Prize-Collecting Steiner Networks via Iterative Rounding

Hajiaghayi, MohammadTaghi; Nasri, Arefeh

doi:10.1007/978-3-642-12200-2_45

Cited by 10 publications

(6 citation statements)

References 19 publications

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“…v k the straight edges in all gadgets have the same x value. To finish the proof use ( 13) and (14).…”

Section: Lagrangian-multiplier Preserving Approximation Algorithms Fo...mentioning

confidence: 99%

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On the Integrality Gap of the Prize-Collecting Steiner Forest LP

Könemann,

Olver,

Pashkovich

et al. 2017

Preprint

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In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph G = (V, E), edge costs {c e ≥ 0} e∈E , terminal pairs {(s i , t i )} k i=1 , and penalties {π i } k i=1 for each terminal pair; the goal is to find a forest F to minimize c(F ) + i:(si,ti) not connected in F π i . The Steiner forest problem can be viewed as the special case where π i = ∞ for all i. It was widely believed that the integrality gap of the natural (and well-studied) linear-programming (LP) relaxation for PCSF (PCSF-LP) is at most 2. We dispel this belief by showing that the integrality gap of this LP is at least 9/4. This holds even for planar graphs. We also show that using this LP, one cannot devise a Lagrangian-multiplierpreserving (LMP) algorithm with approximation guarantee better than 4. Our results thus show a separation between the integrality gaps of the LP-relaxations for prize-collecting and non-prize-collecting (i.e., standard) Steiner forest, as well as the approximation ratios achievable relative to the optimal LP solution by LMP-and non-LMP-approximation algorithms for PCSF. For the special case of prizecollecting Steiner tree (PCST), we prove that the natural LP relaxation admits basic feasible solutions with all coordinates of value at most 1/3 and all edge variables positive. Thus, we rule out the possibility of approximating PCST with guarantee better than 3 using a direct iterative rounding method.

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“…v k the straight edges in all gadgets have the same x value. To finish the proof use ( 13) and (14).…”

Section: Lagrangian-multiplier Preserving Approximation Algorithms Fo...mentioning

confidence: 99%

“…In [14] it was shown, that for every vertex (x, z) of (PCSF-LP) (and hence also (PCST-LP)) where x is positive, there is at least one variable of value at least 1/3. Moreover for (PCSF-LP) this result is tight, i.e.…”

Section: Introductionmentioning

confidence: 99%

On the Integrality Gap of the Prize-Collecting Steiner Forest LP

Könemann,

Olver,

Pashkovich

et al. 2017

Preprint

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“…Hajiaghayi [Haj08] generalizes the iterative rounding approach of Jain to Prize-Collecting Steiner Network when there is a separate non-increasing marginal penalty function for each pair u, v which r uv -connectivity requirement is not satisfied. He obtains an iterative rounding 3-approximation algorithm for this case.…”

Section: Introductionmentioning

confidence: 99%

Prize-collecting steiner network problems

Hajiaghayi

Khandekar

Kortsarz

et al. 2012

ACM Trans. Algorithms

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Abstract. In the Steiner Network problem we are given a graph G with edge-costs and connectivity requirements ruv between node pairs u, v. The goal is to find a minimum-cost subgraph H of G that contains ruv edge-disjoint paths for all u, v ∈ V . In Prize-Collecting Steiner Network problems we do not need to satisfy all requirements, but are given a penalty function for violating the connectivity requirements, and the goal is to find a subgraph H that minimizes the cost plus the penalty. The case when ruv ∈ {0, 1} is the classic Prize-Collecting Steiner Forest problem. In this paper we present a novel linear programming relaxation for the Prize-Collecting Steiner Network problem, and by rounding it, obtain the first constant-factor approximation algorithm for submodular and monotone non-decreasing penalty functions. In particular, our setting includes all-or-nothing penalty functions, which charge the penalty even if the connectivity requirement is slightly violated; this resolves an open question posed in [SSW07]. We further generalize our results for element-connectivity and node-connectivity.

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“…Their approach has been generalized by Sharma, Swamy, and Williamson [53] for network design problems where violated arbitrary 0-1 connectivity constraints are allowed in exchange for a more general penalty function. Hajiaghayi and Nasri [40] show factor 3 for Prize-Collecting Steiner Forest can also be obtained via an iterative rounding approach, first introduced by Jain [44], and indeed factor 3 is the best one can hope via this approach. The work of Hajiaghayi and Jain has also motivated a game-theoretic version of the problem considered by Gupta et al [37].…”

Section: Introductionmentioning

confidence: 99%

Prize-collecting Steiner Problems on Planar Graphs

Bateni¹,

Chekuri²,

Ene³

et al. 2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper, we reduce Prize-Collecting Steiner TSP (PCTSP), Prize-Collecting Stroll (PCS), Prize-Collecting Steiner Tree (PCST), Prize-Collecting Steiner Forest (PCSF) and more generally Submodular Prize-Collecting Steiner Forest (SPCSF) on planar graphs (and more generally bounded-genus graphs) to the same problems on graphs of bounded treewidth. More precisely, we show any α-approximation algorithm for these problems on graphs of bounded treewidth gives an (α + )-approximation algorithm for these problems on planar graphs (and more generally bounded-genus graphs), for any constant > 0. Since PCS, PCTSP, and PCST can be solved exactly on graphs of bounded treewidth using dynamic programming, we obtain PTASs for these problems on planar graphs and bounded-genus graphs. In contrast, we show PCSF is APX-hard to approximate on series-parallel graphs, which are planar graphs of treewidth at most 2. This result is interesting on its own because it gives the first provable hardness separation between prize-collecting and non-prize-collecting (regular) versions of the problems: regular Steiner Forest is known to be polynomially solvable on series-parallel graphs and admits a PTAS on graphs of bounded treewidth. An analogous hardness result can be shown for Euclidian PCSF. This ends the common belief that prize-collecting variants should not add any new hardness to the problems.Prize-collecting problems involve situations where there are various demands that desire to be "served" by some structure and we must find the structure of lowest cost to accomplish this. However, if some of the demands are too expensive to serve, then we can refuse to serve them and instead pay a penalty. In particular, prize-collecting Steiner problems are well-known network design problems with several applications in expanding telecommunications networks (see for example [46,52]), cost sharing, and Lagrangian relaxation techniques (see e.g. [45,21]). A general form of these problems is the Prize-Collecting Steiner Forest (PCSF) problem 1 : given a network (graph) G = (V, E), a set of source-sink pairs 2 D = {{s 1 , t 1 }, {s 2 , t 2 }, . . . , {s k , t k }}, a non-negative cost function c : E → R + , and a non-negative penalty function π : 2 D → R + , our goal is a minimum-cost way of installing (buying) a set of links (edges) and paying the penalty for those pairs which are not connected via installed links. We also consider the problem with a general penalty function called Submodular Prize-Collecting Steiner Forest (SPCSF), in which the penalty function π is a monotone non-negative submodular function 3 of all unsatisfied pairs. In PCSF when all penalties are ∞, the problem is the classic APX-hard Steiner Forest problem, for which the best known approximation ratio is 2 − 2 n (n is the number of nodes of the graph) due to Agrawal, Klein, and Ravi [2] (see also [35] for a more general result and a simpler analysis). The case of Prize-Collecting Steiner Forest problem in which all sinks are identical is the classic (rooted) Prize-Collectin...

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Prize-Collecting Steiner Networks via Iterative Rounding

Cited by 10 publications

References 19 publications

On the Integrality Gap of the Prize-Collecting Steiner Forest LP

On the Integrality Gap of the Prize-Collecting Steiner Forest LP

Prize-collecting steiner network problems

Prize-collecting Steiner Problems on Planar Graphs

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