By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies to ‐regular, bipartite graphs satisfying a weak expansion condition: when is constant, and the graph is a bipartite ‐expander, we obtain an FPTAS for the number of independent sets. Previously such a result for was known only for graphs satisfying the much stronger expansion conditions of random bipartite graphs. The algorithm also applies to weighted independent sets: for a ‐regular, bipartite ‐expander, with fixed, we give an FPTAS for the hard‐core model partition function at fugacity . Finally we present an algorithm that applies to all ‐regular, bipartite graphs, runs in time , and outputs a ‐approximation to the number of independent sets.
Counting Independent Sets in Unbalanced Bipartite Graphs
Understanding the complexity of approximately counting the number of weighted or unweighted independent sets in a bipartite graph (#BIS) is a central open problem in the field of approximate counting. Here we consider a subclass of this problem and give an FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides (L, R) of the bipartition. This includes, among others, the biregular case when λ = 1 (approximating the number of independent sets of G) and Delta_R >= 7 Delta_L log(Delta_L). Our approximation algorithm is based on truncating the cluster expansion of a polymer model partition function that expresses the hard-core partition function in terms of deviations from independent sets that are empty on one side of the bipartition.
Further consequences of this method for unbalanced bipartite graphs include an efficient sampling algorithm for the hard-core model and zero-freeness results for the partition function with complex fugacities. By utilizing connections between the cluster expansion and joint cumulants of certain random variables, we go beyond previous algorithmic applications of the cluster expansion to prove that the hard-core model exhibits exponential decay of correlations for all graphs and fugacities satisfying our conditions. This illustrates the applicability of statistical mechanics tools to algorithmic problems and refines our understanding of the connections between different methods of approximate counting.
more »
« less
- Award ID(s):
- 1803325
- NSF-PAR ID:
- 10332921
- Editor(s):
- Chawla, Shuchi
- Date Published:
- Journal Name:
- Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms
- ISSN:
- 1071-9040
- Page Range / eLocation ID:
- 1456-1466
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
Abstract -
We study the computational complexity of estimating local observables for Gibbs distributions. A simple combinatorial example is the average size of an independent set in a graph. A recent work of Galanis et al (2021) established NP-hardness of approximating the average size of an independent set utilizing hardness of the corresponding optimization problem and the related phase transition behavior. We instead consider settings where the underlying optimization problem is easily solvable. Our main contribution is to classify the complexity of approximating a wide class of observables via a generic reduction from approximate counting to the problem of estimating local observables. The key idea is to use the observables to interpolate the counting problem. Using this new approach, we are able to study observables on bipartite graphs where the underlying optimization problem is easy but the counting problem is believed to be hard. The most-well studied class of graphs that was excluded from previous hardness results were bipartite graphs. We establish hardness for estimating the average size of the independent set in bipartite graphs of maximum degree 6; more generally, we show tight hardness results for general vertex-edge observables for antiferromagnetic 2-spin systems on bipartite graphs. Our techniques go beyond 2-spin systems, and for the ferromagnetic Potts model we establish hardness of approximating the number of monochromatic edges in the same region as known hardness of approximate counting results.more » « less
-
We study the identity testing problem in the context of spin systems or undirected graphical models, where it takes the following form: given the parameter specification of the model M and a sampling oracle for the distribution \mu_{M^*} of an unknown model M^*, can we efficiently determine if the two models M and M^* are the same? We consider identity testing for both soft-constraint and hard-constraint systems. In particular, we prove hardness results in two prototypical cases, the Ising model and proper colorings, and explore whether identity testing is any easier than structure learning. For the ferromagnetic (attractive) Ising model, Daskalasis et al. (2018) presented a polynomial time algorithm for identity testing. We prove hardness results in the antiferromagnetic (repulsive) setting in the same regime of parameters where structure learning is known to require a super-polynomial number of samples. In particular, for n-vertex graphs of maximum degree d, we prove that if |\beta| d = \omega(\log n) (where \beta is the inverse temperature parameter), then there is no identity testing algorithm for the antiferromagnetic Ising model that runs in polynomial time unless RP = NP. We also establish computational lower bounds for a broader set of parameters under the (randomized) exponential time hypothesis. In our proofs, we use random graphs as gadgets; this is inspired by similar constructions in seminal works on the hardness of approximate counting. In the hard-constraint setting, we present hardness results for identity testing for proper colorings. Our results are based on the presumed hardness of #BIS, the problem of (approximately) counting independent sets in bipartite graphs. In particular, we prove that identity testing for colorings is hard in the same range of parameters where structure learning is known to be hard, which in turn matches the parameter regime for NP-hardness of the corresponding decision problem.more » « less
-
We give an FPRAS for counting q-colorings for even on almost every Δ-regular bipartite graph. This improves significantly upon the previous best bound of by Jenssen, Keevash, and Perkins (SODA'19). Analogously, for the hard-core model on independentsets weighted by λ > 0, we present an FPRAS for estimating the partition function when , which improves upon previous results by an Ω(log Δ) factor. Our results for the colorings and hard-core models follow from a general result that applies to arbitrary spin systems. Our main contribution is to show how to elevate probabilistic/analytic bounds on the marginal probabilities for the typical structure of phases on random bipartite regular graphs into efficient algorithms, using the polymer method. We further show evidence that our results for colorings and independent sets are within a constant factor of best possible using current polymer-method approaches.more » « less
-
Czumaj, Artur (Ed.)We study the approximation complexity of the partition function of the eight-vertex model on general 4-regular graphs. For the first time, we relate the approximability of the eight-vertex model to the complexity of approximately counting perfect matchings, a central open problem in this field. Our results extend those in [8]. In a region of the parameter space where no previous approximation complexity was known, we show that approximating the partition function is at least as hard as approximately counting perfect matchings via approximation-preserving reductions. In another region of the parameter space which is larger than the region that is previously known to admit Fully Polynomial Randomized Approximation Scheme (FPRAS), we show that computing the partition function can be reduced to counting perfect matchings (which is valid for both exact and approximate counting). Moreover, we give a complete characterization of nonnegatively weighted (not necessarily planar) 4-ary matchgates, which has been open for several years. The key ingredient of our proof is a geometric lemma. We also identify a region of the parameter space where approximating the partition function on planar 4-regular graphs is feasible but on general 4-regular graphs is equivalent to approximately counting perfect matchings. To our best knowledge, these are the first problems that exhibit this dichotomic behavior between the planar and the nonplanar settings in approximate counting.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1137/1.9781611975994.88
