MPC

MPC 2017, ISSUE 2



Mathematical Programming Computation, Volume 9, Issue 2, June 2017

Dijkstra meets Steiner: a fast exact goal-oriented Steiner tree algorithm

Stefan Hougardy, Jannik Silvanus, Jens Vygen

We present a new exact algorithm for the Steiner tree problem in edge-weighted graphs. Our algorithm improves the classical dynamic programming approach by Dreyfus and Wagner. We achieve a significantly better practical performance via pruning and future costs, a generalization of a well-known concept to speed up shortest path computations. Our algorithm matches the best known worst-case run time and has a fast, often superior, practical performance: on some large instances originating from VLSI design, previous best run times are improved upon by orders of magnitudes. We are also able to solve larger instances of the d-dimensional rectilinear Steiner tree problem for d?{3,4,5}d?{3,4,5}, whose Hanan grids contain up to several millions of edges.

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Mathematical Programming Computation, Volume 9, Issue 2, June 2017

Thinning out Steiner trees: a node-based model for uniform edge costs

Matteo Fischetti, Markus Leitner, Ivana Ljubic, Martin Luipersbeck, Michele Monaci, Max Resch, Domenico Salvagnin, Markus Sinnl

The Steiner tree problem is a challenging NP-hard problem. Many hard instances of this problem are publicly available, that are still unsolved by state-of-the-art branch-and-cut codes. A typical strategy to attack these instances is to enrich the polyhedral description of the problem, and/or to implement more and more sophisticated separation procedures and branching strategies. In this paper we investigate the opposite viewpoint, and try to make the solution method as simple as possible while working on the modeling side. Our working hypothesis is that the extreme hardness of some classes of instances mainly comes from over-modeling, and that some instances can become quite easy to solve when a simpler model is considered. In other words, we aim at “thinning out” the usual models for the sake of getting a more agile framework. In particular, we focus on a model that only involves node variables, which is rather appealing for the “uniform” cases where all edges have the same cost. In our computational study, we first show that this new model allows one to quickly produce very good (sometimes proven optimal) solutions for notoriously hard instances from the literature. In some cases, our approach takes just few seconds to prove optimality for instances never solved (even after days of computation) by the standard methods. Moreover, we report improved solutions for several SteinLib instances, including the (in)famous hypercube ones. We also demonstrate how to build a unified solver on top of the new node-based model and the previous state-of-the-art model (defined in the space of arc and node variables). The solver relies on local branching, initialization heuristics, preprocessing and local search procedures. A filtering mechanism is applied to automatically select the best algorithmic ingredients for each instance individually. The presented solver is the winner of the DIMACS Challenge on Steiner trees in most of the considered categories.

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Mathematical Programming Computation, Volume 9, Issue 2, June 2017

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

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

The Steiner tree problem in graphs is a classical problem that commonly arises in practical applications as one of many variants. While often a strong relationship between different Steiner tree problem variants can be observed, solution approaches employed so far have been prevalently problem-specific. In contrast, this paper introduces a general-purpose solver that can be used to solve both the classical Steiner tree problem and many of its variants without modification. This versatility is achieved by transforming various problem variants into a general form and solving them by using a state-of-the-art MIP-framework. The result is a high-performance solver that can be employed in massively parallel environments and is capable of solving previously unsolved instances.

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Mathematical Programming Computation, Volume 9, Issue 2, June 2017

Swap-vertex based neighborhood for Steiner tree problems

Zhang-Hua Fu, Jin-Kao Hao

Steiner tree problems (STPs) are very important in both theory and practice. In this paper, we introduce a powerful swap-vertex move operator which can be used as a basic element of any neighborhood search heuristic to solve many STP variants. Given the incumbent solution tree T, the swap-vertex move operator exchanges a vertex in T with another vertex out of T, and then attempts to construct a minimum spanning tree, leading to a neighboring solution (if feasible). We develop a series of dynamic data structures, which allow us to efficiently evaluate the feasibility of swap-vertex moves. Additionally, in order to discriminate different swap-vertex moves corresponding to the same objective value, we also develop an auxiliary evaluation function. We present a computational assessment based on a number of challenging problem instances (corresponding to three representative STP variants) which clearly shows the effectiveness of the techniques introduced in this paper. Particularly, as a key element of our KTS algorithm which participated in the 11th DIMACS implementation challenge, the swap-vertex operator as well as the auxiliary evaluation function contributed significantly to the excellent performance of our algorithm.

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