guard games on graphs a team of mobile agents called guards tries to keep an intruder out of an assigned area by blocking all possible attacks in a graph model for this setting the guards and the intruder are located on the vertices of a graph and they move from node to node via connecting edges the area protected by the guards is an induced subgraph of the given graph we investigate the algorithmic aspects of the guarding problem which is to find the minimum number of guards sufficient to patrol the area we show that the guarding problem is pspace hard and provide a set of approximation algorithms all approximation algorithms are based on the study of a variant of the game where the intruder must reach the guarded area in a single step in order to win this variant of the game appears to be a 2 approximation for the guarding problem and for graphs without cycles of length 5 the minimum number of required guards in both games coincides we give a polynomial time algorithm for solving the one step guarding problem in graphs of bounded treewidth and complement this result by showing that the problem is w1 hard parameterized by the treewidth of the input graph we also show that the problem is fixed parameter tractable fpt parameterized by the treewidth and maximum degree of the input graph finally we turn our attention to a large class of sparse graphs including planar graphs and graphs of bounded genus namely apex minor free graphs we prove that the one step guarding problem is fpt and possess a ptas on apex minor free graphs

complexity and approximation of the constrained forest problem given an undirected graph on n vertices with weights on its edges min wcf p consists of computing a covering forest of minimum weight such that each of its tree components contains at least p vertices it has been proved that min wcf p is np hard for any p 4 imielinska et al 1993 10 but 2 1n approximable goemans and williamson 1995 9 while min wcf 2 is polynomial time solvable already the unweighted version of min wcf 3 is np hard even on planar bipartite graphs of maximum degree 3 we prove here that for any p 4 the unweighted version is np hard even for planar bipartite graphs of maximum degree 3 moreover the unweighted version for any p 3 has no ptas for bipartite graphs of maximum degree 3 the latter theorem is the first ever apx hardness result on this problem on the other hand we show that min wcf p is polynomial time solvable on graphs with bounded treewidth and for any p bounded by o lognloglogn it has a ptas on planar graphs

ornstein zernike theory for finite range ising models above t c we derive a precise ornstein zernike asymptotic formula for the decay of the two point function 0 x in the general context of finite range ising type models on d the proof relies in an essential way on the a priori knowledge of the strict exponential decay of the two point function and by the sharp characterization of phase transition due to aizenman barsky and fernandez goes through in the whole of the high temperature region c as a byproduct we obtain that for every c the inverse correlation length is an analytic and strictly convex function of direction

approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and bayesian inference a feedback vertex set of an undirected graph is a subset of vertices that intersects with the vertex set of each cycle in the graph given an undirected graph g with n vertices and weights on its vertices polynomial time algorithms are provided for approximating the problem of finding a feedback vertex set of g with smallest weight when the weights of all vertices in g are equal the performance ratio attained by these algorithms is 4 2 n this improves a previous algorithm which achieved an approximation factor of o sqrt log n for this case for general vertex weights the performance ratio becomes min 2 delta 2 4 log 2 n where delta denotes the maximum degree in g for the special case of planar graphs this ratio is reduced to 10 an interesting special case of weighted graphs where a performance ratio of 4 2 n is achieved is the one where a prescribed subset of the vertices so called blackout vertices is not allowed to participate in any feedback vertex set it is shown how these algorithms can improve the search performance for constraint satisfaction problems an application in the area of bayesian inference of graphs with blackout vertices is also presented

parametric duality and kernelization lower bounds and upper bounds on kernel size determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms theoretically it has been proved that a parameterized problem is kernelizable if and only if it is fixed parameter tractable practically applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance i e a kernel has led to very efficient algorithms and now goes hand in hand with the design of practical algorithms for solving mathcal np hard problems well known examples of such parameterized problems include the vertex cover problem which is kernelizable to a kernel of size bounded by 2k and the planar dominating set problem which is kernelizable to a kernel of size bounded by 335k in this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems in terms of our lower bound results we show for example that unless mathcal p mathcal np planar vertex cover does not have a problem kernel of size smaller than 4k 3 and planar independent set and planar dominating set do not have kernels of size smaller than 2k in terms of our upper bound results we further reduce the upper bound on the kernel size for the planar dominating set problem to 67 k improving significantly the 335 k previous upper bound given by alber fellows and niedermeier j acm 51 2004 pp 363 384 this latter result is obtained by introducing a new set of reduction and coloring rules which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel the paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem

the droplet in the tube a case of phase transition in the canonical ensemble we consider the 2 dimensional ising model with ferromagnetic nearest neighbour interaction at inverse temperature lets n t be the total magnetization inside ann n square box per be the gibbs state in with periodic b c andm be the spontaneous magnetization we show the existence of the limit psi varrho mathop lim limits n to infty left frac 1 beta n right ln mu lambda per s n n varrho for m provided is large enough it turns out that the quantity is closely related to the wulf construction and the dependence of the function on is singular

approximation of minimum weight spanners for sparse graphs a t spanner of a graph g is its spanning subgraph s such that the distance between every pair of vertices in s is at most t times their distance in g the sparsestt spanner problem asks to find for a given graph g and an integer t a t spanner of g with the minimum number of edges the problem is known to be np hard for all t 2 and even more it is np hard to approximate it with ratio o logn for every t 2 for t 5 the problem remains np hard for planar graphs and the approximability status of the problem on planar graphs was open we resolve this open issue by showing that the sparsestt spanner problem admits the efficient polynomial time approximation scheme eptas for every t 1 our result holds for a much wider class of graphs namely the class of apex minor free graphs which contains the classes of planar and bounded genus graphs moreover it is possible to extend our results to weighted apex minor free graphs when the maximum edge weight is bounded by some constant

on the gibbs states of the noncritical potts model on z 2 we prove that all gibbs states of the q state nearest neighbor potts model on mathbb z 2 below the critical temperature are convex combinations of the q pure phases in particular they are all translation invariant to achieve this goal we consider such models in large finite boxes with arbitrary boundary condition and prove that the center of the box lies deeply inside a pure phase with high probability our estimate of the finite volume error term is of essentially optimal order which stems from the brownian scaling of fluctuating interfaces the results hold at any supercritical value of the inverse temperature beta beta c q log left 1 sqrt q right

parameterized and approximation complexity of partial vc dimension abstract we introduce the problem partial vc dimension that asks given a hypergraph h x e and integers k and l whether one can select a set c x of k vertices of h such that the set e c e e of distinct hyperedge intersections with c has size at least l the sets e c define equivalence classes over e partial vc dimension is a generalization of vc dimension which corresponds to the case l 2 k and of distinguishing transversal which corresponds to the case l e the latter is also known as test cover in the dual hypergraph we also introduce the associated fixed cardinality maximization problem max partial vc dimension that aims at maximizing the number of equivalence classes induced by a solution set of k vertices we study the algorithmic complexity of partial vc dimension and max partial vc dimension both on general hypergraphs and on more restricted instances in particular neighborhood hypergraphs of graphs

excluded grid minors and efficient polynomial time approximation schemes two of the most widely used approaches to obtain polynomial time approximation schemes ptass on planar graphs are the lipton tarjan separator based approach and baker s approach in 2005 demaine and hajiaghayi strengthened both approaches using bidimensionality and obtained efficient polynomial time approximation schemes eptass for several problems including c onnected d ominating s et and f eedback v ertex s et in this work we unify the two strengthened approaches to combine the best of both worlds we develop a framework allowing the design of eptas on classes of graphs with the subquadratic grid minor sqgm property roughly speaking a class of graphs has the sqgm property if for every graph g from the class the fact that g contains no t t grid as a minor guarantees that the treewidth of g is subquadratic in t for example the class of planar graphs and more generally classes of graphs excluding some fixed graph as a minor have the sqgm property at the heart of our framework is a decomposition lemma stating that for most bidimensional problems on a graph class g with the sqgm property there is a polynomial time algorithm that given a graph g e g as input and an e 0 outputs a vertex set x of size e opt such that the treewidth of g x is f e here opt is the objective function value of the problem in question and f is a function depending only on e this allows us to obtain eptass on apex minor free graphs for all problems covered by the previous framework as well as for a wide range of packing problems partial covering problems and problems that are neither closed under taking minors nor contractions to the best of our knowledge for many of these problems including c ycle p acking f p acking f d eletion m ax l eaf s panning t ree or p artial r d ominating s et no eptass even on planar graphs were previously known we also prove novel excluded grid theorems in unit disk and map graphs without large cliques using these theorems we show that these classes of graphs have the sqgm property based on the developed framework we design eptass and subexponential time parameterized algorithms for various classes of problems on unit disk and map graphs

approximation algorithms for np complete problems on planar graphs this paper describes a general technique that can be used to obtain approximation schemes for various np complete problems on planar graphs the strategy depends on decompos ing a planar graph into subgraphs of a form we call k outerplanar for fixed k the problems of interest are solvable optimally in linear time on k outerplanar graphs by dynamic programming for general planar graphs if the problem is a maximization problem such as maximum independent set this technique gives for each k a linear time algorithm that produces a solution whose size is at least k k 1 optimal if the problem is a minimization problem such as minimum vertex cover it gives for each k a linear time algorithm that produces a solution whose size is at most k 1 k optimal taking k c log log nl or k c log nl where n is the number of nodes and c is some constant we get polynomial time approximation algorithms whose solution sizes converge toward optimal as n increases the class of problems for which this approach provides approximation schemes includes maximum independent set maximum tile salvage partition into triangles maximum h matching minimum vertex cover minimum dominat ing set and minimum edge dominating set for these and certain other problems the proof of solvability on k outerplanar graphs also enlarges the class of planar gmphs for which the problems are known to be solvable in polynomial time

a 4 31 approximation for the geometric unique coverage problem on unit disks we give an improved approximation algorithm for the unique unit disk coverage problem given a set of points and a set of unit disks both in the plane we wish to find a subset of disks that maximizes the number of points contained in exactly one disk in the subset erlebach and van leeuwen 2008 introduced this problem as the geometric version of the unique coverage problem and gave a polynomial time 18 approximation algorithm in this paper we improve this approximation ratio 18 to 2 4 email protected 0 our algorithm runs in polynomial time which depends exponentially on 1 e the algorithm can be generalized to the budgeted unique unit disk coverage problem in which each point has a profit each disk has a cost and we wish to maximize the total profit of the uniquely covered points under the condition that the total cost is at most a given bound

large deviations for the 2d ising model a lower bound without cluster expansions we show that a lower large deviation bound for the block spin magnetization in the 2d ising model can be pushed all the way forward toward its correct wulff value for all c

applying clique decomposition for computing gromov hyperbolicity given a graph its hyperbolicity is a measure of how close its distance distribution is to the one of a tree this parameter has gained recent attention in the analysis of some graph algorithms and the classification of complex networks we study on practical improvements for the computation of hyperbolicity in large graphs precisely we investigate on relations between the hyperbolicity of a graph g and the hyperbolicity of its atoms that are the subgraphs output by the clique decomposition invented by tarjan 51 65 we prove that the maximum hyperbolicity taken over the atoms is at most one unit off from the hyperbol icity of g and the bound is sharp we also give an algorithm to slightly modify the atoms called the substitution method which is at no extra cost than computing the clique decomposition and so that the maximum hyperbolicity taken over the resulting graphs is exactly the hyperbolicity of the input graph g an experimental evaluation of our method for computing the hyperbolicity of a given graph from its atoms is provided for collaboration networks and biological networks finally on a more theoretical side we deduce from our results the first linear time algorithm for computing the hyperbolicity of an outerplanar graph

the many facets of upper domination this paper studies upper domination i e the problem of computing the maximum cardinality of a minimal dominating set in a graph with respect to classical and parameterised complexity as well as approximability

large deviations and continuum limit in the 2d ising model we study the 2d ising model in a rectangular box l of linear size o l we determine the exact asymptotic behaviour of the large deviations of the magnetization p t2 l ae t when l 1 for values of the parameters of the model corresponding to the phase coexistence region where the order parameter m is strictly positive we study in particular boundary eects due to an arbitrary real valued boundary magnetic field using the self duality of the model a large part of the analysis consists in deriving properties of the covariance function hae 0 ae t i as t 1 at dual values of the parameters of the model to do this analysis we establish new results about the high temperature representation of the model these results are valid for dimensions d 2 and up to the critical temperature they give a complete non perturbative exposition of the high temperature representation we then study the gibbs measure conditioned by p t2 l ae t im l l l ic with 0 c 1 4 and im m m we construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperi metric type

approximating the spanning star forest problem and its application to genomic sequence alignment this paper studies the algorithmic issues of the spanning star forest problem we prove the following results 1 there is a polynomial time approximation scheme for planar graphs 2 there is a polynomial time frac 3 5 approximation algorithm for graphs 3 it is np hard to approximate the problem within ratio frac 259 260 epsilon for graphs 4 there is a linear time algorithm to compute the maximum star forest of a weighted tree 5 there is a polynomial time frac 1 2 approximation algorithm for weighted graphs we also show how to apply this spanning star forest model to aligning multiple genomic sequences over a tandem duplication region

spectral discretization of the axisymmetric vorticity velocity and pressure formulation of the navier stokes problem we deal in this work with the nonlinear navier stokes equations set in a three dimensional axisymmetric bounded domain the boundary conditions that we consider are given on the normal component of the velocity and the tangential component of the vorticity such conditions occur in a large number of flows and we are led to write a vorticity velocity pressure formulation under assumptions on the data of the problem the three dimensional problem is reduced in a two dimensional one for the discretization we use the spectral methods which are well adapted here we prove the well posedness of the obtained formulations and we derive optimal error estimates on the three unknowns the results of the numerical experiments for known functions and a given data corresponding to a poiseuille type flow are coherent with the theoretical ones

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

on the hardness of approximating the network coding capacity this work addresses the computational complexity of achieving the capacity of a general network coding instance it has been shown lehman and lehman soda 2005 that determining the scalar linear capacity of a general network coding instance is np hard in this paper we address the notion of approximation in the context of both linear and nonlinear network coding loosely speaking we show that given an instance of the general network coding problem of capacity c constructing a code of rate c for any universal i e independent of the size of the instance constant 1 is hard specifically finding such network codes would solve a long standing open problem in the field of graph coloring our results refer to scalar linear vector linear and nonlinear encoding functions and are the first results that address the computational complexity of achieving the network coding capacity in both the vector linear and general network coding scenarios in addition we consider the problem of determining the scalar linear capacity of a planar network coding instance i e an instance in which the underlying graph is planar we show that even for planar networks this problem remains np hard

spectral discretization of the vorticity velocity and pressure formulation of the navier stokes equations we consider the navier stokes equations in a two or three dimensional domain provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity we write a variational formulation of this problem with three independent unknowns the vorticity the velocity and the pressure and prove the existence of a solution for this problem next we propose a discretization by spectral methods which relies on this formulation in the two dimensional case we prove quasi optimal error estimates for the three unknowns we conclude with some numerical experiments nous considerons les equations de navier stokes dans un domaine biou tri dimensionnel munies de conditions aux limites non usuelles portant sur la composante normale de la vitesse et la ou les composantes tangentielles du tourbillon nous ecrivons une formulation variationnelle de ce probleme qui comporte trois inconnues independantes le tourbillon la vitesse et la pression nous prouvons que ce probleme admet au moins une solution nous proposons une discretisation par methodes spectrales construite a partir de cette formulation dans le cas bidimensionnel nous etablissons des majorations quasi optimales de l erreur pour les trois inconnues nous concluons par quelques experiences numeriques

new approximations for minimum weighted dominating sets and minimum weighted connected dominating sets on unit disk graphs given a node weighted graph the minimum weighted dominating set mwds problem is to find a minimum weighted vertex subset such that for any vertex it is contained in this subset or it has a neighbor contained in this set and the minimum weighted connected dominating set mwcds problem is to find a mwds such that the graph induced by this subset is connected in this paper we study these two problems on a unit disk graph a 4 e approximation algorithm for an mwds based on a dynamic programming algorithm for a min weight chromatic disk cover is presented meanwhile we also propose a 1 e approximation algorithm for the connecting part by showing a polynomial time approximation scheme for a node weighted steiner tree problem when the given terminal set is c local and thus obtain a 5 e approximation algorithm for an mwcds

a linear time approximation scheme for tsp in undirected planar graphs with edge weights we give an algorithm requiring o c 1 epsilon 2 n time to find an epsilon optimal traveling salesman tour in the shortest path metric defined by an undirected planar graph with nonnegative edge lengths for the case of all lengths equal to 1 the time required is o c 1 epsilon n

approximation schemes for steiner forest on planar graphs and graphs of bounded treewidth we give the first polynomial time approximation scheme ptas for the steiner forest problem on planar graphs and more generally on graphs of bounded genus as a first step we show how to build a steiner forest spanner for such graphs the crux of the process is a clustering procedure called prize collecting clustering that breaks down the input instance into separate subinstances which are easier to handle moreover the terminals in different subinstances are far from each other each subinstance has a relatively inexpensive steiner tree connecting all its terminals and the subinstances can be solved almost separately another building block is a ptas for steiner forest on graphs of bounded treewidth surprisingly steiner forest is np hard even on graphs of treewidth 3 therefore our ptas for bounded treewidth graphs needs a nontrivial combination of approximation arguments and dynamic programming on the tree decomposition we further show that steiner forest can be solved in polynomial time for series parallel graphs graphs of treewidth at most two by a novel combination of dynamic programming and minimum cut computations completing our thorough complexity study of steiner forest in the range of bounded treewidth graphs planar graphs and bounded genus graphs

entropic repulsion of an interface in an external field we consider an interface above an attractive hard wall in the complete wetting regime and submitted to the action of an external increasing convex potential and study its delocalization as the intensity of this potential vanishes our main motivation is the analysis of critical prewetting which corresponds to the choice of a linear external potential we also present partial results on critical prewetting in the two dimensional ising model as well as a few weak results on pathwise estimates for the pure wetting problem for effective interface models

surface tension in ising systems with kac potentials we consider an ising spin system with kac potentials in a torus of d d 2 and fix the temperature below its lebowitz penrose critical value we prove that when the kac scaling parameter vanishes the log of the probability of an interface becomes proportional to its area and the surface tension related to the proportionality constant converges to the van der waals surface tension the results are based on the analysis of the rate functionals for gibbsian large deviations and on the proof that they converge to the perimeter functional of geometric measure theory which extends the notion of area our considerations include nonsmooth interfaces proving that the gibbsian probability of an interface depends only on its area and not on its regularity

non cooperative facility location and covering games we consider a general class of non cooperative games related to combinatorial covering and facility location problems a game is based on an integer programming formulation of the corresponding optimization problem and each of the k players wants to satisfy a subset of the constraints for that purpose resources available in integer units must be bought and their cost can be shared arbitrarily between players we consider the existence and cost of exact and approximate pure strategy nash equilibria in general the prices of anarchy and stability are in q k and deciding the existence of a pure nash equilibrium is np hard under certain conditions however cheap nash equilibria exist in particular if the integrality gap of the underlying integer program is 1 or in the case of single constraint players we also present algorithms that compute simultaneously near stable and near optimal approximate nash equilibria in polynomial time

independent dominating set problem revisited an independent dominating set of a graph g is a subset d of v such that every vertex not in d is adjacent to at least one vertex of d and no two vertices in d are adjacent the independent dominating set ids problem asks for an independent dominating set with minimum cardinality first we show that the independent dominating set problem and the dominating set problem on cubic bipartite graphs are both np complete as an additional result we give an alternative and more direct proof for the np completeness of both the independent dominating set problem and the dominating set problem on at most cubic grid graphs next we show that there are fixed parameter tractable algorithms for the independent dominating set problem and the dominating set problem on at most cubic graphs which run in o 3 3028 k n and o 4 2361 k n time respectively moreover we consider the weighted independent dominating set problem on k graphs we show that the problem on 2 1 graphs is np complete we also show that the problem can be solved in linear time for 1 1 graphs and in polynomial time for 1 graphs for constant respectively

second order large deviation estimates for ferromagnetic systems in the phase coexistence region we consider thed dimensional ising model with ferromagnetic nearest neighbor interaction at inverse temperature let m lambda lambda 1 sum limits i in lambda sigma i be the magnetization inside ad dimensional hyper cube be the gibbs state andm be the spontaneous magnetization for such thatm 0 we find a sufficient condition easily verified to hold for large for m a b to decay exponentially with d 1 d when m b m 1 a b ford 2 this sufficient condition is the exponential decay of a connectivity function we also prove a partial converse to this result obtain a sharper result for the magnetization ond 1 dimensional cross sections of the model and prove a similar result ford 2 m a b m and large when free boundary conditions are chosen outside

rigorous probabilistic analysis of equilibrium crystal shapes the rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade we review here the main results that have been obtained both in two and higher dimensions in particular we describe how the phenomenological wulff and winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases we focus on the main conceptual issues and describe the central ideas of the existing approaches

vorticity velocity pressure formulation for stokes problem we propose a three field formulation for efficiently solving a twodimensional stokes problem in the case of nonstandard boundary conditions more specifically we consider the case where the pressure and either normal or tangential components of the velocity are prescribed at some given parts of the boundary the proposed computational methodology consists in reformulating the considered boundary value problem via a mixed type formulation where the pressure and the vorticity are the principal unknowns while the velocity is the lagrange multiplier the obtained formulation is then discretized and a convergence analysis is performed a priori error estimates are established and some numerical results are presented to highlight the perfomance of the proposed computational methodology

planar orientations with low out degree and compaction of adjacency matrices abstract we consider the problem of orienting the edges of a planar graph in such a way that the out degree of each vertex is minimized if for each vertex v the out degree is at most d then we say that such an orientation is d bounded we prove the following results each planar graph has a 5 bounded acyclic orientation which can be constructed in linear time each planar graph has a 3 bounded orientation which can be constructed in linear time a 6 bounded acyclic orientation and a 3 bounded orientation of each planar graph can each be constructed in parallel time o log n log n on an erew pram using o n log n log n processors as an application of these results we present a data structure such that each entry in the adjacency matrix of a planar graph can be looked up in constant time the data structure uses linear storage and can be constructed in linear time

critical region for droplet formation in the two dimensional ising model we study the formation dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence specifically we consider the 2d ising model in volumes of size l2 inverse temperature c and overall magnetization conditioned to take the value m l2 2m vl where c 1 is the critical temperature m m is the spontaneous magnetization and vl is a sequence of positive numbers we find that the critical scaling for droplet formation dissolution is when vl3 2l 2 tends to a definite limit specifically we identify a dimensionless parameter proportional to this limit a non trivial critical value c and a function such that the following holds for c there is a single wulff shaped droplet containing a fraction c 2 3 of the magnetization deficit and there are no other droplets beyond the scale of log l moreover and are related via a universal equation that apparently is independent of the details of the system

stabilized finite element method for navier stokes equations with physical boundary conditions this paper deals with the numerical approximation of the 2d and 3d navier stokes equations satisfying nonstandard boundary conditions this lays on the finite element discretisation of the corresponding stokes problem which is achieved through a three fields stabilized mixed formulation a priori and a posteriori error bounds are established for the nonlinear problem ascertaining the convergence of the method finally numerical tests are presented including mesh refinement via error indicators

a finite volume version of aizenman higuchi theorem for the 2d ising model in the late 1970s in two celebrated papers aizenman and higuchi independently established that all infinite volume gibbs measures of the two dimensional ferromagnetic nearest neighbor ising model at inverse temperature beta geq 0 are of the form alpha mu beta 1 alpha mu beta where mu beta and mu beta are the two pure phases and 0 leq alpha leq 1 we present here a new approach to this result with a number of advantages a we obtain an optimal finite volume quantitative analogue implying the classical claim b the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon c this new approach might be applicable to systems for which the classical method fails

parallel recognition of series parallel graphs recently he and yesha gave an algorithm for recognizing directed series parallel graphs in time o log2 n with linearly many erew processors we give a new algorithm for this problem based on a structural characterization of series parallel graphs in terms of their ear decompositions our algorithm can recognize undirected as well as directed series parallel graphs it can be implemented in the crcw model of parallel computation to take time o log n in the erew model the time is o log2 n but the number of processors required improves the bounds of the previous algorithm

optimal parallel 5 colouring of planar graphs we show that a 5 colouring of the vertices of an n vertex planar graph may be computed in o log n log n time by an exclusive read exclusive write parallel ram with o n log n log n processors our algorithm while faster than all previously known methods is at the same time the first parallel 5 colouring algorithm to exhibit an optimal speedup optimality is achieved through a method based on the accelerating cascades technique and of independent interest it should be emphasized that although input to the algorithm is a planar graph we do not require a planar embedding to be given as part of the input other results concern the colouring of graphs of bounded genus and the construction of search structures for triangular planar subdivisions

spectral discretization of the vorticity velocity and pressure formulation of the stokes problem we consider the stokes problem in a square or a cube provided with nonstandard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity we write a variational formulation of this problem with three independent unknowns the vorticity the velocity and the pressure next we propose a discretization by spectral methods which relies on this formulation and since it leads to an inf sup condition on the pressure in a natural way we prove optimal error estimates for the three unknowns we present numerical experiments which are in perfect coherence with the analysis

wulff droplets and the metastable relaxation of kinetic ising models we consider the kinetic ising models glauber dynamics corresponding to the infinite volume ising model in dimension 2 with nearest neighbor ferromagnetic interaction and under a positive external magnetic field h minimal conditions on the flip rates are assumed so that all the common choices are being considered we study the relaxation towards equilibrium when the system is at an arbitrary subcritical temperature t and the evolution is started from a distribution which is stochastically lower than the phase we show that as h 0 the relaxation time blows up as exp c t h with lgr c t w t 2 12 t m t here m t is the spontaneous magnetization and w t is the integrated surface tension of the wulff body of unit volume moreover for 0 c the state of the process at time exp h is shown to be close when h is small to the phase the difference between this state and the phase can be described in terms of an asymptotic expansion in powers of the external field this expansion can be interpreted as describing a set of continuations in h of the family of gibbs distributions with the negative magnetic fields into the region of positive fields

a novel parameterised approximation algorithm for minimum vertex cover parameterised approximation is a relatively new but growing field of interest it merges two ways of dealing with np hard optimisation problems namely polynomial approximation and exact parameterised exponential time algorithms we exemplify this idea by designing and analysing parameterised approximation algorithms for minimum vertex cover more specifically we provide a simple algorithm that works on any approximation ratio of the form 2l 1l 1 l 1 2 and has complexity that outperforms previously published algorithms based on sophisticated exact parameterised algorithms in particular for l 1 factor 1 5 approximation our algorithm runs in time o 1 0883 k where parameter k 23 t and t is the size of a minimum vertex cover additionally we present an improved polynomial time approximation algorithm for graphs of average degree at most four and a limited number of vertices with degree less than two

graph orientation with splits abstract the minimum maximum outdegree problem mmo is to assign a direction to every edge in an input undirected edge weighted graph so that the maximum weighted outdegree taken over all vertices becomes as small as possible in this paper we introduce a new variant of mmo called the p split minimum maximum outdegree problem p split mmo in which one is allowed to perform a sequence of p split operations on the vertices before orienting the edges for some specified non negative integer p and study its computational complexity

arboricity and subgraph listing algorithms in this paper we introduce a new simple strategy into edge searching of a graph which is useful to the various subgraph listing problems applying the strategy we obtain the following four algorithms the first one lists all the triangles in a graph g in o a g m time where m is the number of edges of g and a g the arboricity of g the second finds all the quadrangles in o a g m time since a g is at most three for a planar graph g both run in linear time for a planar graph the third lists all the complete subgraphs k l of order l in o la g l 2 m time the fourth lists all the cliques in o a g m time per clique all the algorithms require linear space we also establish an upper bound on a g for a graph g a g leqq lceil 2m n 1 2 rceil where n is the number of vertices in g

storage capacity as an information theoretic vertex cover and the index coding rate motivated by applications in distributed storage the storage capacity of a graph was recently defined to be the maximum amount of information that can be stored across the vertices of a graph such that the information at any vertex can be recovered from the information stored at the neighboring vertices computing the storage capacity is a fundamental problem in network coding and is related or equivalent to some well studied problems such as index coding with side information and generalized guessing games in this paper we consider storage capacity as a natural information theoretic analogue of the minimum vertex cover of a graph indeed while it was known that storage capacity is upper bounded by minimum vertex cover we show that by treating it as such we can get a 3 2 approximation for planar graphs and a 4 3 approximation for triangle free planar graphs since the storage capacity is intimately related to the index coding rate we get a 2 approximation of index coding rate for planar graphs and 3 2 approximation for triangle free planar graphs previously only a trivial 4 approximation of the index coding rate was known for planar graphs we also show a polynomial time approximation scheme for the index coding rate when the alphabet size is constant we then develop a general method of gadget covering to upper bound the storage capacity in terms of the average of a set of vertex covers this method is intuitive and leads to the exact characterization of storage capacity for various families of graphs as an illustrative example we use this approach to derive the exact storage capacity of cycles with chords a family of graphs related to outerplanar graphs finally we generalize the storage capacity notion to include recovery from partial node failures in distributed storage we show tight upper and lower bounds on this partial recovery capacity that scales nicely with the fraction of failures in a vertex

concentration under scaling limits for weakly pinned gaussian random walks we study scaling limits for d dimensional gaussian random walks perturbed by an attractive force toward a certain subspace of mathbb r d especially under the critical situation that the rate functional of the corresponding large deviation principle admits two minimizers we obtain different type of limits in a positive recurrent regime depending on the co dimension of the subspace and the conditions imposed at the final time under the presence or absence of a wall the motivation comes from the study of polymers or 1 1 dimensional interfaces with pinning

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

the approximability of the weighted hamiltonian path completion problem on a tree given a graph the hamiltonian path completion problem is to find an augmenting edge set such that the augmented graph has a hamiltonian path in this paper we show that the hamiltonian path completion problem will unlikely have any constant ratio approximation algorithm unless np p this problem remains hard to approximate even when the given subgraph is a tree moreover if the edge weights are restricted to be either 1 or 2 the hamiltonian path completion problem on a tree is still np hard then it is observed that this problem is strongly np hard so it does not have any fully polynomial time approximation scheme fptas unless np p when the given tree is a k tree we give an approximation algorithm with performance ratio 1 5

a metric for rooted trees with unlabeled vertices based on nested parentheses in this paper we propose a new metric for rooted trees with unlabeled vertices based on alignments of nested parenthesis strings we prove that the time complexity for computing this metric is np hard and present a 1 5 approximation algorithm for it

approximation algorithms for pspace hard hierarchically and periodically specified problems we study the efficient approximability of basic graph and logic problems in the literature when instances are specified hierarchically as in t lengauer j assoc comput mach 36 1989 pp 474 509 or are specified by one dimensional finite narrow periodic specifications as in e wanke paths and cycles in finite periodic graphs in lecture notes in comp sci 711 springer verlag new york 1993 pp 751 760 we show that for most of the problems pi considered when specified using k level restricted hierarchical specifications or k narrow periodic specifications the following hold let rho be any performance guarantee of a polynomial time approximation algorithm for pi when instances are specified using standard specifications then forall epsilon 0 pi has a polynomial time approximation algorithm with performance guarantee 1 epsilon rho pi has a polynomial time approximation scheme when restricted to planar instances these are the first polynomial time approximation schemes for pspace hard hierarchically or periodically specified problems since several of the problems considered are pspace hard our results provide the first examples of natural pspace hard optimization problems that have polynomial time approximation schemes this answers an open question in condon et al chicago j theoret comput sci 1995 article 4

degree constrained decompositions of graphs bounded treewidth and planarity we study the problem of decomposing the vertex set v of a graph into two nonempty parts v1 v2 which induce subgraphs where each vertex v v1 has degree at least a v inside v1 and each v v2 has degree at least b v inside v2 we give a polynomial time algorithm for graphs with bounded treewidth which decides if a graph admits a decomposition and gives such a decomposition if it exists this result and its variants are then applied to designing polynomial time approximation schemes for planar graphs where a decomposition does not necessarily exist but the local degree conditions should be met for as many vertices as possible

approximation schemes for np hard geometric optimization problems a survey traveling salesman steiner tree and many other famous geometric optimization problems are np hard since we do not expect to design efficient algorithms that solve these problems optimally researchers have tried to design approximation algorithms which can compute a provably near optimal solution in polynomial time we survey such algorithms in particular a new technique developed over the past few years that allows us to design approximation schemes for many of these problems for any fixed constant c 0 the algorithm can compute a solution whose cost is at most 1 c times the optimum the running time is polynomial for every fixed c 0 and in many cases is even nearly linear we describe how these schemes are designed and survey the status of a large number of problems

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

planar graphs have bounded queue number we show that planar graphs have bounded queue number thus proving a conjecture of heath et al 66 from 1992 the key to the proof is a new structural tool called layered partitions and the result that every planar graph has a vertex partition and a layering such that each part has a bounded number of vertices in each layer and the quotient graph has bounded treewidth this result generalises for graphs of bounded euler genus moreover we prove that every graph in a minor closed class has such a layered partition if and only if the class excludes some apex graph building on this work and using the graph minor structure theorem we prove that every proper minor closed class of graphs has bounded queue number layered partitions have strong connections to other topics including the following two examples first they can be interpreted in terms of strong products we show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth similar statements hold for all proper minor closed classes second we give a simple proof of the result by devos et al 31 that graphs in a proper minor closed class have low treewidth colourings

parallel approximation schemes for a class of planar and near planar combinatorial optimization problems define a g almost planar graph to be a graph g v e consisting of vertex set v and a genus g layout with at most v crossover nodes we study a class of combinatorial optimization problems formulated as follows let x x1 x2 xn be a set of variables each of which has a finite domain d 0 1 poly n also let s be a fixed finite set of finite arity relations r1 rq the optimization problem max relation s is the following given a set of terms t1 t2 tm where each term ti is of the form f xi1 xi2 xir for some f s assign values to each xi 1 i n so as to maximize the number of satisfied terms we show that for each fixed finite set s and fixed g 0 there is an nc approximation scheme ncas for the problem max relation s when restricted to instances whose bipartite graphs that represent the variable term relationship are g almost planar this result in conjunction with approximation preserving reductions to max relation s enables us to obtain ncass for a number of graph theoretic and satisfiability problems when restricted to g almost planar instances our results provide a characterization of a class of problems having an ncas and hence a ptas

approximation algorithms for maximum two dimensional pattern matching we introduce the following optimization version of the classical pattern matching problem referred to as the maximum pattern matching problem given a two dimensional rectangular text and a two dimensional rectangular pattern find the maximum number of non overlapping occurrences of the pattern in the text unlike the classical two dimensional pattern matching problem the maximum two dimensional pattern matching problem is np complete we devise polynomial time approximation algorithms and approximation schemes for this problem we also briefly discuss how the approximation algorithms can be extended to include a number of other variants of the problem

completeness in standard and differential approximation classes poly d apx and d ptas completeness several problems are known to be apx dapx ptas or poly apx pb complete under suitably defined approximation preserving reductions but to our knowledge no natural problem is known to be ptas complete and no problem at all is known to be poly apx complete on the other hand dptas and poly dapx completeness have not been studied until now we first prove in this paper the existence of natural poly apx and poly dapx complete problems under the well known ptas reduction and under the dptas reduction defined in g ausiello c bazgan m demange and v th paschos completeness in differential approximation classes mfcs 03 respectively next we deal with ptas and dptas completeness we introduce approximation preserving reductions called ft and dft respectively and prove that under these new reductions natural problems are ptas complete or dptas complete then we deal with the existence of intermediate problems under our reductions and we partially answer this question showing that the existence of npo intermediate problems under turing reductions is a sufficient condition for the existence of intermediate problems under both ft and dft reductions finally we show that min coloring is dapx complete under dptas reductions this is the first dapx complete problem that is not simultaneously apx complete

polynomial time approximation schemes for geometric intersection graphs a disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane for the case that the disk representation is given we present polynomial time approximation schemes ptass for the maximum weight independent set problem selecting disjoint disks of maximum total weight and for the minimum weight vertex cover problem in disk graphs these are the first known ptass for mathcal np hard optimization problems on disk graphs they are based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously so that a dynamic programming approach becomes feasible the ptass for disk graphs represent a common generalization of previous results for planar graphs and unit disk graphs they can be extended to intersection graphs of other disk like geometric objects such as squares or regular polygons also in higher dimensions

beyond bidimensionality parameterized subexponential algorithms on directed graphs in this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs we develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs we exemplify our approaches with two well studied problems for the first problem k leaf out branching which is to find an oriented spanning tree with at least k leaves we obtain an algorithm solving the problem in time 2 o k l o g k n n o 1 on directed graphs whose underlying undirected graph excludes some fixed graph h as a minor for the special case when the input directed graph is planar the running time can be improved to 2 o k n n o 1 the second example is a generalization of the directed hamiltonian path problem namely k internal out branching which is to find an oriented spanning tree with at least k internal vertices we obtain an algorithm solving the problem in time 2 o k l o g k n o 1 on directed graphs whose underlying undirected graph excludes some fixed apex graph h as a minor finally we observe that on these classes of graphs the k directed path problem is solvable in time o email protected kn f e for any e 0 where f is some function of e our methods are based on non trivial combinations of obstruction theorems for undirected graphs kernelization problem specific combinatorial structures and a layering technique similar to the one employed by baker to obtain ptas for planar graphs

mutual exclusion scheduling abstract mutual exclusion scheduling is the problem of scheduling unit time tasks non preemptively on m processors subject to constraints represented by a graph g such that tasks represented by adjacent vertices in g must run in disjoint time intervals this problem arises in load balancing the parallel solution of partial differential equations by domain decomposition minimizing the completion time is np hard even if either the number of processors or the completion time is fixed but greater than two however polynomial time is sufficient to produce optimal schedules for forests and simple heuristics perform well on certain classes of graphs for graphs derived from the two dimensional domain decomposition problem heuristics yield solutions within 4 c 7 time units of optimal where c is the maximal number of regions that touch each other at a single point in the domain decomposition these solutions are within a constant factor of optimal

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

digraph measures kelly decompositions games and orderings we consider various well known equivalent complexity measures for graphs such as elimination orderings k trees and cops and robber games and study their natural translations to digraphs we show that on digraphs the translations of these measures are also equivalent and induce a natural connectivity measure we introduce a decomposition for digraphs and an associated width kelly width which is equivalent to the aforementioned measure we demonstrate its usefulness by exhibiting potential applications including polynomial time algorithms for np complete problems on graphs of bounded kelly width and complexity analysis of asymmetric matrix factorization finally we compare the new width to other known decompositions of digraphs

an approximation algorithm for the maximum independent set problem on planar graphs in this paper we consider the maximum independent set problem in which one would like to find a maximum set of independent i e pairwise nonadjacent vertices in a given graph the problem is np complete and still remains so even if we restrict ourselves to the class of planar graphs it has been conjectured that there exist no polynomial time exact algorithms for any np complete problems we present a polynomial time approximation algorithm for the maximum independent set problem on planar graphs for a given planar graph having any number n of vertices our algorithm finds in o n log n time an independent set that is necessarily larger in size than half a maximum independent set thus the absolute worst case ratio of our algorithm is greater than tfrac 1 2

max ones generalized to larger domains we study a family of problems called maximum solution where the objective is to maximize a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints when the domain is boolean i e restricted to 0 1 the maximum solution problem is identical to the well studied max ones problem and the approximability is completely understood for all restrictions on the underlying constraints s khanna m sudan l trevisan and d p williamson siam j comput 30 2001 pp 1863 1920 we continue this line of research by considering domains containing more than two elements we present two main results a complete classification for the approximability of all maximal constraint languages over domains of cardinality at most 4 and a complete classification of the approximability of the problem when the set of allowed constraints contains all permutation constraints under the assumption that a conjecture due to szczepara minimal clones generated by groupoids ph d thesis universite de montreal montreal qc 1996 holds we give a complete classification for all maximal constraint languages these classes of languages are well studied in universal algebra and computer science they have for instance been considered in connection with machine learning and constraint satisfaction our results are proved by using algebraic results from clone theory and the results indicate that this approach is very powerful for classifying the approximability of certain optimization problems

polynomial time approximation schemes for max bisection on planar and geometric graphs the max bisection and min bisection problems are to find a partition of the vertices of a graph into two equal size subsets that respectively maximizes or minimizes the number of edges with endpoints in both subsets we design the first polynomial time approximation scheme for the max bisection problem on arbitrary planar graphs solving a long standing open problem the method of solution involves designing exact polynomial time algorithms for computing optimal partitions of bounded treewidth graphs in particular max and min bisection which could be of independent interest using a similar method we design also the first polynomial time approximation scheme for max bisection on unit disk graphs which could also be easily extended to other geometrically defined graphs

deciding first order properties of locally tree decomposable structures we introduce the concept of a class of graphs or more generally relational structures being locally tree decomposable there are numerous examples of locally tree decomposable classes among them the class of planar graphs and all classes of bounded valence or of bounded tree width we also consider a slightly more general concept of a class of structures having bounded local tree width we show that for each property of structures that is definable in first order logic and for each locally tree decomposable class c of structures there is a linear time algorithm deciding whether a given structure a c has property for classes c of bounded local tree width we show that for every k 1 there is an algorithm solving the same problem in time o n1 1 k where n is the cardinality of the input structure

partitioning planar graphs a common problem in graph theory is that of dividing the vertices of a graph into two sets of prescribed size while cutting a minimum number of edges in this paper this problem is considered as it is restricted to the class of planar graphs let g be a planar graph on n vertices and s in 0 n be given an s partition of g is a partition of the vertex set of g into sets of size s and n s an optimals partition is an s partition that cuts the fewest number of edges the main result of this paper is an algorithm that finds an optimal s partition in time o b 2 n 3 2 4 5b where b is the number of edges cut by an optimal s partition in particular by letting s lfloor n 2 rfloor the immediate corollary that any planar graph with small o log n bisection width may be bisected in polynomial time is obtained furthermore suppose that a planar embedding hat g of g is also given such that the embedding of each biconnected component in hat g is at most m outerplanar such an embedding

on maximum independent set of categorical product and ultimate categorical ratios of graphs we first present polynomial algorithms to compute the independence number of the categorical product for two cographs or two splitgraphs respectively then we prove that computing the maximum independent set of the categorical product of a planar graph of maximum degree three and a k 4 is np complete the ultimate categorical independence ratio of a graph g is defined as lim k g k n k the ultimate categorical independence ratio can be computed in polynomial time for cographs splitgraphs permutation graphs interval graphs and graphs of bounded treewidth also we present an o 3 n 3 time exact exponential algorithm for the ultimate categorical independence ratio of general graphs we further present a ptas for the ultimate categorical independence ratio of planar graphs lastly we show that the ultimate categorical independent domination ratio for complete multipartite graphs is zero except when the graph is complete bipartite with color classes of equal size in which case it is 1 2

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

computational study on planar dominating set problem recently there has been significant theoretical progress towards fixed parameter algorithms for the dominating set problem of planar graphs it is known that the problem on a planar graph with n vertices and dominating number k can be solved in o 2 o k n time using tree branch decomposition based algorithms in this paper we report computational results of fomin and thilikos algorithm which uses the branch decomposition based approach the computational results show that the algorithm can solve the dominating set problem of large planar graphs in a practical time and memory space for the class of graphs with small branchwidth for the class of graphs with large branchwidth the size of instances that can be solved by the algorithm in practice is limited to about one thousand edges due to a memory space bottleneck the practical performances of the algorithm coincide with the theoretical analysis of the algorithm the results of this paper suggest that the branch decomposition based algorithms can be practical for some applications on planar graphs

interface surface tension and reentrant pinning transition in the 2d ising model we develop a new way to look at the high temperature representation of the ising model up to the critical temperature and obtain a number of interesting consequences in the two dimensional case it is possible to use these tools to prove results on phase separation lines in the whole phase coexistence regime by way of a duality transformation we illustrate the power of these techniques by studying an ising model with a boundary magnetic field in which a reentrant pinning transition takes place more precisely we show that the typical configurations of the model can be described at the macroscopic level by interfaces which are solutions of the corresponding thermodynamic variational problem this variational problem is solved explicitly there exist values of the boundary magnetic field and temperatures 0 t 2 but is pinned for t 1 t 3 it is not pinned an important property of the surface tension which is used in this paper is the sharp triangle inequality about which we report some new results the techniques used in this work are robust and can be used in a variety of different situations

faster approximation schemes and parameterized algorithms on odd h minor free graphs we improve the running time of the general algorithmic technique known as baker s approach 1994 1 on h minor free graphs from o n f h to o f h n o 1 the numerous applications include e g a 2 approximation for coloring and ptases for various problems such as dominating set and max cut where we obtain similar improvements on classes of odd minor free graphs which have gained significant attention in recent time we obtain a similar acceleration for a variant of the structural decomposition theorem proved by demaine et al 2010 20 we use these algorithms to derive faster 2 approximations furthermore we present the first ptases and subexponential fpt algorithms for independent set and vertex cover on these graph classes using a novel dynamic programming technique we also introduce a technique to derive nearly subexponential parameterized algorithms on h minor free graphs our technique applies in particular to problems such as steiner tree directed subgraph with a property directed longest path and connected independent dominating set on some or all proper minor closed graph classes we obtain as a corollary that all problems with a minor monotone subexponential kernel and amenable to our technique can be solved in subexponential fpt time onh minor free graphs this results in a general methodology for subexponential parameterized algorithms outside the framework of bidimensionality

satisfactory graph partition variants and generalizations the satisfactory partition problem asks for deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part this problem was introduced by gerber and kobler m gerber d kobler algorithmic approach to the satisfactory graph partitioning problem european journal of operational research 125 2000 283 291 and studied further by other authors in this paper we first review some applications and related problems then we survey structural complexity and approximation results obtained for satisfactory partition and for some of its variants and generalizations a list of open questions concludes this survey

max np completeness made easy we introduce a simple technique to obtain reductions between optimization constraint satisfaction problems with respect to an approximation preserving reducibility more powerful than the commonly used l reducibility the technique applies the probabilistic method to reduce the size of disjunctions as a first application we prove the max np completeness of max 3sat without using the pcp theorem thus solving an open question posed in khanna et al 1994 successively we show that the planar restrictions of several optimization constraint satisfaction problems admit linear time approximation schemes thus improving the results of khanna and motwani 1996

polynomial time data reduction for dominating set dealing with the np complete dominating set problem on graphs we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side in particular we prove that dominating set restricted to planar graphs has a so called problem kernel of linear size achieved by two simple and easy to implement reduction rules moreover having implemented our reduction rules first experiments indicate the impressive practical potential of these rules thus this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization

fluctuations of the phase boundary in the 2d ising ferromagnet we discuss some statistical properties of the phase boundary in the 2d low temperature ising ferromagnet in a box with the two component boundary conditions we prove the weak convergence in c 0 1 of measures describing the fluctuations of phase boundaries in the canonical ensemble of interfaces with fixed endpoints and area enclosed below them the limiting gaussian measure coincides with the conditional distribution of certain gaussian process obtained by the integral transformation of the white noise