Elementary approximation algorithms for prize collecting Steiner tree problems

Gutner, Shai

doi:10.1016/j.ipl.2007.12.010

Cited by 3 publications

(1 citation statement)

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“…The general form of the PCSF problem first has been formulated by Hajiaghayi and Jain [12]. They showed how by a primal-dual algorithm to a novel integer programming formulation of the problem with doubly-exponential variables, we can obtain a 3-approximation algorithm for the problem (see also [10]). In addition, they show that the factor 3 in the analysis of their algorithm is tight.…”

Section: Introductionmentioning

confidence: 99%

Prize-Collecting Steiner Networks via Iterative Rounding

Hajiaghayi¹,

Nasri

2010

LATIN 2010: Theoretical Informatics

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Abstract. In this paper we design an iterative rounding approach for the classic prize-collecting Steiner forest problem and more generally the prize-collecting survivable Steiner network design problem. We show as an structural result that in each iteration of our algorithm there is an LP variable in a basic feasible solution which is at least one-third-integral resulting a 3-approximation algorithm for this problem. In addition, we show this factor 3 in our structural result is indeed tight for prize-collecting Steiner forest and thus prize-collecting survivable Steiner network design. This especially answers negatively the previous belief that one might be able to obtain an approximation factor better than 3 for these problems using a natural iterative rounding approach. Our structural result is extending the celebrated iterative rounding approach of Jain [13] by using several new ideas some from more complicated linear algebra. The approach of this paper can be also applied to get a constant factor (bicriteria-)approximation algorithm for degree constrained prize-collecting network design problems.We emphasize that though in theory we can prove existence of only an LP variable of at least one-third-integral, in practice very often in each iteration there exists a variable of integral or almost integral which results in a much better approximation factor than provable factor 3 in this paper (see patent application [11]). This is indeed the advantage of our algorithm in this paper over previous approximation algorithms for prize-collecting Steiner forest with the same or slightly better provable approximation factors.

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Section: Introductionmentioning

confidence: 99%