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For the minimum cardinality vertex cover and maximum cardinality matching problems, the max-product form of belief propagation (BP) is known to perform poorly on general graphs. In this paper, we present an iterative loopy annealing BP (LABP) algorithm which is shown to converge and to solve a Linear Programming relaxation of the vertex cover or matching problem on general graphs. LABP finds (asymptotically) a minimum half-integral vertex cover (hence provides a 2-approximation) and a maximum fractional matching on any graph. We also show that LABP finds (asymptotically) a minimum size vertex cover for any bipartite graph and as a consequence compute the matching number of the graph. Our proof relies on some subtle monotonicity arguments for the local iteration. We also show that the Bethe free entropy is concave and that LABP maximizes it. Using loop calculus, we also give an exact (also intractable for general graphs) expression of the partition function for matching in term of the LABP messages which can be used to improve mean-field approximations.
A common approach for designing scalable algorithms for massive data sets is to distribute the computation across, say $k$, machines and process the data using limited communication between them. A particularly appealing framework here is the simulta
We present a near-tight analysis of the average query complexity -- `a la Nguyen and Onak [FOCS08] -- of the randomized greedy maximal matching algorithm, improving over the bound of Yoshida, Yamamoto and Ito [STOC09]. For any $n$-vertex graph of ave
We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: the $p$-Edge-Connected and $p$-Vertex-Connected VC problem (where $p geq 2$ is a fixed integer). Like Connected VC, both new VC problems are FPT, but do not
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Gaussian belief propagation (GaBP) is an iterative message-passing algorithm for inference in Gaussian graphical models. It is known that when GaBP converges it converges to the correct MAP estimate of the Gaussian random vector and simple sufficient