Abstract
We consider graph three-partitions with the objective of minimizing the number of edges between the first two partition sets while keeping the size of the third block small. We review most of the existing relaxations for this min-cut problem and focus on a new class of semidefinite relaxations, based on matrices of order 2n+1 which provide a good compromise between quality of the bound and computational effort to actually compute it. Here, n is the order of the graph. Our numerical results indicate that the new bounds are quite strong and can be computed for graphs of medium size (n approx 300) with reasonable effort of a few minutes of computation time. Further, we exploit those bounds to obtain bounds on the size of the vertex separators. A vertex separator is a subset of the vertex set of a graph whose removal splits the graph into two disconnected subsets. We also present an elegant way of convexifying non-convex quadratic problems by using semidefinite programming. This approach results with bounds that can be computed with any standard convex quadratic programming solver.
Highlights
The vertex separator problem (VSP) for a graph is to find a subset of vertices whose removal disconnects the graph into two components of roughly equal size
We prove that the eigenvalue bound from [12] equals the optimal value of the semidefinite programming (SDP) relaxation from Theorem 5, with matrix variable of order 2n
We prove that the SDP relaxation S D P1 is equivalent to the SDP relaxation (36) from [18], see Theorem 12
Summary
The vertex separator problem (VSP) for a graph is to find a subset of vertices (called vertex separator) whose removal disconnects the graph into two components of roughly equal size. It asks to find a vertex partition (S1, S2, S3) with specified cardinalities, such that the number of edges joining vertices in S1 and S2 is minimized We remark that this min-cut problem is known to be NP-hard [8]. These will be used to find vertex separators of small size. Before we present our new relaxations for (MC) we find it useful to give a short overview of existing relaxation techniques This allows us to set the stage for our own results and to describe the rich structure of the problem which gives rise to a variety of relaxations
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