# Approximation of the Stability Number of a Graph via Copositive Programming

@article{Klerk2002ApproximationOT, title={Approximation of the Stability Number of a Graph via Copositive Programming}, author={Etienne de Klerk and Dmitrii V. Pasechnik}, journal={SIAM J. Optim.}, year={2002}, volume={12}, pages={875-892} }

Lovasz and Schrijver [SIAM J. Optim., 1 (1991), pp. 166--190] showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs (SDPs) of increasing size (lift-and-project method). In this paper we present a similar idea. We show how the stability number can be computed as the solution of a conic linear program (LP) over the cone of copositive matrices. Subsequently, we show how to approximate the copositive cone ever more closely…

## 313 Citations

### Computing the Stability Number of a Graph Via Linear and Semidefinite Programming

- Computer Science, MathematicsSIAM J. Optim.
- 2007

This work is based on and refines de Klerk and Pasechnik’s approach to approximating the stability number via copositive programming and provides a closed-form expression for the values computed by the linear programming approximations.

### Semidefinite bounds for the stability number of a graph via sums of squares of polynomials

- MathematicsMath. Program.
- 2005

The hierarchy of Lasserre is known to converge in α(G) steps as it refines the hierarchy of Lovasz and Schrijver, and the de Klerk and Pasechnik conjecture that their hierarchy also finds the stability number after α( G) steps is proved.

### Computing the stability number of a graph via semidefinite and linear programming

- Computer Science, Mathematics
- 2005

This work is based on and refines De Klerk and Pasechnik’s approach to approximating the stability number via copositive programming and gives a closed-form expression for the values computed by the linear programming approximations.

### Finite Convergence of Sum-of-Squares Hierarchies for the Stability Number of a Graph

- MathematicsSIAM J. Optim.
- 2022

We investigate a hierarchy of semidefinite bounds θ(G) for the stability number α(G) of a graph G, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM J.…

### Copositive programming motivated bounds on the stability and the chromatic numbers

- MathematicsMath. Program.
- 2010

The Parrilo hierarchy is investigated to approximate this cone and computational simplifications for the approximation of the chromatic number of vertex transitive graphs are provided.

### Contribution of copositive formulations to the graph partitioning problem

- Computer Science
- 2013

It is proved that the copositive formulations based on results from Burer and Wolkowicz and Zhao are equivalent and that they both imply semidefinite relaxations which are stronger than the Donath–Hoffman eigenvalue lower bound.

### Semidefinite relaxations for copositive optimization

- Computer Science, Mathematics
- 2016

This work studies two nested sequences of spectrahedral cones that approximate C*.

### A characterization of the weighted version of McEliece-Rodemich-Rumsey-Schrijver number based on convex quadratic programming

- MathematicsDiscret. Math. Algorithms Appl.
- 2015

A class of graphs for which the weighted version of ϑ′(G) coincides with the weighted stability number is characterized and the Lovasz bound is established.

### A completely positive formulation of the graph isomorphism problem and its positive semidefinite relaxation

- MathematicsJ. Comb. Optim.
- 2020

This paper provides a natural heuristic that uses the SDP to solve the graph isomorphism problem and runs this heuristic on several pairs of non-isomorphic strongly regular graphs and finds the results to be encouraging.

## References

SHOWING 1-10 OF 24 REFERENCES

### Tighter Linear and Semidefinite Relaxations for Max-Cut Based on the Lov[a-acute]sz--Schrijver Lift-and-Project Procedure

- MathematicsSIAM J. Optim.
- 2002

It is shown that the cut polytope of a graph can be found in k iterations if there exist k edges whose contraction produces a graph with no K5-minor.

### Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization

- Mathematics
- 2000

In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where the…

### A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0-1 Programming

- MathematicsMath. Oper. Res.
- 2003

The three methods for constructing hierarchies of successive linear or semidefinite relaxations of a polytope are presented in a common elementary framework and it is shown that the Lasserre construction provides the tightest Relaxations of P.

### On Copositive Programming and Standard Quadratic Optimization Problems

- MathematicsJ. Glob. Optim.
- 2000

The primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase, and are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached.

### Some NP-complete problems in quadratic and nonlinear programming

- Computer Science, MathematicsMath. Program.
- 1987

A special class of indefinite quadratic programs is constructed, with simple constraints and integer data, and it is shown that checking (a) or (b) on this class is NP-complete.

### Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming

- Computer ScienceJACM
- 1995

This algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.

### A new bound for Pólya's Theorem with applications to polynomials positive on polyhedra

- Mathematics
- 2001

### Copositive realxation for genera quadratic programming

- Computer Science
- 1998

We consider general, typically nonconvex, Quadratic programming Problem. The Semidefinite relaxation proposed by Shor provides bounds on the optimal solution, but it does not always provide…

### Interior-point polynomial algorithms in convex programming

- MathematicsSiam studies in applied mathematics
- 1994

This book describes the first unified theory of polynomial-time interior-point methods, and describes several of the new algorithms described, e.g., the projective method, which have been implemented, tested on "real world" problems, and found to be extremely efficient in practice.

### Global Optimization with Polynomials and the Problem of Moments

- MathematicsSIAM J. Optim.
- 2001

It is shown that the problem of finding the unconstrained global minimum of a real-valued polynomial p(x): R n to R, in a compact set K defined byPolynomial inequalities reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems.