Abstract. 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.

Abstract. 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.

We consider the problem of sampling uniformly at random from the set of proper k-colorings of a graph with maximum degree Δ. Our main result is the design of a simple Markov chain that converges in O(nk log n) time to the desired distribution when k>116Δ.

We study two widely used algorithms, Glauber dynamics and the Swendsen-Wang algorithm, on rectangular subsets of the hypercubic lattice Z d . We prove that under certain circumstances, the mixing time in a box of side length L with periodic boundary conditions can be exponential in L d,1 . In other words, under these circumstances, the mixing in these widely used algorithms is not rapid; instead, it is torpid. The models we study are the independent set model and the q-state Potts model. For both models, we prove that Glauber dynamics is torpid in the region with phase coexistence. For the Potts model, we prove that Swendsen-Wang is torpid at the phase transition point.

ABSTRACT:The paper considers spin systems on the d-dimensional integer lattice ‫ޚ‬ d with nearest-neighbor interactions. A sharp equivalence is proved between decay with distance of spin correlations (a spatial property of the equilibrium state) and rapid mixing of the Glauber dynamics (a temporal property of a Markov chain Monte Carlo algorithm). Specifically, we show that if the mixing time of the Glauber dynamics is O(n log n) then spin correlations decay exponentially fast with distance. We also prove the converse implication for monotone systems, and for general systems we prove that exponential decay of correlations implies O(n log n) mixing time of a dynamics that updates sufficiently large blocks (rather than single sites). While the above equivalence was already known to hold in various forms, we give proofs that are purely combinatorial and avoid the functional analysis machinery employed in previous proofs.

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(β * ) at some desired inverse temperature β * is to define a sequence, which we call a cooling schedule,where Z(0) is trivial to compute and the ratios Z(β i+1 )/Z(β i ) are easy to estimate by sampling from the distribution corresponding to Z(β i ). Previous approaches required a cooling schedule of length O * (ln A) where A = Z(0), thereby ensuring that each ratio Z(β i+1 )/Z(β i ) is bounded. We present a cooling schedule of length ℓ = O * ( √ ln A). 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 * ( √ n), which implies an overall savings of O * (n) in the running time of the approximate counting algorithm (since roughly ℓ samples are needed to estimate each ratio).A similar improvement in the length of the cooling schedule was recently obtained by Lovász 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, i. e., the schedule depends on Z. More precisely, we prove any non-adaptive cooling schedule has length at least O * (ln A), and we present an algorithm to find an adaptive schedule of length O * ( √ ln A).

Recent inapproximability results of Sly (2010), together with an approximation algorithm presented by Weitz (2006) establish a beautiful picture for the computational complexity of approximating the partition function of the hard-core model. Let λc(T∆) denote the critical activity for the hard-model on the infinite ∆-regular tree. Weitz presented an FPTAS for the partition function when λ < λc(T∆) for graphs with constant maximum degree ∆. In contrast, Sly showed that for all ∆ ≥ 3, there exists ε∆ > 0 such that (unless RP = N P ) there is no FPRAS for approximating the partition function on graphs of maximum degree ∆ for activities λ satisfying λc(T∆) < λ < λc(T∆) + ε∆.We prove that a similar phenomenon holds for the antiferromagnetic Ising model. Sinclair, Srivastava, and Thurley (2014) extended Weitz's approach to the antiferromagnetic Ising model, yielding an FPTAS for the partition function for all graphs of constant maximum degree ∆ when the parameters of the model lie in the uniqueness region of the infinite ∆-regular tree. We prove the complementary result for the antiferrogmanetic Ising model without external field, namely, that unless RP = N P , for all ∆ ≥ 3, there is no FPRAS for approximating the partition function on graphs of maximum degree ∆ when the inverse temperature lies in the non-uniqueness region of the infinite tree T∆. Our proof works by relating certain second moment calculations for random ∆-regular bipartite graphs to the tree recursions used to establish the critical points on the infinite tree.

Recent results establish for the hard-core model (and more generally for 2-spin antiferromagnetic systems) that the computational complexity of approximating the partition function on graphs of maximum degree ∆ undergoes a phase transition that coincides with the uniqueness/nonuniqueness phase transition on the infinite ∆-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs, so-called #BIS-hardness. We improve this hardness result by establishing it for bipartite graphs of maximum degree ∆. To this end, we first present a detailed picture for the phase diagram for the infinite ∆-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature (corresponding to the region where the ordered phase "dominates") that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree ∆.The #BIS-hardness result uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent results establishing connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. In this paper we extend these connections to random regular graphs for all ferromagnetic models. Using these connections, we establish the Bethe prediction for every ferromagnetic spin system on random regular graphs, which says roughly that the expectation of the log of the partition function Z is the same as the log of the expectation of Z. As a further consequence of our results, we prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing (i. e., exponentially slow convergence to its stationary distribution) on random ∆-regular graphs at the critical temperature for sufficiently large q.