Independent Sets In Bipartite Graphs Research Articles

Approximately counting independent sets in bipartite graphs via graph containers

AbstractBy 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.

A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs.

This paper studies the problem of upper bounding the number of independent sets in a graph, expressed in terms of its degree distribution. For bipartite regular graphs, Kahn (2001) established a tight upper bound using an information-theoretic approach, and he also conjectured an upper bound for general graphs. His conjectured bound was recently proved by Sah et al. (2019), using different techniques not involving information theory. The main contribution of this work is the extension of Kahn’s information-theoretic proof technique to handle irregular bipartite graphs. In particular, when the bipartite graph is regular on one side, but may be irregular on the other, the extended entropy-based proof technique yields the same bound as was conjectured by Kahn (2001) and proved by Sah et al. (2019).

Approximating partition functions of bounded-degree Boolean counting Constraint Satisfaction Problems

We study the complexity of #CSPΔ(Γ), which is the problem of counting satisfying assignments to CSP instances with constraints from Γ and whose variables can appear at most Δ times. Our main result shows that: (i) if every function in Γ is affine, then #CSPΔ(Γ) is in FP for all Δ, (ii) otherwise, if every function in Γ is in a class called IM2, then for large Δ, #CSPΔ(Γ) is equivalent under approximation-preserving reductions to the problem of counting independent sets in bipartite graphs, (iii) otherwise, for large Δ, it is NP-hard to approximate #CSPΔ(Γ), even within an exponential factor.

Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP

Jaeger et al. (Math Proc Camb Philos Soc 108(1):35–53, 1990) proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: the evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén (in: ICALP 2010, vol. 6198, pp. 426–437, Springer, Berlin, Heidelberg, 2010) and Husfeldt and Taslaman (in: IPEC 2010, vol. 6478, pp. 192–203, Springer, Berlin, Heidelberg, 2010) in combination with the results of Curticapean (in: ICALP 2015, pp. 380–392, Springer, 2015), extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line $$y=1$$ , which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given n-vertex graph cannot be determined in time unless #ETH fails. Another dichotomy theorem we strengthen is the one of Creignou and Hermann (Inf Comput 125(1):1–12, 1996) for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that the #P-hard cases cannot be solved in time unless #ETH fails. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time unless #ETH fails. In order to prove our results, we use the block interpolation idea by Curticapean and transfer it to systems of linear equations that might not directly correspond to interpolation.

Linear-time algorithms for counting independent sets in bipartite permutation graphs

The counting of independent sets in bipartite graphs is a #P-complete problem but counting those in permutation graphs is a problem that can be solved in polynomial-time. This paper develops some linear-time algorithms for counting independent sets, maximal independent sets, and independent perfect dominating sets in the class of bipartite permutation graphs, which is the intersection class between permutation and bipartite graphs.

Approximately counting locally-optimal structures

In general, constructing a locally-optimal structure is a little harder than constructing an arbitrary structure, but significantly easier than constructing a globally-optimal structure. A similar situation arises in listing. In counting, most problems are #P-complete, but in approximate counting we observe an interesting reversal of the pattern. Assuming that #BIS is not equivalent to #SAT under AP-reductions, we show that counting maximal independent sets in bipartite graphs is harder than counting maximum independent sets. Motivated by this, we show that various counting problems involving minimal separators are #SAT-hard to approximate. These problems have applications for constructing triangulations and phylogenetic trees.

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-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 (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-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 that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.

The complexity of approximating conservative counting CSPs

We study the complexity of the approximate weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, a classification is known over the Boolean domain; we extend this to arbitrary finite domains. We show that if F is “weakly log-modular”, then #CSP(F) is in FP. Otherwise, it is at least as difficult to approximate as #BIS (counting independent sets in bipartite graphs). #BIS is complete for the complexity class #RHΠ1, and believed to be intractable. We further sub-divide the #BIS-hard case: if F is “weakly log-supermodular”, #CSP(F) is as easy as a Boolean log-supermodular weighted #CSP; otherwise, we show that it is NP-hard to approximate. Finally, we give a full trichotomy for the arity-2 case: #CSP(F) is in FP, is #BIS-equivalent, or is equivalent to #SAT (approximately counting the satisfying assignments of Boolean CNF formulas). We also discuss algorithmic aspects of our classification.

Maximal independent sets in bipartite graphs with at least one cycle

Graph Theory A maximal independent set is an independent set that is not a proper subset of any other independent set. Liu [J.Q. Liu, Maximal independent sets of bipartite graphs, J. Graph Theory, 17 (4) (1993) 495-507] determined the largest number of maximal independent sets among all n-vertex bipartite graphs. The corresponding extremal graphs are forests. It is natural and interesting for us to consider this problem on bipartite graphs with cycles. Let \mathscrBₙ (resp. \mathscrBₙ') be the set of all n-vertex bipartite graphs with at least one cycle for even (resp. odd) n. In this paper, the largest number of maximal independent sets of graphs in \mathscrBₙ (resp. \mathscrBₙ') is considered. Among \mathscrBₙ the disconnected graphs with the first-, second-, \ldots, \fracn-22-th largest number of maximal independent sets are characterized, while the connected graphs in \mathscrBₙ having the largest, the second largest number of maximal independent sets are determined. Among \mathscrBₙ' graphs have the largest number of maximal independent sets are identified.

The complexity of approximating bounded-degree Boolean #CSP

The degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP=RP. For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs.

Maximal independent sets in bipartite graphs obtained from Boolean lattices

Attempts to enumerate maximal antichains in Boolean lattices give rise to problems involving maximal independent sets in bipartite graphs whose vertex sets are comprised of adjacent levels of the lattice and whose edges correspond to proper containment. In this paper, we find bounds on the numbers of maximal independent sets in these graphs.

The Complexity of Ferromagnetic Ising with Local Fields

We consider the complexity of approximating the partition function of the ferromagnetic Ising model with varying interaction energies and local external magnetic fields. Jerrum and Sinclair provided a fully polynomial randomized approximation scheme for the case in which the system is consistent in the sense that the local external fields all favour the same spin. We characterize the complexity of the general problem by showing that it is equivalent in complexity to the problem of approximately counting independent sets in bipartite graphs, thus it is complete in a logically defined subclass of #P previously studied by Dyer, Goldberg, Greenhill and Jerrum. By contrast, we show that the corresponding computational task for the is complete for all of #P with respect to approximation-preserving reductions.

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.

Neighborhood conditions for balanced independent sets in bipartite graphs

Let G be a balanced bipartite graph of order 2 n and minimum degree δ( G)⩾3. If, for every balanced independent set S of four vertices, | N( S)| > n then G is traceable, the circumference is at least 2 n − 2 and G contains a 2-factor (with only small order exceptional graphs for the latter statement). If the neighborhood union condition is replaced by | N( S)| > n + 2 then G is hamiltonian.

Maximal independent sets in bipartite graphs

AbstractA maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. In this paper, we determine the maximum number of maximal independent sets among all bipartite graphs of order n and the extremal graphs as well as the corresponding results for connected bipartite graphs. © 1993 John Wiley &amp; Sons, Inc.

An O( n log n + m log log n) maximum weight clique algorithm for circular-arc graphs

Let F be a family of arcs in a circle, where each arc is associated with a weight. The maximum weight clique problem is to find a subset of pairwise overlapping arcs in F such that their total weight is maximum. This problem was originally solved by Hsu in O( m· n) time. Recently, an O( n 2 log log n) algorithm was discovered by Apostolico and Hambrusch for the unweighted case, whereas the weighted case was posed as an open problem. We extend their approach to solve the weighted case in O( n log n + m log log n) time. Our algorithm is based on a characterization of maximum weight independent sets in bipartite graphs, which is not related to bipartite matching.