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.
Supported by the Joint Berlin/Zürich Graduate Program “Combinatorics, Geometry, and Computation” (CGC), financed by ETH Zurich and the German Science Foundation (DFG).
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References
L. Cai, D. Corneil and A. Proskurowski: A generalization of line graphs: (X, Y)-intersection graphs. Journal of Graph Theory 21 (1996) 267–287.
R. Diestel: Graph Theory (2nd Edition). Springer Verlag, New York, 2000.
J. Edmonds: Matroids and the greedy algorithm. Mathematical Programming 1 (1971) 127–136.
U. Faigle: Matroids in combinatorial optimization. In: Combinatorial Geometries (N. White, ed.), Cambridge University Press, Cambridge, 1987, pp. 161–210.
S.P. Fekete, R.T. Firla and B. Spille: Characterizing matchings as the intersection of matroids. Preprint, December 2002, arXiv:math.CO/0212235.
A. Frank: A weighted matroid intersection algorithm. Journal of Algorithms 2 (1981) 328–336.
T.A. Jenkyns: The efficacy of the “greedy” algorithm. Proceedings of the 7th Southeastern Conference on Combinatorics, Graph Theory, and Computing, Utilitas Mathematica, Winnipeg, 1976, pp. 341–350.
B. Korte and D. Hausmann: An analysis of the greedy algorithm for independence systems. In: Algorithmic Aspects of Combinatorics; Annals of Discrete Mathematic 2 (B. Alspach et al., eds.), North-Holland, Amsterdam, 1978, pp. 65–74.
B. Korte and J. Vygen: Combinatorial Optimization (2nd Edition). Springer Verlag, Berlin Heidelberg, 2002.
Y. Okamoto: Submodularity of some classes of the combinatorial optimization games. Mathematical Methods of Operations Research 58 (2003), to appear.
J. Oxley: Matroid Theory. Oxford University Press, New York, 1992.
F. Protti and J.L. Szwarcfiter: Clique-inverse graphs of bipartite graphs. Journal of Combinatorial Mathematics and Combinatorial Computing 40 (2002) 193–203.
R. Rado: Note on independence functions. Proceedings of the London Mathematical Society 7 (1957) 300–320.
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Kashiwabara, K., Okamoto, Y., Uno, T. (2003). Matroid Representation of Clique Complexes. In: Warnow, T., Zhu, B. (eds) Computing and Combinatorics. COCOON 2003. Lecture Notes in Computer Science, vol 2697. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45071-8_21
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DOI: https://doi.org/10.1007/3-540-45071-8_21
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