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A new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs

Joćo Gouveia (jgouveia***at***u.washington.edu)
Monique Laurent (M.Laurent***at***cwi.nl)
Pablo A. Parrilo (parrilo***at***mit.edu)
Rekha Thomas (thomas***at***math.washington.edu)

Abstract: The theta bodies of a polynomial ideal are a series of semidefinite programming relaxations of the convex hull of the real variety of the ideal. In this paper we construct the theta bodies of the vanishing ideal of cycles in a binary matroid. Applied to cuts in graphs, this yields a new hierarchy of semidefinite programming relaxations of the cut polytope of the graph. If the binary matroid avoids certain minors we can characterize when the first theta body in the hierarchy equals the cycle polytope of the matroid. Specialized to cuts in graphs, this result solves a problem posed by Lov{\'a}sz.

Keywords: Semidefinite programming, theta bodies, max-cut problem, binary matroid

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Combinatorial Optimization

Citation: Preprint, July/09

Download: [PDF]

Entry Submitted: 07/26/2009
Entry Accepted: 07/26/2009
Entry Last Modified: 10/15/2010

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