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

  • Authors:

  • João Gouveia;Monique Laurent;Pablo A. Parrilo;Rekha Thomas

  • Affiliations:

  • University of Washington, Department of Mathematics, Box 354350, 98195, Seattle, WA, USA and University of Coimbra, CMUC, Department of Mathematics, 3001-454, Coimbra, Portugal;CWI, Science Park 123, 1098 XG, Amsterdam, The Netherlands and Tilburg University, Department of Econometrics and Operations Research, Tilburg, The Netherlands;Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, Laboratory for Information and Decision Systems, 77 Massachusetts Avenue, 02139-4307, Cambridge, M ...;University of Washington, Department of Mathematics, Box 354350, 98195, Seattle, WA, USA

  • Venue:
  • Mathematical Programming: Series A and B
  • Year:
  • 2012

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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 Lovász.