## Strengthening Integrality Gaps for Capacitated Network Design and Covering Problems

Citations: | 63 - 1 self |

### BibTeX

@MISC{Carr_strengtheningintegrality,

author = {Robert D. Carr and et al.},

title = {Strengthening Integrality Gaps for Capacitated Network Design and Covering Problems},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

A capacitated covering IP is an integer program of the form min{cx|Ux ≥ d, 0 ≤ x ≤ b, x ∈ Z +}, where all entries of c, U, and d are nonnegative. Given such a formulation, the ratio between the optimal integer solution and the optimal solution to the linear program relaxation can be as bad as ||d||∞, even when U consists of a single row. We show that by adding additional inequalities, this ratio can be improved significantly. In the general case, we show that the improved ratio is bounded by the maximum number of non-zero coefficients in a row of U, and provide a polynomial-time approximation algorithm to achieve this bound. This improves the previous best approximation algorithm which guaranteed a solution within the maximum row sum times optimum. We also show that for particular instances of capacitated covering problems, including the minimum knapsack problem and the capacitated network design problem, these additional inequalities yield even stronger improvements in the IP/LP ratio. For the minimum knapsack, we show that this improved ratio is at most 2. This is the first non-trivial IP/LP ratio for this basic problem. Capacitated network design generalizes the classical network design problem by introducing capacities on the edges, whereas previous work only considers the case when all capacities equal 1. For capacitated network design problems, we show that this improved ratio depends on a parameter of the graph, and we also provide polynomial-time approximation algorithms to match this bound. This improves on the best previous m-approximation, where m is the number of edges in the graph. We also discuss improvements for some other special capacitated covering problems, including the fixed charge network flow problem. Finally, for the capacitated network design problem, we give some stronger results and algorithms for series parallel graphs and strengthen these further for outerplanar graphs. Most of our approximation algorithms rely on solving a single LP. When the original LP (before adding our strengthening inequalities) has a polynomial number of constraints, we describe a combinatorial FPTAS for the LP with our (exponentially-many) inequalities added. Our contribution here is to describe an appropriate

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Citation Context ... Û-Õ Ö ×=Ù Ð Ñ Ò-ú ø ù é ù Õ Ö × Ø=Õ Ö × ÝAßlò é=Õ ñ × í Ð ñ × óAå í=ã ô=û'ã õ ü ý/ð ßlòxé=Õ A different form of flow cover inequalities were introduced in a polyhedral study of fixed charge problems =-=[29]-=-. They presented them as packing, not covering, inequalities; and they did not consider their effect at tightening the IP/LP ratio. THEOREM 2.10. The integrality gap for 2-node fixedcharge network flo... |

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Citation Context ...escribed by Garg and Könemann [10]. For solving the LP using the ellipsoid method, or simplex method, we describe simpler separation routines.Ô When there is only one demand pair, Schwarz and Krumke =-=[32]-=- describe an FPTAS on series parallel graphs, if this demand pair corresponds to the defining nodes of the series parallel graph. An outerplanar graph is a planar graph that can be embedded so that al... |

1 | to � � and �¦� � � demand to - demand |