Unification of lower-bound analyses of the lift-and-project rank of combinatorial optimization polyhedra

Authors:
Sung-Pil Hong;Levent Tunçel
Affiliations:
Department of Industrial Engineering, Seoul National University, Seoul 151-744, South Korea;Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ont., Canada N2L 3G1
Venue:
Discrete Applied Mathematics
Year:
2008

Citing 18
Cited 3

A hierarchy of relaxation between the continuous and convex hull representations

SIAM Journal on Discrete Mathematics
A lift-and-project cutting plane algorithm for mixed 0-1 programs

Mathematical Programming: Series A and B
Tighter representations for set partitioning problems

Discrete Applied Mathematics
Disjunctive programming: properties of the convex hull of feasible points

Discrete Applied Mathematics
On a Representation of the Matching Polytope Via Semidefinite Liftings

Mathematics of Operations Research
Semidefinite programs and association schemes

Computing - Special issue on combinatorial optimization
When Does the Positive Semidefiniteness Constraint Help in Lifting Procedures?

Mathematics of Operations Research
Tighter Linear and Semidefinite Relaxations for Max-Cut Based on the Lovász--Schrijver Lift-and-Project Procedure

SIAM Journal on Optimization
Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets

SIAM Journal on Optimization
The Probable Value of the Lovász-Schrijver Relaxations for Maximum Independent Set

SIAM Journal on Computing
Proving Integrality Gaps without Knowing the Linear Program

FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
On the Matrix-Cut Rank of Polyhedra

Mathematics of Operations Research
Lift and project relaxations for the matching and related polytopes

Discrete Applied Mathematics
Lower Bound for the Number of Iterations in Semidefinite Hierarchies for the Cut Polytope

Mathematics of Operations Research
Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube

Combinatorica
Subset Algebra Lift Operators for 0-1 Integer Programming

SIAM Journal on Optimization
On Lovász-Schrijver Lift-and-Project Procedures on the Dantzig-Fulkerson-Johnson Relaxation of the TSP

SIAM Journal on Optimization
Tree-width and the Sherali-Adams operator

Discrete Optimization

Sherali-adams relaxations of the matching polytope

Proceedings of the forty-first annual ACM symposium on Theory of computing
Complexity analyses of Bienstock-Zuckerberg and lasserre relaxations on the matching and stable set polytopes

IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Handelman rank of zero-diagonal quadratic programs over a hypercube and its applications

Journal of Global Optimization

Quantified Score

Hi-index	0.04

Visualization

Abstract

We present a unifying framework to establish a lower bound on the number of semidefinite-programming-based lift-and-project iterations (rank) for computing the convex hull of the feasible solutions of various combinatorial optimization problems. This framework is based on the maps which are commutative with the lift-and-project operators. Some special commutative maps were originally observed by Lovasz and Schrijver and have been used usually implicitly in the previous lower-bound analyses. In this paper, we formalize the lift-and-project commutative maps and propose a general framework for lower-bound analysis, in which we can recapture many of the previous lower-bound results on the lift-and-project ranks.