growth temperatures ($T_{g}\leq$120$^{\circ}$C), nanocolumns are made of almost 4 independent elongated magnetic nanostructures  For $T_{g}\geq$120$^{\circ}$C, these independent nanostructures percolate into a single nanocolumn sharply leading to higher blocking temperatures  Increasing $T_{g}$ leads to larger columns with a wider size distribution as evidenced by ZFC-FC measurements and given by TEM observations  In parallel, some Ge$_{3}$Mn$_{5}$ clusters start to form and their contribution increases when increasing $T_{g}$  Results on magnetic anisotropy seems counter-intuitive  Indeed Ge$_{3}$Mn$_{5}$ clusters exhibit strong out-of-plane anisotropy whereas nanocolumns which are highly elongated magnetic structures are almost isotropic  This effect is probably due to compensating in-plane magnetoelastic coupling (due to the columns compression) and out-of-plane shape anisotropy  

\section{Conclusion}

In this paper, we have investigated the structural and magnetic properties of thin Ge$_{1-x}$Mn$_{x}$ films grown by low temperature molecular beam epitaxy  A wide range of growth temperatures and Mn concentrations have been explored  All the samples contain Mn-rich nanocolumns as a consequence of 2D-spinodal decomposition  However their size, crystalline structure and magnetic properties depend on growth temperature and Mn concentration  For low growth temperatures, nanocolumns are very small (their diameter ranges between 1 8 nm for 1 3 \% of Mn and 2 8 nm for 11 3 \% of Mn), their Curie temperature is rather low ($<$ 170 K) and they behave as almost four uncorrelated superparamagnetic nanostructures  Increasing Mn concentration leads to higher columns densities while diameters remain nearly unchanged  For higher growth temperatures, the nanocolumns mean diameter increases and their size distribution widens  Moreover the 4 independent magnetic nanostructures percolate into a single magnetic nanocolumn  Some columns are ferromagnetic even if Curie temperatures remain quite low  In this regime, increasing Mn concentration leads to larger columns while their density remains nearly the same  In parallel, Ge$_{3}$Mn$_{5}$ nanoclusters start to form in the film with their $c$-axis perpendicular to the film plane  In both temperature regimes, the Mn incorporation mechanism in the nanocolumns and/or in the matrix changes above 5 \% of Mn and nanocolumns exhibit an isotropic magnetic behaviour due to the competing effects of out-of-plane shape anisotropy and in-plane magnetoelastic coupling  Finally for a narrow range of growth temperatures around 130$^{\circ}$C, nanocolumns exhibit Curie temperatures higher than 400 K  Our goal is now to investigate the crystalline structure inside the nanocolumns, in particular the position of Mn atoms in the distorted diamond structure, which is essential to understand magnetic and future transport properties in Ge$_{1-x}$Mn$_{x}$ films 

\section{Aknowledgements}
The authors would like to thank Dr  F  Rieutord for grazing incidence x-ray diffraction measurements performed on the GMT station of BM32 beamline at the European Synchrotron Radiation Facility 





ground_truth: 1425 $\mu_{B}$.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are the three teams that used conflict optimization in the challenge?

Read the following text and answer briefly.

Context 3:  While the original TABUCOL algorithm includes a "tabu-list" mechanism to avoid cycling, it is not always sufficient, and requires some hyper-parameter tuning in order to obtain a good performance on a large variety of instances  To overcome this issue, we use a neighbourhood, but replace the "tabu-list" by the conflict optimizer scheme presented above 
PARTIALCOL inspired neighbourhood PARTIALCOL another local search algorithm solving the vertex coloring problem was introduced in 2008  This algorithm proposes a new local search scheme that allows partial coloring (thus allowing uncolored vertices)  The goal is to minimize the number of uncolored vertices 
Similarly to TABUCOL, PARTIALCOL starts with an initial solution, removes one color (unassigning its vertices), and performs local search iterations until no vertex is left uncolored  When coloring a vertex, the adjacent conflicting vertices are uncolored  Then, the algorithm repeats the process until all vertices are colored, or the stopping criterion is met 
This neighbourhood was also introduced alongside a tabu-search procedure  The tabu-search scheme is also replaced by a conflict-optimization scheme  Note that this neighbourhood was predominantly used by the other teams 

Finding Initial Solutions

Lasa team used two approaches to find initial solutions: 1  DSATUR is the classical graph coloring algorithm presented in Section 1  2  Orientation greedy is almost the only algorithm where the geometry of the segments is used  If segments are almost parallel, it is likely that they do not intersect (thus forming an independent set) 
This greedy algorithm first sorts the segments by orientation, ranging from − π 2 to π 2   For each segment in this order, the algorithm tries to color it using the first available color  If no color has been found, a new color is created for coloring the considered segment  This algorithm is efficient, produces interesting initial solutions and takes into account the specificities of the competition 

Solution Initialization

The gitastrophe team uses the traditional greedy algorithm of Welsh and Powell to obtain initial solutions: order the vertices in decreasing order of degree, and assign each vertex the minimum-label color not used by its neighbors  During the challenge Gitastrophe attempted to use different orderings for the greedy algorithm, such as sorting by the slope of the line segment associated with each vertex (as the orientation greedy initialization presented in Section 3), and also tried numerous other strategies 
Ultimately, after running the solution optimizer for approximately the same amount of time, all initializations resulted in an equal number of colors 

Modifications to the Conflict Optimizer

Taking inspiration from memetic algorithms, which alternate between an intensification and a diversification stage, the algorithm continually switched between a phase using the above conflict score, and one minimizing only the number of conflicts  Thus during the conflict-minimization phase, the random variables f (C j ) and w(u) are both fixed equal to 1 leading to a conflict score
Each phase lasted for 10 5 iterations  Adding the conflict-minimization phase gave minor improvements to some of the challenge instances 

Shadoks

In this section, we describe the choices used by the Shadoks team for the options described in Section 2 1  The Shadoks generally chose to eliminate the color with the smallest number of elements  However, if the multistart option is toggled on, then a random color is used each time  The conflict set S is stored in a queue 
The Shadoks tried other strategies, but found that the queue gives the best results  The weight function used is w(u) = 1 + q(u) p , mostly with p = 1 2  The effect of the parameter p is shown in Fig    Notice that in all figures, the number of colors shown is the average of ten executions of the code using different random seeds 
The algorithm uses σ = 0 15, easy vertices, q max = 59022, but does not use the BDFS nor any clique  If q(u) is larger than a threshold q max , the Shadoks set w(u) = ∞ so that the vertex u never reenters S  If at some point an uncolored vertex v is adjacent to some vertex u of infinite weight in every color class, then the conflict optimizer is restarted 
When restarting, the initial coloring is shuffled by moving some vertices from their initial color class to a new one  Looking at Fig  , the value of q max does not seem to have much influence as long as it is not too small  Throughout the challenge the Shadoks almost exclusively used q max = 2000 • (75000/m) 2 , where m is the number of vertices 
This value roughly ensures a restart every few hours  q max =0 5k q max =5k q max =50k q max =100k q max =250k The Shadoks use the function f as a Gaussian random variable of mean 1 and variance σ  A good default value is σ = 0 15  The effect of the variance is shown in Fig    Notice that setting σ = 0 gives much worse results 
Option (e) The goal of BDFS is to further optimize very good solutions that the conflict optimizer is not able to improve otherwise  Fig  shows the influence of BDFS  While on this figure, the advantages of BDFS cannot be noticed, its use near the end of the challenge improved about 30 solutions  The bounded depth-first search (BDFS) algorithm tries to improve the dequeuing process 
The goal is to prevent a vertex in conflict with some adjacent colored vertices from entering in the conflict set  At the first level, the algorithm searches for a recoloring of some adjacent vertices which allows us to directly recolor the conflict vertex  If no solution is found, the algorithm  In both figures the algorithm uses p = 1 2, easy vertices, q max = 59022, but does not use the BDFS nor any clique 
For σ ≥ 0 25, no solution better than 248 colors is found  could recolor some vertices at larger distances from the conflict vertex  To do so, a local search is performed by trying to recolor vertices at a bounded distance from the conflict vertex in the current partial solution  The BDFS algorithm has two parameters: adjacency bound a max and depth d 
In order to recolor a vertex v, BDFS gets the set C of color classes with at most a max neighbors of v  If a class in C has no neighbor of v, v is assigned to C  Otherwise, for each class C ∈ C, BDFS tries to recolor the vertices in C which are adjacent to v by recursively calling itself with depth d − 1 
At depth d = 0 the algorithm stops trying to color the vertices  During the challenge the Shadoks used BDFS with parameters a max = 3 and d = 3  The depth was increased to 5 (resp  7) when the number of vertices in the queue was 2 (resp  1)  Degeneracy order Given a target number of colors k, we call easy vertices a set of vertices Y such that, if the remainder of the vertices of G are colored using k colors, then we are guaranteed to be able to color all vertices of G with k colors 
This is obtained using the degeneracy order Y   To obtain Y we iteratively remove from the graph a vertex v that has at most k − 1 neighbors, appending v to the end of Y   We repeat until no other vertex can be added to Y   Notice that, once we color the remainder of the graph with at least k colors, we can use a greedy coloring for Y in order from last to first without increasing the number of colors used 
Removing the easy vertices reduces the total number of vertices, making the conflict optimizer more effective  The Shadoks always toggle this option on (the challenge instances contain from 0 to 23% easy vertices) 

Results

We provide the results of the experiments performed with the code from the three teams on two classes of instances  First, we present the results on some selected CG:SHOP 2022 instances  These instances are intersection graphs of line segments  Second, we execute the code on graphs that are not intersection graphs, namely the classic DIMACS graphs , comparing the results of our conflict optimizer implementations to previous solutions 
The source code for the three teams is available at: • Lasa: https://github com/librallu/dogs-color • Gitastrophe: https://github com/jacketsj/cgshop2022-gitastrophe • Shadoks: https://github com/gfonsecabr/shadoks-CGSHOP2022

CG:SHOP 2022 Instances

We selected 14 instances (out of 225) covering the different types of instances given in the CG:SHOP 2022 challenge  The results are presented in Table   For comparison, we executed the HEAD code on some instances using the default parameters  The table shows the smallest number of colors for which HEAD found a solution 
We ran HEAD for 1 hour of repetitions for each target number of colors on a single CPU core (the HEAD solver takes the target number of colors as a parameter and we increased this parameter one by one)  At the end of the challenge, 8 colorings computed by Lasa, 11 colorings computed by Gitastrophe, and 23 colorings computed by Shadoks over 225 instances have been proved optimal (their number of colors is equal to the size of a clique) 
In order to compare the efficiency of the algorithms, we executed the different implementations on the CG:SHOP instance vispecn13806  The edge density of this graph is 19%, the largest clique that we found has 177 vertices and the best coloring found during the challenge uses 218 colors  Notice that vispecn13806 is the same instance used in other Shadoks experiments in Section 5  Notice also that HEAD algorithm provides 283 colors after one hour compared to less than 240 colors for the conflict optimizers 
We ran the three implementations on three different servers and compared the results shown in Figure   For each implementation, the x coordinate is the running time in hours, while the y coordinate is the smallest number of colors found at that time 

Results on DIMACS Graphs

We tested the implementation of each team on the DIMACS instances to gauge the performance of the conflict optimizer on other classes of graphs  We compared our results to the best known bounds and to the state of the art coloring algorithms HEAD and QACOL   The time limit for Lasa's algorithms is 1 hour 
CWLS is Lasa's conflict optimizer with the neighbourhood presented in TABUCOL , while PWLS is the optimizer with the neighbourhood presented in PARTIALCOL   Gitastrophe algorithm ran 10 minutes after which the number of colors no longer decreases  Shadoks algorithm ran for 1 hour without the BDFS option (results with BDFS are worse) 
Results are presented in Table   We only kept the difficult DIMACS instances  For the other instances, all the results match the best known bounds  The DIMACS instances had comparatively few edges (on the order of thousands or millions); the largest intersection graphs considered in the CG:SHOP challenge had over 1 5 billion edges 
We notice that the conflict optimizer works extremely poorly on random graphs, but it is fast and appears to perform well on geometric graphs (r250 5, r1000 1c, r1000 5, dsjr500 1c and dsjr500 5), matching the best-known results   Interestingly, these geometric graphs are not intersection graphs as in the CG:SHOP challenge, but are generated based on a distance threshold 
On the DIMACS graphs, Lasa implementation shows better performance than the other implementations 



ground_truth: Lasa, Gitastrophe, and Shadoks.