# Graph bisection revisited

Research output: Contribution to journalArticleScientificpeer-review

### Abstract

The graph bisection problem is the problem of partitioning the vertex set of a graph into two sets of given sizes such that the sum of weights of edges joining these two sets is optimized. We present a semidefinite programming relaxation for the graph bisection problem with a matrix variable of order n—the number of vertices of the graph—that is equivalent to the currently strongest semidefinite programming relaxation obtained by using vector lifting. The reduction in the size of the matrix variable enables us to impose additional valid inequalities to the relaxation in order to further strengthen it. The numerical results confirm that our simplified and strengthened semidefinite relaxation provides the currently strongest bound for the graph bisection problem in reasonable time.
Original language English 143-154 Annals of Operations Research 265 1 https://doi.org/10.1007/s10479-017-2575-3 Published - Jun 2018

### Fingerprint

Graph
Semidefinite programming
Valid inequalities
Partitioning

### Keywords

• graph bisection
• graph partition
• semidefinite programming

### Cite this

Sotirov, Renata. / Graph bisection revisited. In: Annals of Operations Research. 2018 ; Vol. 265, No. 1. pp. 143-154.
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In: Annals of Operations Research, Vol. 265, No. 1, 06.2018, p. 143-154.

Research output: Contribution to journalArticleScientificpeer-review

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N2 - The graph bisection problem is the problem of partitioning the vertex set of a graph into two sets of given sizes such that the sum of weights of edges joining these two sets is optimized. We present a semidefinite programming relaxation for the graph bisection problem with a matrix variable of order n—the number of vertices of the graph—that is equivalent to the currently strongest semidefinite programming relaxation obtained by using vector lifting. The reduction in the size of the matrix variable enables us to impose additional valid inequalities to the relaxation in order to further strengthen it. The numerical results confirm that our simplified and strengthened semidefinite relaxation provides the currently strongest bound for the graph bisection problem in reasonable time.

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KW - graph partition

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