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The complexity of combinatorial optimization problems on d-dimensional boxes

journal contribution
posted on 2023-06-08, 09:42 authored by Miroslav ChlebikMiroslav Chlebik, Janka Chlebíková
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.

History

Publication status

  • Published

Journal

SIAM Journal on Discrete Mathematics

ISSN

0895-4801

Issue

1

Volume

21

Page range

158-169

Pages

12.0

Department affiliated with

  • Mathematics Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2012-02-06

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