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A new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs |
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| Authors: | Jo?o Gouveia Monique Laurent Pablo A Parrilo Rekha Thomas |
| Institution: | 1. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195, USA 2. CMUC, Department of Mathematics, University of Coimbra, 3001-454, Coimbra, Portugal 3. CWI, Science Park 123, 1098 XG, Amsterdam, The Netherlands 4. Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands 5. Department of Electrical Engineering and Computer Science, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA |
| 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. |
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