The complexity of combinatorial optimization problems on d-dimensional boxes

Chlebík, Miroslav and Chlebíková, Janka (2007) The complexity of combinatorial optimization problems on d-dimensional boxes. SIAM Journal on Discrete Mathematics, 21 (1). pp. 158-169. ISSN 0895-4801

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Abstract

The Maximum Independent Set problem in d-box graphs, i.e., in intersection graphs of axis-parallel rectangles in R-d, is known to be NP-hard for any fixed d >= 2. A challenging open problem is that of how closely the solution can be approximated by a polynomial time algorithm. For the restricted case of d-boxes with bounded aspect ratio a PTAS exists [T. Erlebach, K. Jansen, and E. Seidel, SIAM J. Comput., 34 (2005), pp. 1302-1323]. In the general case no polynomial time algorithm with approximation ratio o(log(d-1)n) for a set of n d-boxes is known. In this paper we prove APX-hardness of the MAXIMUM INDEPENDENT SET problem in d-box graphs for any fixed d >= 3. We give an explicit lower bound 245/244 on efficient approximability for this problem unless P = NP. Additionally, we provide a generic method how to prove APX-hardness for other graph optimization problems in d-box graphs for any fixed d >= 3.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Depositing User: Miroslav Chlebik
Date Deposited: 06 Feb 2012 21:20
Last Modified: 11 Apr 2012 09:55
URI: http://sro.sussex.ac.uk/id/eprint/30820
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