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

@article{Gouveia2012ANS,
  title={A new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs},
  author={Jo{\~a}o Gouveia and Monique Laurent and Pablo A. Parrilo and Rekha R. Thomas},
  journal={Mathematical Programming},
  year={2012},
  volume={133},
  pages={203-225}
}
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… 
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21 Citations
Highly Influential Citations
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Background Citations
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Methods Citations
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21 Citations

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References

SHOWING 1-10 OF 40 REFERENCES
Lower Bound for the Number of Iterations in Semidefinite Hierarchies for the Cut Polytope
TLDR
It is shown that [ n/2] iterations are needed for finding the cut polytope of the complete graphK n and that such a relaxation can be expressed as a projection of such a semidefinite relaxation after at mostn steps.
Theta Bodies for Polynomial Ideals
TLDR
A hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal called theta bodies of the ideal is introduced and a geometric description of the first theta body for all ideals is given.
Decomposition and optimization over cycles in binary matroids
On the cycle polytope of a binary matroid
Semidefinite programming relaxations for semialgebraic problems
  • P. Parrilo
  • Mathematics, Computer Science
    Math. Program.
  • 2003
TLDR
It is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility and provide a constructive approach for finding bounded degree solutions to the Positivstellensatz.
Semidefinite programming and integer programming
Semidefinite representations for finite varieties
  • M. Laurent
  • Mathematics, Computer Science
    Math. Program.
  • 2007
TLDR
This work proves the finite convergence of a hierarchy of semidefinite relaxations introduced by Lasserre, which involves combinatorial moment matrices indexed by a basis of the quotient vector space ℝ[x1, . . . ,xn]/I.
On the cut polytope
TLDR
It is shown that inequalities associated with chordless cycles define facets of this polytope; moreover, for these inequalities a polynomial algorithm to solve the separation problem is presented.
Semidefinite Programming
TLDR
A survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution are given.
Master polytopes for cycles of binary matroids
...
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