A Copositive Programming Approach to Graph Partitioning

Authors:
Janez Povh;Franz Rendl
Affiliations:
-;-
Venue:
SIAM Journal on Optimization
Year:
2007

Citing 0
Cited 5

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Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization

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Copositive and semidefinite relaxations of the quadratic assignment problem

Discrete Optimization
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Abstract

We consider 3-partitioning the vertices of a graph into sets $S_1$, $S_2$, and $S_3$ of specified cardinalities, such that the total weight of all edges joining $S_1$ and $S_2$ is minimized. This problem is closely related to several NP-hard problems like determining the bandwidth or finding a vertex separator in a graph. We show that this problem can be formulated as a linear program over the cone of completely positive matrices, leading in a natural way to semidefinite relaxations of the problem. We show in particular that the spectral relaxation introduced by Helmberg &etal; (1995) can equivalently be formulated as a semidefinite program. Finally we propose a tightened version of this semidefinite program and show on some small instances that this new bound is a significant improvement over the spectral bound.