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تسجيل مستخدم جديدCounting perfect matchings and the switch chain
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We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham and Holmes. We determine the largest hereditary class for which the chain is ergodic, and define a large new hereditary class of graphs for which it is rapidly mixing. We go on to show that the chain has exponential mixing time for a slightly larger class. We also examine the question of ergodicity of the switch chain in a arbitrary graph. Finally, we give exact counting algorithms for three classes.
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We study a simple Markov chain, the switch chain, on the set of all perfect matchings in a bipartite graph. This Markov chain was proposed by Diaconis, Graham and Holmes as a possible approach to a sampling problem arising in Statistics. We ask: for
which classes of graphs is the Markov chain ergodic and for which is it rapidly mixing? We provide a precise answer to the ergodicity question and close bounds on the mixing question. We show for the first time that the mixing time of the switch chain is polynomial in the case of monotone graphs, a class that includes examples of interest in the statistical setting.
In a recent paper, Beniamini and Nisan gave a closed-form formula for the unique multilinear polynomial for the Boolean function determining whether a given bipartite graph $G subseteq K_{n,n}$ has a perfect matching, together with an efficient algor
ithm for computing the coefficients of the monomials of this polynomial. We give the following generalization: Given an arbitrary non-negative weight function $w$ on the edges of $K_{n,n}$, consider its set of minimum weight perfect matchings. We give the real multilinear polynomial for the Boolean function which determines if a graph $G subseteq K_{n,n}$ contains one of these minimum weight perfect matchings.
The min-cost matching problem suffers from being very sensitive to small changes of the input. Even in a simple setting, e.g., when the costs come from the metric on the line, adding two nodes to the input might change the optimal solution completely
. On the other hand, one expects that small changes in the input should incur only small changes on the constructed solutions, measured as the number of modified edges. We introduce a two-stage model where we study the trade-off between quality and robustness of solutions. In the first stage we are given a set of nodes in a metric space and we must compute a perfect matching. In the second stage $2k$ new nodes appear and we must adapt the solution to a perfect matching for the new instance. We say that an algorithm is $(alpha,beta)$-robust if the solutions constructed in both stages are $alpha$-approximate with respect to min-cost perfect matchings, and if the number of edges deleted from the first stage matching is at most $beta k$. Hence, $alpha$ measures the quality of the algorithm and $beta$ its robustness. In this setting we aim to balance both measures by deriving algorithms for constant $alpha$ and $beta$. We show that there exists an algorithm that is $(3,1)$-robust for any metric if one knows the number $2k$ of arriving nodes in advance. For the case that $k$ is unknown the situation is significantly more involved. We study this setting under the metric on the line and devise a $(10,2)$-robust algorithm that constructs a solution with a recursive structure that carefully balances cost and redundancy.
A matching $M$ in a graph $G$ is said to be uniquely restricted if there is no other matching in $G$ that matches the same set of vertices as $M$. We describe a polynomial-time algorithm to compute a maximum cardinality uniquely restricted matching i
n an interval graph, thereby answering a question of Golumbic et al. (Uniquely restricted matchings, M. C. Golumbic, T. Hirst and M. Lewenstein, Algorithmica, 31:139--154, 2001). Our algorithm actually solves the more general problem of computing a maximum cardinality strong independent set in an interval nest digraph, which may be of independent interest. Further, we give linear-time algorithms for computing maximum cardinality uniquely restricted matchings in proper interval graphs and bipartite permutation graphs.
We show fully polynomial time randomized approximation schemes (FPRAS) for counting matchings of a given size, or more generally sampling/counting monomer-dimer systems in planar, not-necessarily-bipartite, graphs. While perfect matchings on planar g
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