Approximability of Sparse Integer Programs

  • David Pritchard
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5757)

Abstract

The main focus of this paper is a pair of new approximation algorithms for sparse 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 2 k k 2-approximation algorithm. This is the first polynomial-time approximation algorithm for this problem with approximation ratio depending only on k, for any k > 1. Our approach starts from iterated LP relaxation, and then uses probabilistic and greedy methods to recover a feasible solution.

Keywords

Feasible Solution Approximation Algorithm Integer Program Knapsack Problem Vertex Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • David Pritchard
    • 1
  1. 1.Department of Combinatorics & OptimizationUniversity of WaterlooCanada

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