An integer linear programming approach to the steiner problem in graphs

Aneja, Y. P.

doi:10.1002/net.3230100207

Cited by 125 publications

(65 citation statements)

References 9 publications

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“…Of particular importance, we show that ifx is integral, then (4c) and (6) can efficiently be separated exactly, which ensures correctness of a branch-and-cut algorithm for formulations (4) and (15), respectively.…”

Section: Separationmentioning

confidence: 83%

Branch-and-cut approaches for chance-constrained formulations of reliable network design problems

Song

Luedtke

2013

Math. Prog. Comp.

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We study solution approaches for the design of reliably connected networks. Specifically, given a network with arcs that may fail at random, the goal is to select a minimum cost subset of arcs such the probability that a connectivity requirement is satisfied is at least 1− , where is a risk tolerance. We consider two types of connectivity requirements. We first study the problem of requiring an s-t path to exist with high probability in a directed graph. Then we consider undirected graphs, where we require the graph to be fully connected with high probability. We model each problem as a stochastic integer program with a joint chance constraint, and present two formulations that can be solved by a branch-and-cut algorithm. The first formulation uses binary variables to represent whether or not the connectivity requirement is satisfied in each scenario of arc failures and is based on inequalities derived from graph cuts in individual scenarios. We derive additional valid inequalities for this formulation and study their facetinducing properties. The second formulation is based on probabilistic graph cuts, an extension of graph cuts to graphs with random arc failures. Inequalities corresponding to probabilistic graph cuts are sufficient to define the set of feasible solutions and can be separated efficiently at integer solutions, allowing this formulation to be solved by a branch-and-cut algorithm. Computational results demonstrate that the approaches can effectively solve instances on large graphs with many failure scenarios. In addition, we demonstrate that, by varying the risk tolerance, our model yields a rich set of solutions on the efficient frontier of cost and reliability.

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Section: Separationmentioning

confidence: 83%

Branch-and-cut approaches for chance-constrained formulations of reliable network design problems

Song

Luedtke

2013

Math. Prog. Comp.

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“…For example, the Steiner tree problem --the problem of connecting at minimum cost a subset S of terminals possibly using some Steiner vertices in V~S ~ can be formulated as (IPo(ls)) where (ls)ij= 1 if ideS and 0 otherwise (or, (IS)i= 1 ifieSand 0 otherwise). The formulation (IPo(ls)) is known as the set covering formulation [2 ]. When r~j= k for all ide Vwe obtain the minimum-cost k-edge-conected network design problem.…”

Section: Eeg~(s) (Ij)cs(s)mentioning

confidence: 99%

Survivable networks, linear programming relaxations and the parsimonious property

Goemans

Bertsimas

1993

Mathematical Programming

152

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We consider the survivable network design problem --the problem of designing, at minimum cost, a network with edge-connectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and the k-edge-connected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worst-case analyses of two heuristics for the survivable network design problem.

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“…The obvious question left open by [12], that we answer affirmatively in this paper The algorithms in [1,9] are based on the classical undirected cut formulation for Steiner forests [2]. The integrality gap of this relaxation is known to be (2 − 1/k) and the results in [1,9] are therefore tight.…”

Section: Introductionmentioning

confidence: 69%

“…It therefore follows that t must be the responsible terminal forS and henceS ∈ S t . Finally, notice that δ (S) = δ (S) and hence we can increase yS and decrease y S at the same rate without violating any of the constraints of type (2). Continuing this procedure will lead to a symmetric dual y that is feasible for (D).…”

Section: This Implies That Optmentioning

confidence: 97%

From Primal-Dual to Cost Shares and Back: A Stronger LP Relaxation for the Steiner Forest Problem

Könemann

Leonardi

Schäfer

et al. 2005

Automata, Languages and Programming

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In this paper we consider a game theoretical variant of the Steiner forest problem. An instance of this game consists of an undirected graph G = (V, E), non-negative costs c(e) for all edges e in E, and k players. Each player i has an associated pair of terminals s i and t i . Consider a forest F in G. We say that player i is serviced if s i and t i are connected in F. Player i derives a private utility u i for receiving service. In a recent paper, Könemann, Leonardi, and Schäfer [12] showed that a natural primal-dual algorithm, KLS, gives rise to a 2-approximate budget balanced and group-strategyproof cost sharing method for the above game.In this paper we show that the techniques used in [12] yield a new linear programming relaxation for the Steiner forest problem: the lifted-cut relaxation. First, we give an alternate proof of the approximate budget-balance result in [12] by showing that the cost shares computed by algorithm KLS are feasible for the dual of this relaxation. Second, we are able to show that this new undirected relaxation for Steiner forests is strictly stronger than the well-studied undirected cut relaxation.We conclude the paper with a negative result, arguing that no cross-monotonic cost sharing method can achieve a budget balance factor of less than 2 for the Steiner tree and Steiner forest games. This shows that the results of [11,12] are essentially tight.

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An integer linear programming approach to the steiner problem in graphs

Cited by 125 publications

References 9 publications

Branch-and-cut approaches for chance-constrained formulations of reliable network design problems

Branch-and-cut approaches for chance-constrained formulations of reliable network design problems

Survivable networks, linear programming relaxations and the parsimonious property

From Primal-Dual to Cost Shares and Back: A Stronger LP Relaxation for the Steiner Forest Problem

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