arXiv subjects

Mark Jerrum

Publications and source records attributed to Mark Jerrum.

54 records · Page 3Linked to original sources

arXiv2012

Approximating the partition function of the ferromagnetic Potts model

We provide evidence that it is computationally difficult to approximate the partition function of the ferromagnetic q-state Potts model when q>2. Specifically we show that the partition function is hard for the complexity class #RHPi_1 under approximation-preserving reducibility. Thus, it is as hard to approximate the partition function as it is to find approximate solutions to a wide range of counting problems, including that of determining the number of independent sets in a bipartite graph. Our proof exploits the first order phase transition of the "random cluster" model, which is a probability distribution on graphs that is closely related to the q-state Potts model.

cs.CC↗

arXiv2012

The expressibility of functions on the Boolean domain, with applications to Counting CSPs

An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post's lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs. Motivated by a desire to understand the computational complexity of (weighted) counting CSPs, we develop an analogous notion of functional clones and study the landscape of these clones. One of these clones is the collection of log-supermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In the conservative case (where all nonnegative unary functions are available), we show that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial non-lsm function is computationally as hard to approximate as any problem in #P. Furthermore, we show that any non-trivial functional clone (in a sense that will be made precise) contains the binary function "implies". As a consequence, in the conservative case, all non-trivial counting CSPs are as hard as #BIS, the problem of counting independent sets in a bipartite graph. Given the complexity-theoretic results, it is natural to ask whether the "implies" clone is equivalent to the clone of lsm functions. We use the Mobius transform and the Fourier transform to show that these clones coincide precisely up to arity 3. It is an intriguing open question whether the lsm clone is finitely generated. Finally, we investigate functional clones in which only restricted classes of unary functions are available.

cs.CC↗

arXiv2012

Approximating the Tutte polynomial of a binary matroid and other related combinatorial polynomials

We consider the problem of approximating certain combinatorial polynomials. First, we consider the problem of approximating the Tutte polynomial of a binary matroid with parameters q>= 2 and gamma. (Relative to the classical (x,y) parameterisation, q=(x-1)(y-1) and gamma=y-1.) A graph is a special case of a binary matroid, so earlier work by the authors shows inapproximability (subject to certain complexity assumptions) for q>2, apart from the trivial case gamma=0. The situation for q=2 is different. Previous results for graphs imply inapproximability in the region -2<=gamma<0, apart from at two "special points" where the polynomial can be computed exactly in polynomial time. For binary matroids, we extend this result by showing (i) there is no FPRAS in the region gamma<-2 unless NP=RP, and (ii) in the region gamma>0, the approximation problem is hard for the complexity class #RHPi_1 under approximation-preserving (AP) reducibility. The latter result indicates a gap in approximation complexity at q=2: whereas an FPRAS is known in the graphical case, there can be none in the binary matroid case, unless there is an FPRAS for all of #RHPi_1. The result also implies that it is computationally difficult to approximate the weight enumerator of a binary linear code, apart from at the special weights at which the problem is exactly solvable in polynomial time. As a consequence, we show that approximating the cycle index polynomial of a permutation group is hard for #RHPi_1 under AP-reducibility, partially resolving a question that we first posed in 1992.

cs.CC↗

arXiv2011

The complexity of weighted and unweighted #CSP

We give some reductions among problems in (nonnegative) weighted #CSP which restrict the class of functions that needs to be considered in computational complexity studies. Our reductions can be applied to both exact and approximate computation. In particular, we show that a recent dichotomy for unweighted #CSP can be extended to rational-weighted #CSP.

cs.CC↗

arXiv2011

A Counterexample to rapid mixing of the Ge-Stefankovic Process

Ge and Stefankovic have recently introduced a novel two-variable graph polynomial. When specialised to a bipartite graphs G and evaluated at the point (1/2,1) this polynomial gives the number of independent sets in the graph. Inspired by this polynomial, they also introduced a Markov chain which, if rapidly mixing, would provide an efficient sampling procedure for independent sets in G. This sampling procedure in turn would imply the existence of efficient approximation algorithms for a number of significant counting problems whose complexity is so far unresolved. The proposed Markov chain is promising, in the sense that it overcomes the most obvious barrier to mixing. However, we show here, by exhibiting a sequence of counterexamples, that the mixing time of their Markov chain is exponential in the size of the input when the input is chosen from a particular infinite family of bipartite graphs.

math.PR↗

arXiv2011

Inapproximability of the Tutte polynomial of a planar graph

The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: given as input a planar graph G, determine T(G;x,y). Vertigan completely mapped the complexity of exactly computing the Tutte polynomial of a planar graph. He showed that the problem can be solved in polynomial time if (x,y) is on the hyperbola H_q given by (x-1)(y-1)=q for q=1 or q=2 or if (x,y) is one of the two special points (x,y)=(-1,-1) or (x,y)=(1,1). Otherwise, the problem is #P-hard. In this paper, we consider the problem of approximating T(G;x,y), in the usual sense of "fully polynomial randomised approximation scheme" or FPRAS. Roughly speaking, an FPRAS is required to produce, in polynomial time and with high probability, an answer that has small relative error. Assuming that NP is different from RP, we show that there is no FPRAS for the Tutte polynomial in a large portion of the (x,y) plane. In particular, there is no FPRAS if x>1, y<-1 or if y>1, x<-1 or if x<0, y<0 and q>5. Also, there is no FPRAS if x<1, y<1 and q=3. For q>5, our result is intriguing because it shows that there is no FPRAS at (x,y)=(1-q/(1+epsilon),-epsilon) for any positive epsilon but it leaves open the limit point epsilon=0, which corresponds to approximately counting q-colourings of a planar graph.

cs.CC↗

arXiv2010

A complexity dichotomy for hypergraph partition functions

We consider the complexity of counting homomorphisms from an $r$-uniform hypergraph $G$ to a symmetric $r$-ary relation $H$. We give a dichotomy theorem for $r>2$, showing for which $H$ this problem is in FP and for which $H$ it is #P-complete. This generalises a theorem of Dyer and Greenhill (2000) for the case $r=2$, which corresponds to counting graph homomorphisms. Our dichotomy theorem extends to the case in which the relation $H$ is weighted, and the goal is to compute the \emph{partition function}, which is the sum of weights of the homomorphisms. This problem is motivated by statistical physics, where it arises as computing the partition function for particle models in which certain combinations of $r$ sites interact symmetrically. In the weighted case, our dichotomy theorem generalises a result of Bulatov and Grohe (2005) for graphs, where $r=2$. When $r=2$, the polynomial time cases of the dichotomy correspond simply to rank-1 weights. Surprisingly, for all $r>2$ the polynomial time cases of the dichotomy have rather more structure. It turns out that the weights must be superimposed on a combinatorial structure defined by solutions of an equation over an Abelian group. Our result also gives a dichotomy for a closely related constraint satisfaction problem.

cs.CC↗

arXiv2009

An approximation trichotomy for Boolean #CSP

We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language specifying the relations that may be used in constraints. If every relation in the constraint language is affine then the number of satisfying assignments can be exactly counted in polynomial time. Otherwise, if every relation in the constraint language is in the co-clone IM_2 from Post's lattice, then the problem of counting satisfying assignments is complete with respect to approximation-preserving reductions in the complexity class #RHΠ_1. This means that the problem of approximately counting satisfying assignments of such a CSP instance is equivalent in complexity to several other known counting problems, including the problem of approximately counting the number of independent sets in a bipartite graph. For every other fixed constraint language, the problem is complete for #P with respect to approximation-preserving reductions, meaning that there is no fully polynomial randomised approximation scheme for counting satisfying assignments unless NP=RP.

cs.CC↗

arXiv2009

A complexity dichotomy for partition functions with mixed signs

Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of k-colourings or the number of independent sets of a graph and also the partition functions of certain "spin glass" models of statistical physics such as the Ising model. Building on earlier work by Dyer, Greenhill and Bulatov, Grohe, we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or #P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or #P-complete. While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by a Hadamard matrices -- these turn out to be central in our proofs -- we obtain a simple algebraic tractability criterion, which says that the tractable cases are those "representable" by a quadratic polynomial over the field GF(2).

cs.CC↗

arXiv2009

Matrix norms and rapid mixing for spin systems

We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is $\mathbf{0}$ (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for coloring various classes of graphs. We give a general method for estimating a norm of a symmetric nonregular matrix. This leads to improved mixing times for any class of graphs which is hereditary and sufficiently sparse including several classes of degree-bounded graphs such as nonregular graphs, trees, planar graphs and graphs with given tree-width and genus.

math.PR↗

arXiv2008

The Complexity of Weighted Boolean #CSP

This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterised by a finite set F of non-negative functions that may be used to assign weights to the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems correspond to the special case of 0,1-valued functions. We show that the partition function, i.e. the sum of the weights of all configurations, can be computed in polynomial time if either (1) every function in F is of ``product type'', or (2) every function in F is ``pure affine''. For every other fixed set F, computing the partition function is FP^{#P}-complete.

cs.CC↗

arXiv2008

The Mixing Time of Glauber Dynamics for Colouring Regular Trees

We consider Metropolis Glauber dynamics for sampling proper $q$-colourings of the $n$-vertex complete $b$-ary tree when $3\leq q\leq b/2\ln(b)$. We give both upper and lower bounds on the mixing time. For fixed $q$ and $b$, our upper bound is $n^{O(b/\log b)}$ and our lower bound is $n^{Ω(b/q \log(b))}$, where the constants implicit in the $O()$ and $Ω()$ notation do not depend upon $n$, $q$ or $b$.

cs.CC↗

arXiv2007

Inapproximability of the Tutte polynomial

The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: take as input a graph G, and output a value which is a good approximation to T(G;x,y). Jaeger, Vertigan and Welsh have completely mapped the complexity of exactly computing the Tutte polynomial. They have shown that this is #P-hard, except along the hyperbola (x-1)(y-1)=1 and at four special points. We are interested in determining for which points (x,y) there is a "fully polynomial randomised approximation scheme" (FPRAS) for T(G;x,y). Under the assumption RP is not equal to NP, we prove that there is no FPRAS at (x,y) if (x,y) is in one of the half-planes x<-1 or y<-1 (excluding the easy-to-compute cases mentioned above). Two exceptions to this result are the half-line x<-1, y=1 (which is still open) and the portion of the hyperbola (x-1)(y-1)=2 corresponding to y<-1 which we show to be equivalent in difficulty to approximately counting perfect matchings. We give further intractability results for (x,y) in the vicinity of the origin. A corollary of our results is that, under the assumption RP is not equal to NP, there is no FPRAS at the point (x,y)=(0,1--lambda) when λ>2 is a positive integer. Thus there is no FPRAS for counting nowhere-zero λflows for λ>2. This is an interesting consequence of our work since the corresponding decision problem is in P for example for λ=6.

cs.CC↗

arXiv2006

Markov chain comparison

This is an expository paper, focussing on the following scenario. We have two Markov chains, $\mathcal {M}$ and $\mathcal {M}'$. By some means, we have obtained a bound on the mixing time of $\mathcal {M}'$. We wish to compare $\mathcal {M}$ with $\mathcal {M}'$ in order to derive a corresponding bound on the mixing time of $\mathcal {M}$. We investigate the application of the comparison method of Diaconis and Saloff-Coste to this scenario, giving a number of theorems which characterize the applicability of the method. We focus particularly on the case in which the chains are not reversible. The purpose of the paper is to provide a catalogue of theorems which can be easily applied to bound mixing times.

math.PR↗

arXiv2006

Systematic scan for sampling colorings

We address the problem of sampling colorings of a graph $G$ by Markov chain simulation. For most of the article we restrict attention to proper $q$-colorings of a path on $n$ vertices (in statistical physics terms, the one-dimensional $q$-state Potts model at zero temperature), though in later sections we widen our scope to general ``$H$-colorings'' of arbitrary graphs $G$. Existing theoretical analyses of the mixing time of such simulations relate mainly to a dynamics in which a random vertex is selected for updating at each step. However, experimental work is often carried out using systematic strategies that cycle through coordinates in a deterministic manner, a dynamics sometimes known as systematic scan. The mixing time of systematic scan seems more difficult to analyze than that of random updates, and little is currently known. In this article we go some way toward correcting this imbalance. By adapting a variety of techniques, we derive upper and lower bounds (often tight) on the mixing time of systematic scan. An unusual feature of systematic scan as far as the analysis is concerned is that it fails to be time reversible.

math.PR↗

arXiv2005

Elementary bounds on Poincare and log-Sobolev constants for decomposable Markov chains

We consider finite-state Markov chains that can be naturally decomposed into smaller ``projection'' and ``restriction'' chains. Possibly this decomposition will be inductive, in that the restriction chains will be smaller copies of the initial chain. We provide expressions for Poincare (resp. log-Sobolev) constants of the initial Markov chain in terms of Poincare (resp. log-Sobolev) constants of the projection and restriction chains, together with further a parameter. In the case of the Poincare constant, our bound is always at least as good as existing ones and, depending on the value of the extra parameter, may be much better. There appears to be no previously published decomposition result for the log-Sobolev constant. Our proofs are elementary and self-contained.

math.PR↗

arXiv2004

Two remarks concerning balanced matroids

The property of balance (in the sense of Feder and Mihail) is investigated in the context of paving matroids. The following examples are exhibited: (a) a class of ``sparse'' paving matroids that are balanced, but at the same time rich enough combinatorially to permit the encoding of hard counting problems; and (b) a paving matroid that is not balanced. The computational significance of (a) is the following. As a consequence of balance, there is an efficient algorithm for approximating the number of bases of a sparse paving matroid within specified relative error. On the other hand, determining the number of bases exactly is likely to be computationally intractable.

math.CO↗

arXiv2003

On the approximation of one Markov chain by another

Motivated by applications in Markov chain Monte Carlo, we discuss what it means for one Markov chain to be an approximation to another. Specifically included in that discussion are situations in which a Markov chain with continuous state space is approximated by one with finite state space. A simple sufficient condition for close approximation is derived, which indicates the existence of three distinct approximation regimes. Counterexamples are presented to show that these regimes are real and not artifacts of the proof technique. An application to the ``ball walk'' of Lovász and Simonovits is provided as an illustrative example.

math.PR↗