Abstract
The “exact subgraph” approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational challenge because of the potentially large number of violated subgraph constraints. We introduce a computational framework for these relaxations designed to cope with these difficulties. We suggest a partial Lagrangian dual, and exploit the fact that its evaluation decomposes into several independent subproblems. This opens the way to use the bundle method from non-smooth optimization to minimize the dual function. Finally computational experiments on the Max-Cut, stable set and coloring problem show the excellent quality of the bounds obtained with this approach.
Highlights
The study of NP-hard problems has led to the introduction of various hierarchies of relaxations, which typically involve several levels
Adams, Anjos, Rendl and Wiegele [1] introduced a hierarchy of semidefinite programming (SDP) relaxations which act in the space of symmetric n × n matrices and at level k of the hierarchy all submatrices of order k have to be “exact” in a well-defined sense, i.e. they have to fulfill an exact subgraph constraint (ESC)
The focus of this paper lies in computational results, so we omit further extensive theoretical investigations, but we want to draw the attention to a major structural difference between the Max-Cut problem and the stable set and the coloring problem
Summary
The study of NP-hard problems has led to the introduction of various hierarchies of relaxations, which typically involve several levels. Adams, Anjos, Rendl and Wiegele [1] introduced a hierarchy of SDP relaxations which act in the space of symmetric n × n matrices and at level k of the hierarchy all submatrices of order k have to be “exact” in a well-defined sense, i.e. they have to fulfill an exact subgraph constraint (ESC). It is the main purpose of this paper to describe an efficient way to optimize over level k of this hierarchy for small values of k, e.g. k 7, and demonstrate the efficiency of our approach for the Max-Cut, stable set and coloring problem. A spectrahedron is a set that is obtained as the intersection of the cone of positive semidefinite matrices with some linear affine subspace
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