NDLI: A New Approximation Method for Set Covering Problems, with Applications to Multidimensional Bin Packing

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Caprara, Alberto Sviridenko, Maxim Bansal, Nikhil
Copyright Year	2009
Abstract	In this paper we introduce a new general approximation method for set covering problems, based on the combination of randomized rounding of the (near-) optimal solution of the linear programming (LP) relaxation, leading to a partial integer solution and the application of a well-behaved approximation algorithm to complete this solution. If the value of the solution returned by the latter can be bounded in a suitable way, as is the case for the most relevant generalizations of bin packing, the method leads to improved approximation guarantees, along with a proof of tighter integrality gaps for the LP relaxation. For d-dimensional vector packing, we obtain a polynomial-time randomized algorithm with asymptotic approximation guarantee arbitrarily close to $\ln d + 1$. For $d=2$, this value is $1.693\dots$; i.e., we break the natural 2 barrier for this case. Moreover, for small values of d this is a notable improvement over the previously known $O(\ln d)$ guarantee by Chekuri and Khanna [SIAM J. Comput., 33 (2004), pp. 837851]. For two-dimensional bin packing with and without rotations, we obtain polynomial-time randomized algorithms with asymptotic approximation guarantee $1.525\dots$, improving upon previous algorithms with asymptotic performance guarantees arbitrarily close to 2 by Jansen and van Stee [On strip packing with rotations, in Proceedings of the 37th Annual ACM Symposium on the Theory of Computing, 2005, pp. 755761] for the problem with rotations and $1.691\ldots$ by Caprara [Math. Oper. Res., 33 (2008), pp. 203215] for the problem without rotations. The previously unknown key property used in our proofs follows from a retrospective analysis of the implications of the landmark bin packing approximation scheme by Fernandez de la Vega and Lueker [Combinatorica, 1 (1981), pp. 349355]. We prove that their approximation scheme is subset oblivious, which leads to numerous applications.
Starting Page	1256
Ending Page	1278
Page Count	23
File Format	PDF
ISSN	00975397
DOI	10.1137/080736831
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-10-01
Access Restriction	Subscribed
Subject Keyword	set cover Applications approximation algorithm Approximation algorithms Scheduling theory, deterministic bin packing
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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