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}
}
  • M. Laurent, Z. Sun
  • Published 31 May 2013
  • Mathematics
  • Journal of Global Optimization
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… 
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References

SHOWING 1-10 OF 32 REFERENCES
Approximation of the Stability Number of a Graph via Copositive Programming
TLDR
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
TLDR
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
TLDR
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
TLDR
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.
Rank of Handelman hierarchy for Max-Cut
Computing the Stability Number of a Graph Via Linear and Semidefinite Programming
TLDR
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
TLDR
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
TLDR
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
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
  • J. Lasserre
  • Mathematics, Computer Science
    Math. Oper. Res.
  • 2002
TLDR
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.
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