TY - GEN

T1 - 1-sparsity Approximation Bounds for Packing Integer Programs

AU - Chekuri, Chandra

AU - Quanrud, Kent

AU - Torres, Manuel R.

N1 - Funding Information:
C. Chekuri and K. Quanrud supported in part by NSF grant CCF-1526799. M. Torres supported in part by fellowships from NSF and the Sloan Foundation.

PY - 2019

Y1 - 2019

N2 - We consider approximation algorithms for packing integer programs (PIPs) of the form where c, A, and b are nonnegative. We let denote the width of A which is at least 1. Previous work by Bansal et al. [1] obtained an -approximation ratio where is the maximum number of nonzeroes in any column of A (in other words the -column sparsity of A). They raised the question of obtaining approximation ratios based on the -column sparsity of A (denoted by) which can be much smaller than Motivated by recent work on covering integer programs (CIPs) [4, 7] we show that simple algorithms based on randomized rounding followed by alteration, similar to those of Bansal et al. [1] (but with a twist), yield approximation ratios for PIPs based on First, following an integrality gap example from [1], we observe that the case of is as hard as maximum independent set even when In sharp contrast to this negative result, as soon as width is strictly larger than one, we obtain positive results via the natural LP relaxation. For PIPs with width where we obtain an -approximation. In the large width regime, when we obtain an -approximation. We also obtain a -approximation when.

AB - We consider approximation algorithms for packing integer programs (PIPs) of the form where c, A, and b are nonnegative. We let denote the width of A which is at least 1. Previous work by Bansal et al. [1] obtained an -approximation ratio where is the maximum number of nonzeroes in any column of A (in other words the -column sparsity of A). They raised the question of obtaining approximation ratios based on the -column sparsity of A (denoted by) which can be much smaller than Motivated by recent work on covering integer programs (CIPs) [4, 7] we show that simple algorithms based on randomized rounding followed by alteration, similar to those of Bansal et al. [1] (but with a twist), yield approximation ratios for PIPs based on First, following an integrality gap example from [1], we observe that the case of is as hard as maximum independent set even when In sharp contrast to this negative result, as soon as width is strictly larger than one, we obtain positive results via the natural LP relaxation. For PIPs with width where we obtain an -approximation. In the large width regime, when we obtain an -approximation. We also obtain a -approximation when.

KW - Approximation algorithms

KW - Packing integer programs

KW - column sparsity

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U2 - 10.1007/978-3-030-17953-3_10

DO - 10.1007/978-3-030-17953-3_10

M3 - Conference contribution

AN - SCOPUS:85065881347

SN - 9783030179526

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 128

EP - 140

BT - Integer Programming and Combinatorial Optimization - 20th International Conference, IPCO 2019, Proceedings

A2 - Lodi, Andrea

A2 - Nagarajan, Viswanath

PB - Springer-Verlag Berlin Heidelberg

T2 - 20th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2019

Y2 - 22 May 2019 through 24 May 2019

ER -