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.
We present an improved "cooling schedule" for simulated annealing algorithms for combinatorial counting problems. Under our new schedule the rate of cooling accelerates as the temperature decreases. Thus, fewer intermediate temperatures are needed as the simulated annealing algorithm moves from the high temperature (easy region) to the low temperature (difficult region). We present applications of our technique to colorings and the permanent (perfect matchings of bipartite graphs). Moreover, for the permanent, we improve the analysis of the Markov chain underlying the simulated annealing algorithm. This improved analysis, combined with the faster cooling schedule, results in an O(n 7 log 4 n) time algorithm for approximating the permanent of a 0/1 matrix.
We study the complexity of approximating the value of the independent set polynomial Z G ( ) of a graph G with maximum degree when the activity is a complex number.When is real, the complexity picture is well-understood, and is captured by two real-valued thresholds ⇤ and c , which depend on and satisfy 0 < ⇤ < c . It is known that if is a real number in the interval ( ⇤ , c ) then there is an FPTAS for approximating Z G ( ) on graphs G with maximum degree at most . On the other hand, if is a real number outside of the (closed) interval, then approximation is NP-hard. The key to establishing this picture was the interpretation of the thresholds ⇤ and c on the -regular tree. The "occupation ratio" of a -regular tree T is the contribution to Z T ( ) from independent sets containing the root of the tree, divided by Z T ( ) itself. This occupation ratio converges to a limit, as the height of the tree grows, if and only if 2Unsurprisingly, the case where is complex is more challenging. It is known that there is an FPTAS when is a complex number with norm at most ⇤ and also when is in a small strip surrounding the real interval [0, c ). However, neither of these results is believed to fully capture the truth about when approximation is possible. Peters and Regts identified the values of for which the occupation ratio of the -regular tree converges. These values carve a cardioid-shaped region ⇤ in the complex plane, whose boundary includes the critical points ⇤ and c . Motivated by the picture in the real case, they asked whether ⇤ marks the true approximability threshold for general complex values .Our main result shows that for every outside of ⇤ , the problem of approximating Z G ( ) on graphs G with maximum degree at most is indeed NP-hard. In fact, when is outside of ⇤ and is not a positive real number, we give the stronger result that approximating Z G ( ) is actually #P-hard. Further, on the negative real axis, when < ⇤ , we show that it is #P-hard to even decide whether Z G ( ) > 0, resolving in the affirmative a conjecture of Harvey, Srivastava and Vondrák.Our proof techniques are based around tools from complex analysis -specifically the study of iterative multivariate rational maps. The full version is available at arxiv.org/abs/1711.00282 and is attached as an appendix. The theorem numbering here matches the full version.
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