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Approximability of Sparse Integer Programs

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Abstract

The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min cx:Ax≥b,0≤x≤d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A,b,c,d are nonnegative.) For any k≥2 and ε>0, if P≠NP this ratio cannot be improved to k−1−ε, and under the unique games conjecture this ratio cannot be improved to k−ε. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max cx:Ax≤b,0≤x≤d} where A has at most k nonzeroes per column, we give a (2k 2+2)-approximation algorithm. Our approach builds on the iterated LP relaxation framework. In addition, we obtain improved approximations for the second problem when k=2, and for both problems when every A ij is small compared to b i . Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per column.

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Authors and Affiliations

  1. Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    David Pritchard

  2. Department of Computer and Information Science, University of Pennsylvania, Philadelphia, USA

    Deeparnab Chakrabarty

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  1. David Pritchard
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  2. Deeparnab Chakrabarty
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Correspondence to David Pritchard.

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The first author is partially supported by an NSERC post-doctoral fellowship.

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Pritchard, D., Chakrabarty, D. Approximability of Sparse Integer Programs. Algorithmica 61, 75–93 (2011). https://doi.org/10.1007/s00453-010-9431-z

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