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Targeted Branching for the Maximum Independent Set Problem

Authors Demian Hespe , Sebastian Lamm , Christian Schorr

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Demian Hespe
  • Karlsruhe Institute of Technology, Institute for Theoretical Informatics, Germany
Sebastian Lamm
  • Karlsruhe Institute of Technology, Institute for Theoretical Informatics, Germany
Christian Schorr
  • Karlsruhe Institute of Technology, Institute for Theoretical Informatics, Germany

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Demian Hespe, Sebastian Lamm, and Christian Schorr. Targeted Branching for the Maximum Independent Set Problem. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.SEA.2021.17

Abstract

Finding a maximum independent set is a fundamental NP-hard problem that is used in many real-world applications. Given an unweighted graph, this problem asks for a maximum cardinality set of pairwise non-adjacent vertices. In recent years, some of the most successful algorithms for solving this problem are based on the branch-and-bound or branch-and-reduce paradigms. In particular, branch-and-reduce algorithms, which combine branch-and-bound with reduction rules, have been able to achieve substantial results, solving many previously infeasible real-world instances. These results were to a large part achieved by developing new, more practical reduction rules. However, other components that have been shown to have a significant impact on the performance of these algorithms have not received as much attention. One of these is the branching strategy, which determines what vertex is included or excluded in a potential solution. Even now, the most commonly used strategy selects vertices solely based on their degree and does not take into account other factors that contribute to the performance of the algorithm.
In this work, we develop and evaluate several novel branching strategies for both branch-and-bound and branch-and-reduce algorithms. Our strategies are based on one of two approaches which are motivated by existing research. They either (1) aim to decompose the graph into two or more connected components which can then be solved independently, or (2) try to remove vertices that hinder the application of a reduction rule which can lead to smaller graphs. Our experimental evaluation on a large set of real-world instances indicates that our strategies are able to improve the performance of the state-of-the-art branch-and-reduce algorithm by Akiba and Iwata. To be more specific, our reduction-based packing branching rule is able to outperform the default branching strategy of selecting a vertex of highest degree on 65% of all instances tested. Furthermore, our decomposition-based strategy based on edge cuts is able to achieve a speedup of 2.29 on sparse networks (1.22 on all instances).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Branch-and-bound
  • Mathematics of computing → Combinatorial optimization
Keywords
  • Graphs
  • Combinatorial Optimization
  • Independent Set
  • Vertex Cover
  • Clique
  • Branch-and-Reduce
  • Branch-and-Bound
  • Data Reduction

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