# 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…

## 21 Citations

Cycle algebras and polytopes of matroids

- Mathematics
- 2021

Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes…

Chapter 7: Spectrahedral Approximations of Convex Hulls of Algebraic Sets

- Mathematics
- 2012

This chapter describes a method for finding spectrahedral approximations of the convex hull of a real algebraic variety (the set of real solutions to a finite system of polynomial equations). The…

A Semidefinite Approach to the $K_i$ Cover Problem

- Mathematics
- 2012

We apply theta body relaxations to the $K_i$-cover problem and show polynomial time solvability for certain classes of graphs. In particular, we give an effective relaxation where all $K_i$-$p$-hole…

Equivariant Semidefinite Lifts of Regular Polygons

- MathematicsMath. Oper. Res.
- 2017

This paper shows that one can construct an equivariant psd lift of the regular 2^n-gon of size 2n-1, which is exponentially smaller than the psd Lift of the sum-of-squares hierarchy, and proves that the construction is essentially optimal.

Convex Hulls of Algebraic Sets

- Mathematics
- 2012

This article describes a method to compute successive convex approximations of the convex hull of the solutions to a system of polynomial equations over the reals. The method relies on sums of…

Matrix convex hulls of free semialgebraic sets

- Mathematics
- 2015

This article resides in the realm of the noncommutative (free) analog of real algebraic geometry - the study of polynomial inequalities and equations over the real numbers - with a focus on matrix…

Mincut ideals of two-terminal networks

- Mathematics, Computer ScienceApplicable Algebra in Engineering, Communication and Computing
- 2010

It turns out that some features of the mincut ideals arising from networks such as the Cohen-Macaulay property and the computation of Betti numbers, which are important in tight reliability bounds, have a compact expression for series-parallel networks.

On 2-Level Polytopes Arising in Combinatorial Settings

- MathematicsSIAM J. Discret. Math.
- 2018

A trade-off formula for the number of cliques and stable sets in a graph; a description of stable matching poly topes as affine projections of certain order polytopes; and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree are presented.

## References

SHOWING 1-10 OF 40 REFERENCES

Theta Bodies for Polynomial Ideals

- MathematicsSIAM J. Optim.
- 2010

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

- MathematicsJ. Comb. Theory, Ser. B
- 1989

On the cut polytope

- MathematicsMath. Program.
- 1986

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

- Computer Science, MathematicsSIAM Rev.
- 1996

A survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution are given.

A hierarchy of relaxation between the continuous and convex hull representations

- Mathematics
- 1990

In this paper a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero-one polynomial programming problem, and then relinearizes it into…

A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems

- Mathematics, Computer ScienceSIAM J. Discret. Math.
- 1990

It is shown that the strength of the resulting reformulation depends on the degree of the terms used to produce the polynomial program at the intermediate step of this method, and a hierarchy of sharper representations is obtained with the final relaxation representing the convex hull of feasible solutions.

On the Shannon capacity of a graph

- Mathematics, Computer ScienceIEEE Trans. Inf. Theory
- 1979

It is proved that the Shannon zero-error capacity of the pentagon is \sqrt{5} and a well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases.

Real algebraic geometry

- Mathematics
- 1992

1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilbert's 17th Problem.…