SCIP-Jack—a solver for STP and variants with parallelization extensions

Gamrath, Gerald; Koch, Thorsten; Maher, Stephen J.; Rehfeldt, Daniel; Shinano, Yuji

doi:10.1007/s12532-016-0114-x

Cited by 47 publications

(81 citation statements)

References 29 publications

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“…Each scenario k ∈ K has probability p k ∈ (0, 1], k∈K p k = 1, as well as second-stage edge costs c k : E → R ≥0 and terminals T k ⊆ V, r ∈ T k . The objective is to select first-stage edges E 0 S ⊆ E and second-stage edges E k S ⊆ E for each k ∈ K such that the subgraph induced by E 0 S ∪ E k S , G[E 0 S ∪ E k S ], connects T k and the expected cost Our contribution For the deterministic STP a wealth of theoretical results [6,11,20,29] and empirically successful computational techniques are known [8,10,31]. However, as noted in [2,27], the generalization of results from the STP to the SSTP is not straightforward.…”

Section: Definition 1 (Stochastic Steiner Tree Problem (Sstp))mentioning

confidence: 99%

“…This requires O(|K |(|A|+|V | log |V |)). In order to apply conditions (10) and (11), integrality is relaxed and the so-called Dantzig bound is used. The resulting LP can be solved in O(|K | log |K |) by choosing elements in ascending order based on their utility ratioD k e /( p k c k e ) for all k ∈ K .…”

Section: Proposition 2 Given a Lower Bound Lbmentioning

confidence: 99%

“…In order to benefit from these tests (that are valid on undirected graphs) second-stage arcs are only fixed to zero if both z k i j and z k ji can be fixed to zero. The following definitions are required: The concept of the bottleneck Steiner distance has been first considered in [7], and allows the formulation of an effective edge elimination test, which has been employed in several successful algorithms for the STP [8,10,31]. Using these definitions, the following two rules for fixing variables are stated for the SSTP and formulation (SDC FB 3 ).…”

Section: Data: Sstp Instance I = (G = (V E) R C P T)mentioning

confidence: 99%

“…The colors and line styles are chosen according to the configurations applied in the previous Sections (see Figs. 8,9,10). Note that a time limit of 1 h is applied.…”

Section: Effects Of the Benders Decomposition Approachmentioning

confidence: 99%

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Decomposition methods for the two-stage stochastic Steiner tree problem

et al. 2017

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A new algorithmic approach for solving the stochastic Steiner tree problem based on three procedures for computing lower bounds (dual ascent, Lagrangian relaxation, Benders decomposition) is introduced. Our method is derived from a new integer linear programming formulation, which is shown to be strongest among all known formulations. The resulting method, which relies on an interplay of the dual information retrieved from the respective dual procedures, computes upper and lower bounds and combines them with several rules for fixing variables in order to decrease the size of problem instances. The effectiveness of our method is compared in an extensive computational study with the state-of-the-art exact approach, which employs a Benders decomposition based on two-stage branch-and-cut, and a genetic algorithm introduced during the DIMACS implementation challenge on Steiner trees. Our results indicate that the presented method significantly outperforms existing ones, both on benchmark instances from literature, as well as on large-scale telecommunication networks.

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Section: Definition 1 (Stochastic Steiner Tree Problem (Sstp))mentioning

confidence: 99%

Section: Proposition 2 Given a Lower Bound Lbmentioning

confidence: 99%

Section: Data: Sstp Instance I = (G = (V E) R C P T)mentioning

confidence: 99%

“…The colors and line styles are chosen according to the configurations applied in the previous Sections (see Figs. 8,9,10). Note that a time limit of 1 h is applied.…”

Section: Effects Of the Benders Decomposition Approachmentioning

confidence: 99%

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Decomposition methods for the two-stage stochastic Steiner tree problem

et al. 2017

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“…The Steiner tree problem in graphs (SPG) is a classical  -hard problem [22]: Given an undirected connected graph G = (V, E), costs c : E → Q + and a set T ⊆ V of terminals, the problem is to find a minimum-cost tree S ⊆ G that spans T. Although commonly cited to entail a variety of practical applications [11,18,31,32,36], the SPG rarely arises in pristine shape when it comes to modeling real-world problems [17]. Instead, one predominantly encounters variations of the classical Steiner tree problem.…”

Section: Introductionmentioning

confidence: 99%

Reduction techniques for the prize collecting Steiner tree problem and the maximum‐weight connected subgraph problem

2018

Self Cite

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The concept of reduction has frequently distinguished itself as a pivotal ingredient of exact solving approaches for the Steiner tree problem in graphs. In this article we broaden the focus and consider reduction techniques for three Steiner problem variants that have been extensively discussed in the literature and entail various practical applications: The prize-collecting Steiner tree problem, the rooted prize-collecting Steiner tree problem and the maximum-weight connected subgraph problem. By introducing and subsequently deploying numerous new reduction methods, we are able to drastically decrease the size of a large number of benchmark instances, already solving more than 90% of them to optimality. Furthermore, we demonstrate the impact of these techniques on exact solving, using the example of the state-of-the-art Steiner problem solver SCIP-JACK.Proof. First, note that both of the minima attained in (7) and (8) correspond to a path. Next, let t (1)min ) is minimal. The corresponding path between t i and t (1) min contains an edge {v j , v k }∈ (N(t i )) and since the

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Stronger path‐based extended formulation for the Steiner tree problem

Filipecki

Vyve

2019

Networks

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The Steiner tree problem (STP) is a classical NP-hard combinatorial optimization problem with applications in computational biology and network wiring. The objective of this problem is to find a minimum cost subgraph of a given undirected graph G with edge costs, that spans a subset of vertices called terminals. We present currently used linear programming formulations of the problem based on two different approaches-the bidirected cut relaxation (BCR) and the hypergraphic formulations (HYP), the former offering better computational performance, and the latter better bounds on the integrality gap. As our contribution, we propose a new hierarchy of path-based extended formulations for STP. We show that this hierarchy provides better integrality gaps on graph instances used to define the worst-case lower bounds on the integrality gap for both BCR and HYP. We prove that each consecutive level of our hierarchy is at least as strong as the previous one. Additionally, we also present numerical results showing that several difficult STP instances can be solved to integer optimality by using this hierarchy. Our approach can be adapted to variants of STP or applied to hypergraphic formulations for further potential improvement on the integrality gap bounds, in exchange for additional computational effort. KEYWORDS extended formulation, flow-based formulation, integrality gap, linear programming relaxation, mixed-integer linear programming, Steiner tree Networks. 2020;75:3-17.wileyonlinelibrary.com/journal/net

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SCIP-Jack—a solver for STP and variants with parallelization extensions

Cited by 47 publications

References 29 publications

Decomposition methods for the two-stage stochastic Steiner tree problem

Decomposition methods for the two-stage stochastic Steiner tree problem

Reduction techniques for the prize collecting Steiner tree problem and the maximum‐weight connected subgraph problem

Stronger path‐based extended formulation for the Steiner tree problem

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