Fully Polynomial Randomized Approximation Scheme Research Articles

Approximately Counting Answers to Conjunctive Queries with Disequalities and Negations

We study the complexity of approximating the number of answers to a small query \(\varphi\) in a large database \(\mathcal{D}\) . We establish an exhaustive classification into tractable and intractable cases if \(\varphi\) is a conjunctive query possibly including disequalities and negations: — If there is a constant bound on the arity of \(\varphi\) , and if the randomised Exponential Time Hypothesis (rETH) holds, then the problem has a fixed-parameter tractable approximation scheme (FPTRAS) if and only if the treewidth of \(\varphi\) is bounded. — If the arity is unbounded and \(\varphi\) does not have negations, then the problem has an FPTRAS if and only if the adaptive width of \(\varphi\) (a width measure strictly more general than treewidth) is bounded; the lower bound relies on the rETH as well. Additionally we show that our results cannot be strengthened to achieve a fully polynomial randomised approximation scheme (FPRAS): We observe that, unless \(\mathrm{NP}=\mathrm{RP}\) , there is no FPRAS even if the treewidth (and the adaptive width) is \(1\) . However, if there are neither disequalities nor negations, we prove the existence of an FPRAS for queries of bounded fractional hypertreewidth, strictly generalising the recently established FPRAS for conjunctive queries with bounded hypertreewidth due to Arenas, Croquevielle, Jayaram and Riveros (STOC 2021).

An Approximate Skolem Function Counter

One approach to probabilistic inference involves counting the number of models of a given Boolean formula. Here, we are interested in inferences involving higher-order objects, i.e., functions. We study the following task: Given a Boolean specification between a set of inputs and outputs, count the number of functions of inputs such that the specification is met. Such functions are called Skolem functions. We are motivated by the recent development of scalable approaches to Boolean function synthesis. This stands in relation to our problem analogously to the relationship between Boolean satisfiability and the model counting problem. Yet, counting Skolem functions poses considerable new challenges. From the complexity-theoretic standpoint, counting Skolem functions is not only #P-hard; it is quite unlikely to have an FPRAS (Fully Polynomial Randomized Approximation Scheme) as the problem of synthesizing a Skolem function remains challenging, even given access to an NP oracle. The primary contribution of this work is the first algorithm, SkolemFC, that computes the number of Skolem functions. SkolemFC relies on technical connections between counting functions and propositional model counting: our algorithm makes a linear number of calls to an approximate model counter and computes an estimate of the number of Skolem functions with theoretical guarantees. Our prototype displays impressive scalability, handling benchmarks comparably to state-of-the-art Skolem function synthesis engines, even though counting all such functions ostensibly poses a greater challenge than synthesizing a single function.

Beyond windability: Approximability of the four-vertex model

We study the approximability of the four-vertex model, a special case of the six-vertex model. We prove that, despite being NP-hard to approximate in the worst case, the four-vertex model admits a fully polynomial randomized approximation scheme (FPRAS) when the input satisfies certain linear equation system over GF(2). The FPRAS is given by a Markov chain known as the worm process, whose state space and rapid mixing rely on the solution of the linear equation system. This is the first attempt to design an FPRAS for the six-vertex model with unwindable constraint functions. Additionally, we explore the applications of this technique on planar graphs, providing efficient sampling algorithms.

Approximate counting of standard set-valued tableaux

We present a randomized algorithm for generating standard set-valued tableaux by extending the Green-Nijenhuis-Wilf hook walk algorithm. In the case of asymptotically rank two partitions, we use this algorithm to give a fully polynomial almost uniform sampler (FPAUS) for standard set-valued tableaux. This FPAUS is then used to construct a fully polynomial randomized approximation scheme (FPRAS) for counting the number of standard set-valued tableaux for such shapes. We also construct a FPAUS and FPRAS for standard set-valued tableaux when either the size of the partition or the difference between the maximum value and the size of the partition is fixed. Our methods build on the work of Jerrum-Valiant-Vazirani and provide a framework for constructing FPAUS's and FPRAS's for other counting problems in algebraic combinatorics.

Counting Candy Crush configurations

A k-stable c-coloured Candy Crush grid is a weak proper c-colouring of a particular type of k-uniform hypergraph. In this paper we introduce a fully polynomial randomised approximation scheme (FPRAS) which counts the number of k-stable c-coloured Candy Crush grids of a given size (m,n) for certain values of c and k. We implemented this algorithm on Matlab, and found that in a Candy Crush grid with 7 available colours there are approximately 4.3×1061 3-stable colourings. (Note that, typical Candy Crush games are played with 6 colours and our FPRAS is not guaranteed to work in expected polynomial time with k=3 and c=6.) We also discuss the applicability of this FPRAS to the problem of counting the number of weak c-colourings of other, more general hypergraphs.

A remark on approximating permanents of positive definite matrices

Let A be an n×n positive definite Hermitian matrix with all eigenvalues between 1 and 2. We represent the permanent of A as the integral of some explicit log-concave function on R2n. Consequently, there is a fully polynomial randomized approximation scheme (FPRAS) for perA.

Phase transitions of the Moran process and algorithmic consequences

The Moran process is a random process that models the spread of genetic mutations through graphs. On connected graphs, the process eventually reaches “fixation,” where all vertices are mutants, or “extinction,” where none are. Our main result is an almost‐tight upper bound on expected absorption time. For all ϵ>0, we show that the expected absorption time on an n‐vertex graph is o(n3+ϵ). Specifically, it is at most , and there is a family of graphs where it is Ω(n3). In proving this, we establish a phase transition in the probability of fixation, depending on the mutants' fitness r. We show that no similar phase transition occurs for digraphs, where it is already known that the expected absorption time can be exponential. Finally, we give an improved fully polynomial randomized approximation scheme (FPRAS) for approximating the probability of fixation. On degree‐bounded graphs where some basic properties are given, its running time is independent of the number of vertices.

The Worm Process for the Ising Model is Rapidly Mixing

We prove rapid mixing of the worm process for the zero-field ferromagnetic Ising model, on all finite connected graphs, and at all temperatures. As a corollary, we obtain a fully-polynomial randomized approximation scheme for the Ising susceptibility, and for a certain restriction of the two-point correlation function.

A Polynomial-Time Algorithm for Estimating the Partition Function of the Ferromagnetic Ising Model on a Regular Matroid

We investigate the computational difficulty of approximating the partition function of the ferromagnetic Ising model on a regular matroid. Jerrum and Sinclair have shown that there is a fully polynomial randomised approximation scheme (FPRAS) for the class of graphic matroids. On the other hand, the authors have previously shown, subject to a complexity-theoretic assumption, that there is no FPRAS for the class of binary matroids, which is a proper superset of the class of graphic matroids. In order to map out the region where approximation is feasible, we focus on the class of regular matroids, an important class of matroids which properly includes the class of graphic matroids, and is properly included in the class of binary matroids. Using Seymour's decomposition theorem, we give an FPRAS for the class of regular matroids.

Sampling colourings of the triangular lattice

AbstractWe show that the Glauber dynamics on proper 9‐colourings of the triangular lattice is rapidly mixing, which allows for efficient sampling. Consequently, there is a fully polynomial randomised approximation scheme (FPRAS) for counting proper 9‐colourings of the triangular lattice. Proper colourings correspond to configurations in the zero‐temperature anti‐ferromagnetic Potts model. We show that the spin system consisting of proper 9‐colourings of the triangular lattice has strong spatial mixing. This implies that there is a unique infinite‐volume Gibbs distribution, which is an important property studied in statistical physics. Our results build on previous work by Goldberg, Martin and Paterson, who showed similar results for 10 colours on the triangular lattice. Their work was preceded by Salas and Sokal's 11‐colour result. Both proofs rely on computational assistance, and so does our 9‐colour proof. We have used a randomised heuristic to guide us towards rigourous results. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 501–533, 2012

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 ( log n ) 4 in running time. This class of dense problems is shown to be nontrivial in counting, in the sense that they are #P-Complete.

Approximately Counting Integral Flows and Cell-Bounded Contingency Tables

We consider the problem of approximately counting integral flows in a network. We show that there is a fully polynomial randomized approximation scheme (FPRAS) based on volume estimation if all capacities are sufficiently large, generalizing a result of Dyer, Kannan, and Mount [Random Structures Algorithms, 10 (1997), pp. 487-506]. We apply this to approximating the number of contingency tables with prescribed cell bounds when the number of rows is constant, but the row sums, column sums, and cell bounds may be arbitrary. We provide an FPRAS for this problem via a combination of dynamic programming and volume estimation. This generalizes an algorithm of Cryan and Dyer [J. Comput. System Sci., 67 (2003), pp. 291-310] for standard contingency tables, but the analysis here is considerably more intricate.

Kernels for Generalized Multiple-Instance Learning

The multiple-instance learning (MIL) model has been successful in numerous application areas. Recently, a generalization of this model and an algorithm for it were introduced, showing significant advantages over the conventional MIL model on certain application areas. Unfortunately, that algorithm is not scalable to high dimensions. We adapt that algorithm to one using a support vector machine with our new kernel k\wedge. This reduces the time complexity from exponential in the dimension to polynomial. Computing our new kernel is equivalent to counting the number of boxes in a discrete, bounded space that contain at least one point from each of two multisets. We show that this problem is #P-complete, but then give a fully polynomial randomized approximation scheme (FPRAS) for it. We then extend k\wedge by enriching its representation into a new kernel kmin, and also consider a normalized version of k\wedge that we call k\wedge/\vee (which may or may not not be a kernel, but whose approximation yielded positive semidefinite Gram matrices in practice). We then empirically evaluate all three measures on data from content-based image retrieval, biological sequence analysis, and the musk data sets. We found that our kernels performed well on all data sets relative to algorithms in the conventional MIL model.

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 et al. 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 ≠ 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 ≠ NP , there is no FPRAS at the point ( x , y ) = ( 0 , 1 - λ ) when λ > 2 is a positive integer. Thus, there is no FPRAS for counting nowhere-zero λ flows for λ > 2 . This is an interesting consequence of our work since the corresponding decision problem is in P for example for λ = 6 . Although our main concern is to distinguish regions of the Tutte plane that admit an FPRAS from those that do not, we also note that the latter regions exhibit different levels of intractability. At certain points ( x , y ) , for example the integer points on the x-axis, or any point in the positive quadrant, there is a randomised approximation scheme for T ( G ; x , y ) that runs in polynomial time using an oracle for an NP predicate. On the other hand, we identify a region of points ( x , y ) at which even approximating T ( G ; x , y ) is as hard as #P.

Improved Mixing Bounds for the Anti-Ferromagnetic Potts Model on Z2

AbstractThe authors of this paper consider the anti-ferromagnetic Potts model on the the integer lattice Z2. The model has two parameters: q, the number of spins, and λ = exp(−β), where β 4 is ‘inverse temperature’. It is known that the model has strong spatial mixing if q &gt; 7, or if q = 7 and λ = 0 or λ &gt; 1/8, or if q = 6 and λ = 0 or λ &gt; 1/4. The λ = 0 case corresponds to the model in which configurations are proper q-colourings of Z2. It is shown that the system has strong spatial mixing for q ≥ 6 and any λ. This implies that Glauber dynamics is rapidly mixing (so there is a fully-polynomial randomised approximation scheme for the partition function), and also that there is a unique infinite-volume Gibbs state. We also show that strong spatial mixing occurs for a larger range of λ than was previously known for q = 3, q = 4 and q = 5.

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.

The Complexity of Choosing an H-Coloring (Nearly) Uniformly at Random

Cooper, Dyer, and Frieze [J. Algorithms, 39 (2001), pp. 117--134] studied the problem of sampling H-colorings (nearly) uniformly at random. Special cases of this problem include sampling colorings and independent sets and sampling from statistical physics models such as the Widom--Rowlinson model, the Beach model, the Potts model and the hard-core lattice gas model. Cooper et al. considered the family of cautious ergodic Markov chains with uniform stationary distribution and showed that, for every fixed connected nontrivial graph H, every such chain mixes slowly. In this paper, we give a complexity result for the problem. Namely, we show that for any fixed graph H with no trivial components, there is unlikely to be any polynomial almost uniform sampler (PAUS) for H-colorings. We show that if there were a PAUS for the H-coloring problem, there would also be a PAUS for sampling independent sets in bipartite graphs, and, by the self-reducibility of the latter problem, there would be a fully polynomial randomized approximation scheme (FPRAS) for #BIS---the problem of counting independent sets in bipartite graphs. Dyer, Goldberg, Greenhill, and Jerrum have shown that #BIS is complete in a certain logically defined complexity class. Thus, a PAUS for sampling H-colorings would give an FPRAS for the entire complexity class. In order to achieve our result we introduce the new notion of sampling-preserving reduction which seems to be more useful in certain settings than approximation-preserving reduction.

Clifford algebras and approximating the permanent

We study approximation algorithms for the permanent of an n×n (0,1) matrix A based on the following simple idea: obtain a random matrix B by replacing each 1-entry of A independently by ± e, where e is a random basis element of a suitable algebra; then output |det( B)| 2. This estimator is always unbiased, but it may have exponentially large variance. In our first main result we show that, if we take the algebra to be a Clifford algebra of dimension polynomial in n, then we get an estimator with small variance. Hence, only a constant number of trials suffices to estimate the permanent to good accuracy. The idea of using Clifford algebras is a natural extension of earlier work by Godsil and Gutman, Karmarkar et al., and Barvinok, who used the real numbers, complex numbers and quaternions, respectively. The above result implies that, in principle, this approach gives a fully-polynomial randomized approximation scheme for the permanent, provided |det( B)| 2 can be efficiently computed in the Clifford algebras. Since these algebras are non-commutative it is not clear how to do this. However, our second main result shows how to compute in polynomial time an estimator with the same mean and variance over the 4-dimensional algebra (which is the quaternions, and is non-commutative); in addition to providing some hope that the computations can be performed in higher dimensions, this quaternion algorithm provides an exponential improvement in the variance over that of the 2-dimensional complex version studied by Karmarkar et al.

The computational complexity of two‐state spin systems

AbstractThe subject of this article is spin‐systems as studied in statistical physics. We focus on the case of two spins. This case encompasses models of physical interest, such as the classical Ising model (ferromagnetic or antiferromagnetic, with or without an applied magnetic field) and the hard‐core gas model. There are three degrees of freedom, corresponding to our parameters β, γ, and μ. Informally, β represents the weights of edges joining pairs of “spin blue” sites, γ represents the weight of edges joining pairs of “spin green” sites, and μ represents the weight of “spin green” sites. We study the complexity of (approximately) computing the partition function in terms of these parameters. We pay special attention to the symmetric case μ = 1. Exact computation of the partition function Z is NP‐hard except in the trivial case βγ = 1, so we concentrate on the issue of whether Z can be computed within small relative error in polynomial time. We show that there is a fully polynomial randomised approximation scheme (FPRAS) for the partition function in the “ferromagnetic” region βγ ≥ 1, but (unless RP = NP) there is no FPRAS in the “antiferromagnetic” region corresponding to the square defined by 0 &lt; β &lt; 1 and 0 &lt; γ &lt; 1. Neither of these “natural” regions—neither the hyperbola nor the square—marks the boundary between tractable and intractable. In one direction, we provide an FPRAS for the partition function within a region which extends well away from the hyperbola. In the other direction, we exhibit two tiny, symmetric, intractable regions extending beyond the antiferromagnetic region. We also extend our results to the asymmetric case μ ≠ 1. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 133–154, 2003