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We propose a new algorithmic framework, called partial rejection sampling, to draw samples exactly from a product distribution, conditioned on none of a number of bad events occurring. Our framework builds new connections between the variable framework of the Lovász Local Lemma and some classical sampling algorithms such as the cycle-popping algorithm for rooted spanning trees. Among other applications, we discover new algorithms to sample satisfying assignments of k-CNF formulas with bounded variable occurrences. [ABSTRACT FROM AUTHOR]

We present a simple algorithm that perfectly samples configurations from the unique Gibbs measure of a spin system on a potentially infinite graph G. The sampling algorithm assumes strong spatial mixing together with subexponential growth of G. It produces a finite window onto a perfect sample from the Gibbs distribution. The run-time is linear in the size of the window, in particular it is constant for each vertex. [ABSTRACT FROM AUTHOR]

We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the 'winding' technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514–527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice. [ABSTRACT FROM AUTHOR]

Jerrum, Sinclair, and Vigoda [J. ACM, 51 (2004), pp. 671--697] showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function of edge-weighted matchings in a bipartite graph. Equivalently, this may be viewed as approximating the partition function of vertex-weighted independent sets in the line graph of a bipartite graph. Line graphs of bipartite graphs are perfect graphs and are known to be precisely the class of (claw, diamond, odd hole)-free graphs. So how far does the result of Jerrum, Sinclair, and Vigoda extend? We first show that it extends to (claw, odd hole)-free graphs, and then show that it extends to the even larger class of (fork, odd hole)-free graphs. Our techniques are based on graph decompositions, which have been the focus of much recent work in structural graph theory, and on structural results of Chv'atal and Sbihi [J. Combin. Theory Ser. B, 44 (1988)], Maffray and Reed [J. Combin. Theory Ser. B, 75 (1999)], and Lozin and Milaniv c [J. Discrete Algorithms, 6 (2008), pp. 595--604]. [ABSTRACT FROM AUTHOR]

We give a fully polynomial-time randomized approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently. [ABSTRACT FROM AUTHOR]

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 nontrivial functional clone (in a sense that will be made precise) contains the binary function "implies". As a consequence, in the conservative case, all nontrivial counting CSPs are as hard to approximate 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 Möbius 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. [ABSTRACT FROM AUTHOR]

We study the mixing time of random walks on small-world networks modelled as follows: starting with the 2-dimensional periodic grid, each pair of vertices {u,v} with distance d > 1 is added as a "long-range" edge with probability proportional to d -r, where r = 0 is a parameter of the model. Kleinberg [33] studied a close variant of this network model and proved that the (decentralised) routing time is O((logn)2) when r = 2 and nO(1) when r 2. Here, we prove that the random walk also undergoes a phase transition at r = 2, but in this case, the phase transition is of a different form. We establish that the mixing time is T(logn) for r < 2, O((logn)4) for r = 2, and nO(1) for r > 2. [ABSTRACT FROM AUTHOR]

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 #RHπ1 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. [ABSTRACT FROM AUTHOR]

We give a fully polynomial-time randomized approximation scheme (FPRAS) for the all-terminal network reliability problem, which is to determine the probability that in an undirected graph, assuming each edge fails independently, the remainder of the graph is still connected. Our main contribution is to confirm a conjecture by Gorodezky and Pak [Random Structures Algorithms, 44 (2014), pp. 201--223] that the expected running time of the "cluster-popping" algorithm in bidirected graphs is bounded by a polynomial in the size of the input. [ABSTRACT FROM AUTHOR]

We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that each of the problems admits an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory. [ABSTRACT FROM AUTHOR]