# Handelman’s hierarchy for the maximum stable set problem

@article{Laurent2013HandelmansHF, title={Handelman’s hierarchy for the maximum stable set problem}, author={Monique Laurent and Zhao Sun}, journal={Journal of Global Optimization}, year={2013}, volume={60}, pages={393-423} }

The maximum stable set problem is a well-known NP-hard problem in combinatorial optimization, which can be formulated as the maximization of a quadratic square-free polynomial over the (Boolean) hypercube. We investigate a hierarchy of linear programming relaxations for this problem, based on a result of Handelman showing that a positive polynomial over a polytope with non-empty interior can be represented as conic combination of products of the linear constraints defining the polytope. We…

## 6 Citations

Polynomial optimization: Error analysis and applications

- Mathematics
- 2015

Polynomial optimization is the problem of minimizing a polynomial function subject to polynomial inequality constraints. In this thesis we investigate several hierarchies of relaxations for…

A new algorithm for concave quadratic programming

- MathematicsJ. Glob. Optim.
- 2019

A new dual is presented, in which, the dual variables are affine functions, and it is demonstrated that the dual of the bound is a semi-definite relaxation of quadratic programs.

Approximating the maximum of a polynomial over a polytope: Handelman decomposition and continuous generating functions

- Mathematics
- 2016

Author(s): De Loera, Jesus; Dutra, Brandon; Koppe, Matthias | Abstract: We investigate a way to approximate the maximum of a polynomial over a polytopal region by using Handelman's polynomial…

Learning dynamic polynomial proofs

- Computer Science, MathematicsNeurIPS
- 2019

This work introduces a machine learning based method to search for a dynamic proof within semi-algebraic proof systems that manipulate polynomial inequalities via elementary inference rules that infer new inequalities from the premises.

Decomposition Methods for Nonlinear Optimization and Data Mining

- Computer Science, Mathematics
- 2016

This dissertation focuses on the problem of maximizing a polynomial function over the continuous domain of a polytope, which is NP-hard, but develops approximation methods that run inPolynomial time when the dimension is fixed.

## References

SHOWING 1-10 OF 32 REFERENCES

Approximation of the Stability Number of a Graph via Copositive Programming

- Mathematics, Computer ScienceSIAM J. Optim.
- 2002

This paper shows how the stability number can be computed as the solution of a conic linear program (LP) over the cone of copositive matrices of a graph by solving semidefinite programs (SDPs) of increasing size (lift-and-project method).

Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz

- Mathematics, Computer ScienceCombinatorics, Probability and Computing
- 2009

New polynomial encodings are constructed for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colourable subgraph.

Handelman rank of zero-diagonal quadratic programs over a hypercube and its applications

- Mathematics, Computer ScienceJ. Glob. Optim.
- 2013

An error analysis of Handelman hierarchy applied to the special class of polynomial programs and its implications in the computation of the combinatorial optimization problems are presented.

The stable set problem and the lift-and-project ranks of graphs

- MathematicsMath. Program.
- 2003

Improved bounds for N+-ranks of graphs in terms of the number of nodes in the graph are provided and it is proved that the subdivision of an edge or cloning a vertex can increase the N-rank of a graph.

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.

Properties of vertex packing and independence system polyhedra

- MathematicsMath. Program.
- 1974

A general class of facets of = convex hull{x∈Rn:Ax≤1m,x binary} is described which subsumes a class examined by Padberg [13].

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.

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…

Semidefinite Programming vs. LP Relaxations for Polynomial Programming

- Mathematics, Computer ScienceMath. Oper. Res.
- 2002

Two hierarchies of relaxations are compared, namely, LP relaxations based on products of the original constraints, in the spirit of the RLT procedure of Sherali and Adams (1990), and recent semidefinite programming (SDP) relaxations introduced by the author.