Learning Branching Strategies for Parameterized Vertex Cover

Abstract

The Parameterized Vertex Cover (PVC) problem is a central problem on graphs, where given a graph G and a positive integer k the goal is to decide whether the graph contains a set of at most k vertices whose deletion destroys all edges of the graph. In other words, a set of vertices is called a vertex cover of a graph if after deleting those vertices we obtain an edgeless graph. The problem is one of Karp’s 21 NP-complete problems (Erickson, 2010)(Karp, 1972), meaning that the problem is computationally hard, takes exponential time to solve, and is not expected to be solvable in polynomial time unless P = NP. For most practical purposes, heuristics, approximation, or parameterized algorithms are the only reasonable way to solve large instances of the problem in a reasonable amount of time. We are interested in solving the problem exactly and one method for solving PVC is the Branch & Reduce paradigm. In a Branch & Reduce algorithm, we construct a search tree to solve a given instance by either branching on which vertices to include into a solution or applying reduction rules to reduce the search space. At a high level, the typical algorithm for PVC selects a vertex of highest degree and branches on either including said vertex in a solution or including all of its neighbors. Several reduction rules are also applied whenever possible. In this work, we investigate a new approach based on machine learning for optimizing vertex selection while solving PVC instances. We constructed a system that uses graph features as inputs to make inferences about the best weighting strategies to be applied on the different node features in order to select the best vertex to branch on. In our approach we utilize reinforcement learning technology to train our model. Our results show that we were able to outperform the high degree strategy in 85% of instances.