a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

boundedness of cohomology abstract let d n and let d d denote the class of all pairs r m in which r n n 0 r n is a noetherian homogeneous ring with artinian base ring r 0 and such that m is a finitely generated graded r module of dimension d for such a pair r m let d m i n denote the finite r 0 length of the n th graded component of the i th r transform module d r i m the cohomology table of a pair r m d d is defined as the family of non negative integers d m d m i n i n n z we say that a subclass c of d d is of finite cohomology if the set d m r m c is finite a set s 0 d 1 z is said to bound cohomology if for each family h s of non negative integers the class r m d d d m i n h i n for all i n s is of finite cohomology our main result says that this is the case if and only if s contains a quasi diagonal that is a set of the form i n i i 0 d 1 with integers n 0 n 1 n d 1 we draw a number of conclusions of this boundedness criterion

design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

multiple positive solutions for semilinear elliptic systems with nonlinear boundary condition abstract in this paper we study the nonlinear boundary value problem u u f u u v in v v f v u v in u n g u u v v n g v u v on where is a bounded domain in r n with smooth boundary and the pairs of f u f v and g u g v are subcritical nonlinearities with the help of the nehari manifold we show existence of at least two positive solutions when one of the pairs of f u f v and g u g v is sublinear and the other one superlinear

on birational macaulayfications and cohen macaulay canonical modules abstract the aim of the paper is twofold at first there is a characterization of those local domains admitting a birational macaulayfication it turns out that a local domain a quotient of a gorenstein ring possesses a cohen macaulay birational extension b if and only if its canonical module k a is a cohen macaulay module in this situation it follows that b is isomorphic to the endomorphism ring of k a the second part of the paper is devoted to the question when the canonical module of a local ring is a cohen macaulay module there are applications on sequentially cohen macaulay rings a large class of rings whose canonical module are cohen macaulay is given by the simplicial affine semigroup rings as a technical tool there is a spectral sequence related to the duality with respect to the dualizing complex

the nehari manifold and application to a semilinear elliptic system abstract in this paper we study the nehari manifold and its application to the following semilinear elliptic system u u f x u q 2 u x v v g x v q 2 v x u n h x u 2 u v x v n h x u v 2 v x where is a bounded domain in r n with smooth boundary we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

cooling schedules for optimal annealing a monte carlo optimization technique called simulated annealing is a descent algorithm modified by random ascent moves in order to escape local minima which are not global minima the level of randomization is determined by a control parameter t called temperature which tends to zero according to a deterministic cooling schedule we give a simple necessary and sufficient condition on the cooling schedule for the algorithm state to converge in probability to the set of globally minimum cost states in the special case that the cooling schedule has parametric form tt c log1 t the condition for convergence is that c be greater than or equal to the depth suitably defined of the deepest local minimum which is not a global minimum state

cohomological degrees and hilbert functions of graded modules making use of the recent construction of cohomological degrees functions we give several estimates on the relationship between number of generators and degrees of ideals and modules with applications to hilbert functions they extend results heretofore known from generalized cohen macaulay local rings to nearly arbitrary local rings the rules of computation these functions satisfy enables comparison with castelnuovo mumford s regularity in the graded case as application we derive sharp improvements on predicting the outcome of effecting noether normalizations in tangent cones

on semilinear elliptic equations involving concave convex nonlinearities and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of positive solutions for semilinear elliptic equations with a sign changing weight function with the help of the nehari manifold we prove that there are at least two positive solutions for eq e f in bounded domains

random path method with pivoting for computing permanents of matrices the permanent of matrix is important in mathematics and applications its computation however is p complete randomized algorithms are natural consideration to deal with such kind of problems a monte carlo algorithm for approximating permanents of matrices is proposed in this paper which improves a method by rasmussen mathematical analysis and numerical computations show the efficiency of the method

bounds on the spectrum for markov chains and markov processes a generalization of cheeger s inequality we prove a general version of cheeger s inequality for discrete time markov chains and continuous time markovian jump processes both reversible and nonreversible with general state space we also prove a version of cheeger s inequality for markov chains and processes with killing as an application we prove l 2 exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs

a note on the finite time behavior of simulated annealing simulated annealing has proven to be a very successful heuristic for various combinatorial optimization problems it is a randomized algorithm that attempts to find the global optimum with high probability by local exchanges in this paper we give a new proof of the convergence of simulated annealing by applying results about rapidly mixing markov chains with this proof technique it is possible to obtain better bounds for the finite time behavior of simulated annealing than previously known

expander flows geometric embeddings and graph partitioning we give a o slog n approximation algorithm for the sparsest cut edge expansion balanced separator and graph conductance problems this improves the o log n approximation of leighton and rao 1988 we use a well known semidefinite relaxation with triangle inequality constraints central to our analysis is a geometric theorem about projections of point sets in rd whose proof makes essential use of a phenomenon called measure concentration we also describe an interesting and natural approximate certificate for a graph s expansion which involves embedding an n node expander in it with appropriate dilation and congestion we call this an expander flow

a diagonal bound for cohomological postulation numbers of projective schemes abstract let x be a projective scheme over a field k and let f be a coherent sheaf of o x modules we show that the cohomological postulation numbers f i of f e g the ultimate places at which the cohomological hilbert functions n dim k h i x f n h f i n start to be polynomial for n 0 are polynomially bounded in terms of the cohomology diagonal h f i i i 0 dim f of f as a consequence we obtain that there are only finitely many different cohomological hilbert functions h f i if f runs through all coherent sheaves of o x modules with fixed cohomology diagonal in order to prove these results we extend the regularity bound of bayer and mumford computational algebraic geometry and commutative algebra proc cortona 1991 cambridge univ press 1993 pp 1 48 from graded ideals to graded modules moreover we prove that the castelnuovo mumford regularity of the dual f of a coherent sheaf of o p k r modules f is polynomially bounded in terms of the cohomology diagonal of f

monomer dimer problem on random planar honeycomb lattice we consider the monomer dimer md problem on a random planar honeycomb lattice model namely the random multiple chain this is a lattice system with non periodic boundary condition whose generating process is inspired by the growth of single walled zigzag carbon nanotubes by applying algebraic and combinatorial techniques we establish a calculating expression of the md partition function for bipartite graphs which corresponds to the permanent of a matrix further by using the transfer matrix argument we show that the computing problem of the permanent of high order matrix can be converted into some lower order matrices for this family of lattices based on which we derive an explicit recurrence formula for evaluating the md partition function of multiple chains and random multiple chains finally we analyze the expectation of the number of monomer dimer arrangements on a random multiple chain and the asymptotic behavior of the annealed md entropy when the multiple chain becomes infinite in width and length respectively

the parisi formula in this chapter we prove the parisi formula which gives the limiting value of the free energy per site for the sherrington kirkpatrick model at each temperature starting with the famous result of guerra that the parisi formula provided a bound for this free energy we make full use of poisson dirichlet cascades we then use the same techniques to prove in a strong sense that in presence of an external field the overlaps are positive

liaison and castelnuovo mumford regularity in this article we establish bounds for the castelnuovo mumford regularity of projective schemes in terms of the degrees of their defining equations the main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities when applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals

analyzing glauber dynamics by comparison of markov chains a popular technique for studying random properties of a combinatorial set is to design a markov chain monte carlo algorithm for many problems there are natural markov chains connecting the set of allowable configurations which are based on local moves or glauber dynamics typically these single site update algorithms are difficult to analyze so often the markov chain is modified to update several sites simultaneously recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems in this work we use the comparison technique of diaconis and saloff coste to show that several of the natural single point update algorithms are efficient the strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed this allows us to give polynomial time bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons we also survey several other comparison techniques along with specific applications which have been used in the context of estimating mixing rates of markov chains

a semilinear elliptic system involving nonlinear boundary condition and sign changing weight function abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e involving nonlinear boundary condition and sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

castelnuovo mumford regularity of deficiency modules abstract let d n and let m be a finitely generated graded module of dimension d over a noetherian homogeneous ring r with local artinian base ring r 0 let beg m gendeg m and reg m respectively denote the beginning the generating degree and the castelnuovo mumford regularity of m if i n 0 and n z let d m i n denote the r 0 length of the n th graded component of the i th r transform module d r i m of m and let k i m denote the i th deficiency module of m our main result says that reg k i m is bounded in terms of beg m and the diagonal values d m j j with j 0 d 1 as an application of this we get a number of further bounding results for reg k i m

comparing castelnuovo mumford regularity and extended degree the borderline cases castelnuovo mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules a recent result of doering gunston and vasconcelos shows that both can be compared in the case of a graded algebra we extend this result to modules and analyze when the estimate is in fact an equality a complete classification is obtained if we choose as extended degree the homological or the smallest extended degree the corresponding algebras are characterized in three ways by relations among the algebra generators by using generic initial ideals and by their hilbert series

the nehari manifold for systems of nonlinear elliptic equations abstract this paper deals with the existence and multiplicity results of nonlocal positive solutions to the following system p u u p 1 2 u 1 u u 1 v 1 q v v q 2 v 1 u 1 v 1 v together with dirichlet or mixed boundary conditions under some hypotheses on the parameters p p 1 and q more precisely the system considered corresponds to a perturbed eigenvalue equation combined with a second equation having concave and convex nonlinearities the study is based on the extraction of palais smale sequences in the nehari manifold the behaviour of the energy corresponding to these positive solutions with respect to the real parameters and is established

glauber dynamics on trees and hyperbolic graphs we study continuous time glauber dynamics for random configurations with local constraints e g proper coloring ising and potts models on finite graphs with n vertices and of bounded degree we show that the relaxation time defined as the reciprocal of the spectral gap 1 2 for the dynamics on trees and on planar hyperbolic graphs is polynomial in n for these hyperbolic graphs this yields a general polynomial sampling algorithm for random configurations we then show that for general graphs if the relaxation time 2 satisfies 2 o 1 then the correlation coefficient and the mutual information between any local function which depends only on the configuration in a fixed window and the boundary conditions decays exponentially in the distance between the window and the boundary for the ising model on a regular tree this condition is sharp

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

mixing times of critical two dimensional potts models we study dynamical aspects of the q state potts model on an n n box at its critical c q heat bath glauber dynamics and cluster dynamics such as swendsen wang that circumvent low temperature bottlenecks are all expected to undergo critical slowdowns in the presence of periodic boundary conditions the inverse spectral gap which in the subcritical regime is o 1 should at criticality be polynomial in n for 1 4 in accordance with the predicted discontinuous phase transition this was confirmed for q 2 the ising model by the second author and sly and for sufficiently large q by borgs et al here we show that the following holds for the critical potts model on the torus for q 3 the inverse gap of glauber dynamics is no 1 for q 4 it is at most no log n and for every q 4 in the phase coexistence regime the inverse gaps of both glauber dynamics and swendsen wang dynamics are exponential in n for free or monochromatic boundary conditions and large q we show that the dynamics at criticality is faster than on the torus unlike the ising model where free periodic boundary conditions induce similar dynamical behavior at all temperatures the inverse gap of swendsen wang dynamics is exp no 1 2017 wiley periodicals inc

permanental partition models and markovian gibbs structures we study both time invariant and time varying gibbs distributions for configurations of particles into disjoint clusters specifically we introduce and give some fundamental properties for a class of partition models called permanental partition models whose distributions are expressed in terms of the permanent of a similarity matrix parameter we show that in the time invariant case the permanental partition model is a refinement of the celebrated pitman ewens distribution whereas in the time varying case the permanental model refines the ewens cut and paste markov chains j appl probab 43 3 778 791 2011 by a special property of the permanent the partition function can be computed exactly allowing us to make several precise statements about this general model including a characterization of exchangeable and consistent permanental models

an improved fully polynomial randomized approximation scheme fpras for counting the number of hamiltonian cycles in dense digraphs we propose an improved algorithm for counting the number of hamiltonian cycles in a directed graph the basic idea of the method is sequential acceptance rejection which is successfully used in approximating the number of perfect matchings in dense bipartite graphs as a consequence a new ratio of the number of hamiltonian cycles to the number of 1 factors is proposed based on this ratio we prove that our algorithm runs in expected time of o n 8 5 for dense problems this improves the markov chain monte carlo method the most powerful existing method by a factor of at least n 4 5 logn 4 in running time this class of dense problems is shown to be nontrivial in counting in the sense that they are p complete

on the permanents of circulant and degenerate schur matrices abstract we communicate three formulas for the permanents of degenerate schur and circulant matrices these combinatorial and integral formulas are intended for the analytical and asymptotic evaluation of the permanents as well as for the solution of three dimensional ising model the paper s goal is to draw attention to the open fundamental problem of finding the permanents asymptotics a solution to this problem would be tremendously important for physics of many body systems and critical phenomena as well as for quantum field theory

adaptive simulated annealing a near optimal connection between sampling and counting we present a near optimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set an important application of our work is to approximating the partition function z of a discrete system such as the ising model matchings or colorings of a graph the typical approach to estimating the partition function z at some desired inverse temperature is to define a sequence which we call a cooling schedule 0 e 0 for well studied problems such as estimating the partition function of the ising model or approximating the number of colorings or matchings of a graph our cooling schedule is of length o s n which implies an overall savings of o n in the running time of the approximate counting algorithm since roughly e samples are needed to estimate each ratio a similar improvement in the length of the cooling schedule was recently obtained by lovasz and vempala in the context of estimating the volume of convex bodies while our reduction is inspired by theirs the discrete analogue of their result turns out to be significantly more difficult whereas a fixed schedule suffices in their setting we prove that in the discrete setting we need an adaptive schedule that is the schedule depends on z more precisely we prove any nonadaptive cooling schedule has length at least o ln a and we present an algorithm to find an adaptive schedule of length o s ln a

miscellanea on quantile estimation and markov chain monte carlo convergence summary in this paper we examine the method of raftery lewis 1992 for estimating the convergence of markov chain monte carlo samplers when functionals of interest are in the form of parameter quantiles although popular the method is commonly mis applied to problems where quantiles are not of primary interest we show how the method can be misleading in this case and that it can seriously underestimate the true length of the burn in we provide a number of examples comparing the convergence rate of the chain in respect of a particular quantile with that of the true convergence rate of the original chain in particular we show how in the case of the independence sampler the two convergence rates are identical if the quantile of interest is chosen to be at an extreme of an appropriately reordered state space

the nehari manifold for a semilinear elliptic system involving sign changing weight functions abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e with sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

on the switch markov chain for perfect matchings we study a simple markov chain the switch chain on the set of all perfect matchings in a bipartite graph this markov chain was proposed by diaconis graham and holmes as a possible approach to a sampling problem arising in statistics we ask for which hereditary classes of graphs is the markov chain ergodic and for which is it rapidly mixing we provide a precise answer to the ergodicity question and close bounds on the mixing question we show for the first time that the mixing time of the switch chain is polynomial in the case of monotone graphs a class that includes examples of interest in the statistical setting

t tetrominoes tiling s markov chain mixes fast abstract korn and pak 2007 3 conjectured that there exists a fully polynomial randomized approximation scheme fpras for approximating the number of ways of tiling a 4 n 4 m rectangular lattice with t tetrominoes using a flow argument we prove this conjecture in affirmative by showing that the mixing time of an appropriate markov chain is polynomial in the area of the lattice

on the two dimensional dynamical ising model in the phase coexistence region we consider a glauber dynamics reversible with respect to the two dimensional ising model in a finite square of sidel in the absence of an external field and at large inverse temperature we first consider the gap in the spectrum of the generator of the dynamics in two different cases with plus and open boundary conditions we prove that when the symmetry under global spin flip is broken by the boundary conditions the gap is much larger than the case in which the symmetry is present for this latter we compute exactly the asymptotics of 1 l log gap asl and show that it coincides with the surface tension along one of the coordinate axes as a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin spin time correlation in the infinite volume plus phase our results establish a connection between the dynamical large deviations and those of the equilibrium gibbs measure studied by shlosman in the framework of the rigorous description of the wulff shape for the ising model finally we show that in the case of open boundary conditions it is possible to rescale the time withl in such a way that asl the finite dimensional distributions of the time rescaled magnetization converge to those of a symmetric continuous time markov chain on the two state space m m m being the spontaneous magnetization our methods rely upon a novel combination of techniques for bounding from below the gap of symmetric markov chains on complicated graphs developed by jerrum and sinclair in their markov chain approach to hard computational problems and the idea of introducing block glauber dynamics instead of the standard single site dynamics in order to put in evidence more effectively the effect of the boundary conditions in the approach to equilibrium

distributed random walks performing random walks in networks is a fundamental primitive that has found applications in many areas of computer science including distributed computing in this article we focus on the problem of sampling random walks efficiently in a distributed network and its applications given bandwidth constraints the goal is to minimize the number of rounds required to obtain random walk samples all previous algorithms that compute a random walk sample of length e as a subroutine always do so naively that is in o e rounds the main contribution of this article is a fast distributed algorithm for performing random walks we present a sublinear time distributed algorithm for performing random walks whose time complexity is sublinear in the length of the walk our algorithm performs a random walk of length e in o ed rounds o hides polylog n factors where n is the number of nodes in the network with high probability on an undirected network where d is the diameter of the network for small diameter graphs this is a significant improvement over the naive o e bound furthermore our algorithm is optimal within a poly logarithmic factor as there exists a matching lower bound nanongkai et al 2011 we further extend our algorithms to efficiently perform k independent random walks in o ked k rounds we also show that our algorithm can be applied to speedup the more general metropolis hastings sampling our random walk algorithms can be used to speed up distributed algorithms in applications that use random walks as a subroutine we present two main applications first we give a fast distributed algorithm for computing a random spanning tree rst in an arbitrary undirected unweighted network which runs in o md rounds with high probability m is the number of edges our second application is a fast decentralized algorithm for estimating mixing time and related parameters of the underlying network our algorithm is fully decentralized and can serve as a building block in the design of topologically aware networks

inapproximability for antiferromagnetic spin systems in the tree nonuniqueness region a remarkable connection has been established for antiferromagnetic 2 spin systems including the ising and hard core models showing that the computational complexity of approximating the partition function for graphs with maximum degree undergoes a phase transition that coincides with the statistical physics uniqueness nonuniqueness phase transition on the infinite regular tree despite this clear picture for 2 spin systems there is little known for multispin systems we present the first analog of this in approximability results for multispin systems the main difficulty in previous inapproximability results was analyzing the behavior of the model on random regular bipartite graphs which served as the gadget in the reduction to this end one needs to understand the moments of the partition function our key contribution is connecting i induced matrix norms ii maxima of the expectation of the partition function and iii attractive fixed points of the associated tree recursions belief propagation the view through matrix norms allows a simple and generic analysis of the second moment for any spin system on random regular bipartite graphs this yields concentration results for any spin system in which one can analyze the maxima of the first moment the connection to fixed points of the tree recursions enables an analysis of the maxima of the first moment for specific models of interest for k colorings we prove that for even k in a tree nonuniqueness region which corresponds to k

reconstruction of random colourings reconstruction problems have been studied in a number of contexts including biology information theory and statistical physics we consider the reconstruction problem for random k colourings on the ary tree for large k bhatnagar et al 2 showed non reconstruction when delta leq frac12 k log k o k log k we tighten this result and show non reconstruction when delta leq k log k log log k 1 log 2 o 1 which is very close to the best known bound establishing reconstruction which is delta geq k log k log log k 1 o 1

castelnuovo mumford regularity of canonical and deficiency modules abstract we give two kinds of bounds for the castelnuovo mumford regularity of the canonical module and the deficiency modules of a ring respectively in terms of the homological degree and the castelnuovo mumford regularity of the original ring

computational complexity of loss networks abstract in this paper we examine the theoretical limits on developing algorithms to find blocking probabilities in a general loss network we demonstrate that exactly computing the blocking probabilities of a loss network is a p complete problem we also show that a general algorithm for approximating the blocking probabilities is also intractable unless rp np which seems unlikely according to current common notions in complexity theory given these results we examine implications for designing practical algorithms for finding blocking probabilities in special cases

gibbs rapidly samples colorings of g n d n gibbs sampling also known as glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs of special interest is the behavior of gibbs sampling on the erd s renyi random graph g n d n where each edge is chosen independently with probability d n and d is fixed while the average degree in g n d n is d 1 o 1 it contains many nodes of degree of order log n log log n the existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on g n d n such as uniform coloring with a constant number of colors or the ising model at any fixed inverse temperature the mixing time of gibbs sampling is at least n 1 1 log log n with high probability high degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including coloring almost all known sufficient conditions in terms of number of colors needed for rapid mixing of gibbs samplers are stated in terms of the maximum degree of the underlying graph in this work we consider sampling q colorings and show that for every d 1 where the number of colors does not depend on n they also provide a rare example where one can prove a polynomial time mixing of gibbs sampler in a situation where the actual mixing time is slower than npolylog n in previous work we have shown that similar results hold for the ferromagnetic ising model however the proof for the ising model crucially relied on monotonicity arguments and the weitz tree both of which have no counterparts in the coloring setting our proof presented here exploits in novel ways the local treelike structure of erd s renyi random graphs block dynamics spatial decay properties and coupling arguments our results give the first polynomial time algorithm to approximately sample colorings on g n d n with a constant number of colors they extend to much more general families of graphs which are sparse in some average sense and to much more general interactions in particular they apply to any graph for which there exists an 0 such that every vertex v of the graph has a neighborhood n v of radius o log n in which the induced sub graph is the union of a tree and at most o 1 edges and where each simple path of length o log n satisfies sum u in gamma sum v neq u alpha d u v o rm log n the results also generalize to the hard core model at low fugacity and to general models of soft constraints at high temperatures

on approximately counting colorings of small degree graphs we consider approximate counting of colorings of an n vertex graph using rapidly mixing markov chains it has been shown by jerrum and by salas and sokal that a simple random walk on graph colorings would mix rapidly provided the number of colors k exceeded the maximum degree delta of the graph by a factor of at least 2 we prove that this is not a necessary condition for rapid mixing by considering the simplest case of 5 coloring graphs of maximum degree 3 our proof involves a computer assisted proof technique to establish rapid mixing of a new heat bath markov chain on colorings using the method of path coupling we outline an extension to 7 colorings of triangle free 4 regular graphs since rapid mixing implies approximate counting in polynomial time we show in contrast that exact counting is unlikely to be possible in polynomial time we give a general proof that the problem of exactly counting the number of proper k colorings of graphs with maximum degree delta is p complete whenever k geq 3 and delta geq 3

decentralized dynamics for finite opinion games game theory studies situations in which strategic players can modify the state of a given system in the absence of a central authority solution concepts such as nash equilibrium have been defined in order to predict the outcome of such situations in multi player settings it has been pointed out that to be realistic a solution concept should be obtainable via processes that are decentralized and reasonably simple accordingly we look at the computation of solution concepts by means of decentralized dynamics these are algorithms in which players move in turns to decrease their own cost and the hope is that the system reaches an equilibrium quickly we study these dynamics for the class of opinion games recently introduced by bindel et al 10 these are games important in economics and sociology that model the formation of an opinion in a social network we study best response dynamics and show upper and lower bounds on the convergence to nash equilibria we also study a noisy version of best response dynamics called logit dynamics and prove a host of results about its convergence rate as the noise in the system varies to get these results we use a variety of techniques developed to bound the mixing time of markov chains including coupling spectral characterizations and bottleneck ratio

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

matrix permanent and quantum entanglement of permutation invariant states we point out that a geometric measure of quantum entanglement is related to the matrix permanent when restricted to permutation invariant states this connection allows us to interpret the permanent as an angle between vectors by employing a recently introduced permanent inequality by carlen et al methods appl anal 13 1 2006 we can prove explicit formulas of the geometric measure for permutation invariant basis states in a simple way

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

convergence of an annealing algorithm the annealing algorithm is a stochastic optimization method which has attracted attention because of its success with certain difficult problems including np hard combinatorial problems such as the travelling salesman steiner trees and others there is an appealing physical analogy for its operation but a more formal model seems desirable in this paper we present such a model and prove that the algorithm converges with probability arbitrarily close to 1 we also show that there are cases where convergence takes exponentially long that is it is no better than a deterministic method we study how the convergence rate is affected by the form of the problem finally we describe a version of the algorithm that terminates in polynomial time and allows a good deal of practical confidence in the solution

counting in two spin models on d regular graphs supported in part by alfred p sloan research fellowship supported in part by department of defense ndseg fellowships

approximation algorithms for two state anti ferromagnetic spin systems on bounded degree graphs we show that for the anti ferromagnetic ising model on the bethe lattice weak spatial mixing implies strong spatial mixing as a by product of our analysis we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti ferromagnetic ising model with non zero external field on the bethe lattice following a method due to weitz 15 we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti ferromagnetic ising model with arbitrary field on graphs of degree at most d throughout the uniqueness region of the gibbs measure on the infinite d regular tree by a standard correspondence our results translate to arbitrary two state anti ferromagnetic spin systems with soft constraints subsequent to a preliminary version of this paper sly and sun 13 have shown that our results are optimal in the sense that under standard complexity theoretic assumptions there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree d for parameters outside the uniqueness region taken together the results of 13 and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two state anti ferromagnetic spin systems

on the trace of random walks on random graphs we study graph theoretic properties of the trace of a random walk on a random graph we show that for any e 0 there exists c 1 such that the trace of the simple random walk of length 1 e n lnn on the random graph g g n p for p c lnn n is with high probability hamiltonian and lnn connected in the special case p 1 i e when g kn we show a hitting time result according to which with high probability exactly one step after the last vertex has been visited the trace becomes hamiltonian and one step after the last vertex has been visited for the k th time the trace becomes 2k connected

quantum speedup of monte carlo methods monte carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically one important example is the use in statistical physics of rapidly mixing markov chains to approximately compute partition functions in this work we describe a quantum algorithm which can accelerate monte carlo methods in a very general setting the algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance achieving a near quadratic speedup over the best possible classical algorithm combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions which use multiple stage markov chain monte carlo techniques the quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently

polynomial time approximation algorithms for the ising model the paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic ising system to any specified degree of accuracy the running time of the algorithm in

exact thresholds for ising gibbs samplers on general graphs we establish tight results for rapid mixing of gibbs samplers for the ferromagnetic ising model on general graphs we show that if d 1 tanh 1 then there exists a constant c such that the discrete time mixing time of gibbs samplers for the ferromagnetic ising model on any graph of n vertices and maximal degree d where all interactions are bounded by and arbitrary external fields are bounded by cnlogn moreover the spectral gap is uniformly bounded away from 0 for all such graphs as well as for infinite graphs of maximal degree d we further show that when dtanh 1 with high probability over the erd s renyi random graph g n d n it holds that the mixing time of gibbs samplers is n1 1 loglogn both results are tight as it is known that the mixing time for random regular and erd s renyi random graphs is with high probability exponential in n when d 1 tanh 1 and dtanh 1 respectively to our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs moreover our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees

castelnuovo mumford regularity of ext modules and homological degree bounds for the castelnuovo mumford regularity of ext modules over a polynomial ring over a field are given in terms of the initial degrees castelnuovo mumford regularities and the number of generators of the two graded modules involved these general bounds are refined in the case where the second module is the ring other estimates for instance on the size of graded pieces of these modules are given we also derive a bound on the homological degree in terms of the castelnuovo mumford regularity this answers positively a question raised by vasconcelos

the nehari manifold for a semilinear elliptic equation with a sign changing weight function abstract the nehari manifold for the equation u x a x u x b x u x 1u x for x together with dirichlet boundary conditions is investigated exploiting the relationship between the nehari manifold and fibrering maps i e maps of the form t j tu where j is the euler functional associated with the equation we discuss how the nehari manifold changes as changes and show how existence and non existence results for positive solutions of the equation are linked to properties of the manifold

two dimensional monomer dimer systems are computationally intractable the classic problem of counting monomer dimer arrangements on a two dimensional lattice is analyzed using techniques from theoretical computer science under a certain assumption made precise in the text it can be shown that the general problem is computationally intractable this negative result contrasts with the special case of a system with monomer density zero for which efficient solutions have been known for some time a second much easier result obtained under the same assumption is that the partition function of a three dimensional ising system is computationally intractable again the negative result contrasts with known efficient techniques for evaluating the partition function of a two dimensional system

random walks totally unimodular matrices and a randomised dual simplex algorithm we discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix we also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron

a chernoff bound for random walks on expander graphs we consider a finite random walk on a weighted graph g we show that the fraction of time spent in a set of vertices a converges to the stationary probability pi a with error probability exponentially small in the length of the random walk and the square of the size of the deviation from pi a the exponential bound is in terms of the expansion of g and improves previous results of d aldous probab engrg inform sci 1 1987 pp 33 46 l lovasz and m simonovits it random structures algorithms 4 1993 pp 359 412 m ajtai j komlos and e szemeredi deterministic simulation of logspace in proc 19th acm symp on theory of computing 1987 we show that taking the sample average from one trajectory gives a more efficient estimate of pi a than the standard method of generating independent sample points from several trajectories using this more efficient sampling method we improve the algorithms of jerrum and sinclair for approximating the number of perfect matchings in a dense graph and for approximating the partition function of a ferromagnetic ising system and we give an efficient algorithm to estimate the entropy of a random walk on an unweighted graph

the homological degree of a module a homological degree of a graded module m is an extension of the usual notion of multiplicity tailored to provide a numerical signature for the module even when m is not cohen macaulay we construct a degree hdeg m that behaves well under hyperplane sections and the modding out of elements of finite support when carried out in a local algebra this degree gives a simulacrum of complexity a la castelnuovo mumford s regularity several applications for estimating reduction numbers of ideals and predictions on the outcome of noether normalizations are given

discrete hit and run for sampling points from arbitrary distributions over subsets of integer hyperrectangles we consider the problem of sampling a point from an arbitrary distribution over an arbitrary subset s of an integer hyperrectangle neither the distribution nor the support set s are assumed to be available as explicit mathematical equations but may only be defined through oracles and in particular computer programs this problem commonly occurs in black box discrete optimization as well as counting and estimation problems the generality of this setting and high dimensionality of s precludes the application of conventional random variable generation methods as a result we turn to markov chain monte carlo mcmc sampling where we execute an ergodic markov chain that converges to so that the distribution of the point delivered after sufficiently many steps can be made arbitrarily close to unfortunately classical markov chains such as the nearest neighbor random walk or the coordinate direction random walk fail to converge to because they can get trapped in isolated regions of the support set to surmount this difficulty we propose discrete hit and run dhr a markov chain motivated by the hit and run algorithm known to be the most efficient method for sampling from log concave distributions over convex bodies in rn we prove that the limiting distribution of dhr is as desired thus enabling us to sample approximately from by delivering the last iterate of a sufficiently large number of iterations of dhr in addition to this asymptotic analysis we investigate finite time behavior of dhr and present a variety of examples where dhr exhibits polynomial performance

eigenvalues of random walks on groups in this paper we discuss and apply a novel method for bounding the eigenvalues of a random walk on a group g or equivalently on its cayley graph this method works by looking at the action of an abelian normal subgroup h of g on g we may then choose eigenvectors which fall into representations of h one is then left with a large number one for each representation of h of easier problems to analyze this analysis is carried out by new geometric methods this method allows us to give bounds on the second largest eigenvalue of random walks on nilpotent groups with low class number the method also lets us treat certain very easy solvable groups and to give better bounds for certain nice nilpotent groups with large class number for example we will give sharp bounds for two natural random walks on groups of upper triangular matrices

the nehari manifold for a class of concave convex elliptic systems involving the p laplacian and nonlinear boundary condition abstract the existence and multiplicity of weak solutions is established for a class of concave convex elliptic systems of the form p u m x u p 2 u a x u 2 u x p v m x v p 2 v b x v 2 v x u p 2 u n u 2 u v v p 2 v n u v 2 v x here p denotes the p laplacian operator defined by p z div z p 2 z p 2 r n is a bounded domain with smooth boundary 1 1 2 p p p p n n p if n p p if n p n is the outer normal derivative r 2 0 0 the weight m x is a positive bounded function and a x b x c are functions which change sign in our technical approach is based on the nehari manifold which is similar to the fibering method of drabek and pohozaev 1997 29 together with the recent idea from brown and wu 2008 10

finiteness of hilbert functions and bounds for castelnuovo mumford regularity of initial ideals bounds for the castelnuovo mumford regularity and hilbert coefficients are given in terms of the arithmetic degree if the ring is reduced or in terms of the defining degrees from this it follows that there exists only a finite number of hilbert functions associated with reduced algebras over an algebraically closed field with a given arithmetic degree and dimension a good bound is also given for the castelnuovo mumford regularity of initial ideals which depends neither on term orders nor on the coordinates and holds for any field

counting hypergraph matchings up to uniqueness threshold abstract we study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges with an activity parameter each matching m is assigned a weight m the counting problem is formulated as computing a partition function that gives the sum of the weights of all matchings in a hypergraph this problem unifies two extensively studied statistical physics models in approximate counting the hardcore model graph independent sets and the monomer dimer model graph matchings for this problem the critical activity c d d k d 1 d 1 is the threshold for the uniqueness of gibbs measures on the infinite d 1 uniform k 1 regular hypertree consider hypergraphs of maximum degree at most k 1 and maximum size of hyperedges at most d 1 we show that when c there is an fptas for computing the partition function and when c there is a ptas for computing the log partition function these algorithms are based on the decay of correlation strong spatial mixing property of gibbs distributions when 2 c there is no pras for the partition function or the log partition function unless np rp towards obtaining a sharp transition of computational complexity of approximate counting we study the local convergence from a sequence of finite hypergraphs to the infinite lattice with specified symmetry we show a surprising connection between the local convergence and the reversibility of a natural random walk this leads us to a barrier for the hardness result the non uniqueness of infinite gibbs measure is not realizable by any finite gadgets

censored glauber dynamics for the mean field ising model we study glauber dynamics for the ising model on the complete graph on n vertices known as the curie weiss model it is well known that at high temperature 1 it is exp n recently levin luczak and peres considered a censored version of this dynamics which is restricted to non negative magnetization they proved that for fixed 1 the mixing time of this model is nlog n analogous to the high temperature regime of the original dynamics furthermore they showed cutoff for the original dynamics for fixed 1 the question whether the censored dynamics also exhibits cutoff remained unsettled

self testing algorithms for self avoiding walks we present a markov chain monte carlo algorithm for almost uniformly generating and approximately counting self avoiding walks in rectangular lattices zd these are classical problems that arise for example in the study of long polymer chains while there are a number of monte carlo algorithms used to solve these problems in practice these are heuristic and their correctness relies on unproven conjectures in contrast our algorithm is shown rigorously to produce answers with specified accuracy and confidence only the efficiency of the algorithm relies on a widely believed conjecture and a novel feature is that this conjecture can be tested as the algorithm proceeds with this self testing feature incorporated the algorithm has polynomially bounded running time and is completely reliable in the sense that it either outputs answers that are guaranteed to be within the specified accuracy and confidence bounds or finds a counter example to the conjecture

the complexity of approximately counting in 2 spin systems on k uniform bounded degree hypergraphs one of the most important recent developments in the complexity of approximate counting is the classification of the complexity of approximating the partition functions of antiferromagnetic 2 spin systems on bounded degree graphs this classification is based on a beautiful connection to the so called uniqueness phase transition from statistical physics on the infinite regular tree our objective is to study the impact of this classification on unweighted 2 spin models on k uniform hypergraphs as has already been indicated by yin and zhao the connection between the uniqueness phase transition and the complexity of approximate counting breaks down in the hypergraph setting nevertheless we show that for every non trivial symmetric k ary boolean function f there exists a degree bound 0 so that for all 0 the following problem is np hard given a k uniform hypergraph with maximum degree at most approximate the partition function of the hypergraph 2 spin model associated with f it is np hard to approximate this partition function even within an exponential factor by contrast if f is a trivial symmetric boolean function e g any function f that is excluded from our result then the partition function of the corresponding hypergraph 2 spin model can be computed exactly in polynomial time

fast uniform generation of regular graphs abstract an algorithm is presented which randomly selects a labelled graph with specified vertex degrees from a distribution which is arbitrarily close to uniform the algorithm is based on simulation of a rapidly convergent stochastic process and runs in polynomial time for a wide class of degree sequences including all regular sequences and all n vertex sequences with no degree exceeding n 2 the algorithm can be extended to cover the selection of a graph with given degree sequence which avoids a specified set of edges one consequence of this extension is the existence of a polynomial time algorithm for selecting an f factor in a sufficiently dense graph a companion algorithm for counting degree constrained graphs is also presented this algorithm has exactly the same range of validity as the one for selection

on the complexity of computing mixed volumes this paper gives various positive and negative results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension on the negative side we present several p hardness results that focus on the difference of computing mixed volumes versus computing the volume of polytopes we show that computing the volume of zonotopes is p hard while each corresponding mixed volume can be computed easily but also give examples showing that computing mixed volumes is hard even when computing the volume is easy on the positive side we derive a randomized algorithm for computing the mixed volumes v overbrace k 1 ld k 1 m 1 overbrace k 2 dots k 2 m 2 dots overbrace k s dots k s m s of well presented convex bodies k 1 dots k s where m 1 dots m s in n 0 and m 1 geq n psi n with psi n o frac log n log log n the algorithm is an interpolation method based on polynomial time randomized algorithms for computing the volume of convex bodies this paper concludes with applications of our results to various problems in discrete mathematics combinatorics computational convexity algebraic geometry geometry of numbers and operations research

the time complexity of maximum matching by simulated annealing the random heuristic search algorithm called simulated annealing is considered for the problem of finding the maximum cardinality matching in a graph it is shown that neither a basic form of the algorithm nor any other algorithm in a fairly large related class of algorithms can find maximum cardinality matchings such that the average time required grows as a polynomial in the number of nodes of the graph in contrast it is also shown for arbitrary graphs that a degenerate form of the basic annealing algorithm obtained by letting temperature be a suitably chosen constant produces matchings with nearly maximum cardinality in polynomial average time

approximating the permanent via importance sampling with application to the dimer covering problem we estimate the asymptotic growth rate of the number of dimer covers of a cubic lattice our estimate 3 0 4466 0 0006 is consistent with the lower bounds obtained by hammersley and later schrijver and the more recent improved upper bound obtained by ciucu obtaining this estimate is an important step toward approximating the partition function of the cubic monomer dimer system from the partition function all of the standard thermodynamic quantities can be evaluated it is well known that computing 3 is equivalent to computing the permanent of a certain 0 1 matrix we describe an extremely efficient monte carlo algorithm for approximating the permanent previous work on monte carlo approaches includes the pioneering results of jerrum and sinclair who use a rapidly mixing random walk our method is inspired by results of soules on convergence of sinkhorn balancing to obtain a maximum entropy doubly stochastic matrix we use the sinkhorn balanced matrix to generate an importance function that allows us to do direct random sampling rather than a random walk that converges to a limiting distribution

factor models on locally tree like graphs we consider homogeneous factor models on uniformly sparse graph sequences converging locally to a unimodular random tree t and study the existence of the free energy density the limit of the log partition function divided by the number of vertices n as n tends to infinity we provide a new interpolation scheme and use it to prove existence of and to explicitly compute the quantity subject to uniqueness of a relevant gibbs measure for the factor model on t by way of example we compute for the independent set or hard core model at low fugacity for the ferromagnetic ising model at all parameter values and for the ferromagnetic potts model with both weak enough and strong enough interactions even beyond uniqueness regimes our interpolation provides useful explicit bounds on in the regimes in which we establish existence of the limit we show that it coincides with the bethe free energy functional evaluated at a suitable fixed point of the belief propagation bethe recursions on t in the special case that t has a galton watson law this formula coincides with the nonrigorous bethe prediction obtained by statistical physicists using the replica or cavity methods thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees we also provide a variational characterization for the bethe prediction in this general setting which is of independent interest

on the hardness of sampling independent sets beyond the tree threshold we consider local markov chain monte carlo algorithms for sampling from the weighted distribution of independent sets with activity where the weight of an independent set i is i a recent result has established that gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree d and c it is np hard to approximate the above weighted sum over independent sets to within a factor polynomial in the size of the graph

tilings of rectangles with t tetrominoes we prove that any two tilings of a rectangular region by t tetrominoes are connected by moves involving only 2 and 4 tiles we also show that the number of such tilings is an evaluation of the tutte polynomial the results are extended to a more general class of regions

multiple solutions for a class of biharmonic elliptic systems with sobolev critical exponent abstract in this paper we study the existence and multiplicity of nontrivial solutions for a class of biharmonic elliptic systems with sobolev critical exponent in a bounded domain by using the variational method and the nehari manifold we obtain the existence and multiplicity results of nontrivial solutions for the systems

dynamics of 2 1 dimensional sos surfaces above a wall slow mixing induced by entropic repulsion we study the glauber dynamics for the 2 1 d solid on solid model above a hard wall and below a far away ceiling on an l times l box of z 2 with zero boundary conditions at large inverse temperature beta it was shown by bricmont el mellouki and frohlich 1986 that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height h asymp 1 beta log l as an essential step in understanding the effect of entropic repulsion on the glauber dynamics we determine the equilibrium height h to within an additive constant h 1 4 beta log l o 1 we then show that starting from zero initial conditions the surface rises to its final height h through a sequence of metastable transitions between consecutive levels the time for a transition from height h ah a in 0 1 to height h 1 is roughly exp c l a for some constant c 0 in particular the mixing time of the dynamics is exponentially large in l i e t mix geq e c l we also provide the matching upper bound t mix leq e c l requiring a challenging analysis of the statistics of height contours at low temperature and new coupling ideas and techniques finally to emphasize the role of entropic repulsion we show that without a floor constraint at height zero the mixing time is no longer exponentially large in l