Abstract
Finding the stability number of a graph, i.e., the maximum number of vertices of which no two are adjacent, is a well known NP-hard combinatorial optimization problem. Since this problem has several applications in real life, there is need to find efficient algorithms to solve this problem. Recently, Gaar and Rendl enhanced semidefinite programming approaches to tighten the upper bound given by the Lovász theta function. This is done by carefully selecting some so-called exact subgraph constraints (ESC) and adding them to the semidefinite program of computing the Lovász theta function. First, we provide two new relaxations that allow to compute the bounds faster without substantial loss of the quality of the bounds. One of these two relaxations is based on including violated facets of the polytope representing the ESCs, the other one adds separating hyperplanes for that polytope. Furthermore, we implement a branch and bound (B&B) algorithm using these tightened relaxations in our bounding routine. We compare the efficiency of our B&B algorithm using the different upper bounds. It turns out that already the bounds of Gaar and Rendl drastically reduce the number of nodes to be explored in the B&B tree as compared to the Lovász theta bound. However, this comes with a high computational cost. Our new relaxations improve the run time of the overall B&B algorithm, while keeping the number of nodes in the B&B tree small.
Highlights
The stable set problem is a fundamental combinatorial optimization problem
We expect that we find a large stable set in this branch of the branch and bound (B&B) tree because the difference between the global lower bound and the upper bound for this branch is the highest of all
We have proven the strength of the bounds by showing that the number of nodes in a B&B algorithm reduces drastically by using these bounds, the computational costs are enormous in the original version with the convex hull formulation (CH) and they are still substantial with the bundle approach (BD) from Gaar and Rendl (2019, 2020)
Summary
The stable set problem is a fundamental combinatorial optimization problem. In the 2015 survey (Wu and Hao 2015), no exact algorithms using semidefinite programming (SDP) are mentioned. We formulate new SDP relaxations and develop solution algorithms to compute these bounds with moderate computational expense, making them applicable within a B&B scheme. A set S ⊆ V is called stable if no vertices in S are adjacent. S is called a maximal stable set if it is not possible to add a vertex to S without losing the stability property. For a graph G = (V , E), the set of all stable set vectors S(G) and the stable set polytope STAB(G) are defined as. A branch and bound algorithm that uses these relaxations is described, followed by the discussion of numerical results in Sect.
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