COCOON 2003: Computing and Combinatorics pp 192-201

# Matroid Representation of Clique Complexes

• Kenji Kashiwabara
• Yoshio Okamoto
• Takeaki Uno
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2697)

## Abstract

In this paper, we approach the quality of a greedy algorithm for the maximum weighted clique problem from the viewpoint of matroid theory. More precisely, we consider the clique complex of a graph (the collection of all cliques of the graph) and investigate the minimum number k such that the clique complex of a given graph can be represented as the intersection of k matroids. This number k can be regarded as a measure of “how complex a graph is with respect to the maximum weighted clique problem” since a greedy algorithm is a k-approximation algorithm for this problem. We characterize graphs whose clique complexes can be represented as the intersection of k matroids for any k > 0. Moreover, we determine the necessary and sufficient number of matroids for the representation of all graphs with n vertices. This number turns out to be n − 1. Other related investigations are also given.

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## Authors and Affiliations

• Kenji Kashiwabara
• 1
• Yoshio Okamoto
• 2
• Takeaki Uno
• 3
1. 1.Department of Systems Science, Graduate School of Arts and SciencesThe University of TokyoTokyoJapan
2. 2.Institute of Theoretical Computer Science, Department of Computer ScienceETH Zurich, ETH ZentrumZurichSwitzerland
3. 3.National Institute of InformaticsTokyoJapan