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Register a new userCounting independent sets in graphs with bounded bipartite pathwidth
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Added by
Martin Dyer
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity $lambda$, can be viewed as a strong generalisation of Jerrum and Sinclairs work on approximately counting matchings, that is, independent sets in line graphs. The class of graphs with bounded bipartite pathwidth includes claw-free graphs, which generalise line graphs. We consider two further generalisations of claw-free graphs and prove that these classes have bounded bipartite pathwidth. We also show how to extend all our results to polynomially-bounded vertex weights.
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We show that the number of independent sets in cocomparability graphs can be counted in linear time, as can counting cliques in comparability graphs. By contrast, counting cliques in cocomparabilty graphs and counting independent sets in comparability graphs are #P-complete. We extend these results to counting maximal cliques and independent sets. We also consider the fixed-paramet
The notion of a Riordan graph was introduced recently, and it is a far-reaching generalization of the well-known Pascal graphs and Toeplitz graphs. However, apart from a certain subclass of Toeplitz graphs, nothing was known on independent sets in Riordan graphs. In this paper, we give exact enumeration and lower and upper bounds for the number of independent sets for various classes of Riordan graphs. Remarkably, we offer a variety of methods to solve the problems that range from the structural decomposition theorem to methods in combinatorics on words. Some of our results are valid for any graph.
We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density $alpha_c(Delta)$ and provide (i) for $alpha < alpha_c(Delta)$ randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most $alpha n$ in $n$-vertex graphs of maximum degree $Delta$; and (ii) a proof that unless NP=RP, no such algorithms exist for $alpha>alpha_c(Delta)$. The critical density is the occupancy fraction of hard core model on the clique $K_{Delta+1}$ at the uniqueness threshold on the infinite $Delta$-regular tree, giving $alpha_c(Delta)simfrac{e}{1+e}frac{1}{Delta}$ as $Deltatoinfty$.
The extension complexity $mathsf{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that affinely projects to $P$. Let $G$ be a bipartite graph with $n$ vertices, $m$ edges, and no isolated vertices. Let $mathsf{STAB}(G)$ be the convex hull of the stable sets of $G$. It is easy to see that $n leqslant mathsf{xc} (mathsf{STAB}(G)) leqslant n+m$. We improve both of these bounds. For the upper bound, we show that $mathsf{xc} (mathsf{STAB}(G))$ is $O(frac{n^2}{log n})$, which is an improvement when $G$ has quadratically many edges. For the lower bound, we prove that $mathsf{xc} (mathsf{STAB}(G))$ is $Omega(n log n)$ when $G$ is the incidence graph of a finite projective plane. We also provide examples of $3$-regular bipartite graphs $G$ such that the edge vs stable set matrix of $G$ has a fooling set of size $|E(G)|$.
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