Martin Luipersbeck

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

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.

A Dual Ascent-Based Branch-and-Bound Framework for the Prize-Collecting Steiner Tree and Related Problems

We present a branch-and-bound (B&B) framework for the asymmetric prize-collecting Steiner tree problem (APCSTP). Several well-known network design problems can be transformed to the APCSTP, including the Steiner tree problem (STP), prize-collecting Steiner tree problem (PCSTP), maximum-weight connected subgraph problem (MWCS), and node-weighted Steiner tree problem (NWSTP). The main component of our framework is a new dual ascent algorithm for the rooted APCSTP, which generalizes Wong’s dual ascent algorithm for the Steiner arborescence problem. The lower bounds and dual information obtained from the algorithm are exploited within powerful bound-based reduction tests and for guiding primal heuristics. The framework is complemented by additional alternative-based reduction tests. Extensive computational results on benchmark instances for the PCSTP, MWCS, and NWSTP indicate the framework’s effectiveness, as most instances from literature are solved to optimality within seconds, including most of the (previo...

A Partition-Based Heuristic for the Steiner Tree Problem in Large Graphs

This paper deals with a new heuristic for the Steiner tree problem (STP) in graphs which aims for the efficient construction of approximate solutions in very large graphs. The algorithm is based on a partitioning approach in which instances are divided into several subinstances that are small enough to be solved to optimality. A heuristic solution of the complete instance can then be constructed through the combination of the subinstances’ solutions. To this end, a new STP-specific partitioning scheme based on the concept of Voronoi diagrams is introduced. This partitioning scheme is then combined with state-of-the-art exact and heuristic methods for the STP. The implemented algorithms are also embedded into a memetic algorithm, which incorporates reduction tests, an algorithm for solution recombination and a variable neighborhood descent that uses best-performing neighborhood structures from the literature. All implemented algorithms are evaluated using previously existing benchmark instances and by using a set of new very large-scale real-world instances. The results show that our approach yields good quality solutions within relatively short time.

A Benders decomposition based framework for solving cable trench problems

Algorithmic framework based on Benders decomposition applicable to different variants of the cable trench problem (CTP).Benders decomposition approach including stabilization and primal heuristics.Computational study addressing three variants of the CTP. In this work, we present an algorithmic framework based on Benders decomposition for the Capacitated p-Cable Trench Problem with Covering. We show that our approach can be applied to most variants of the Cable Trench Problem (CTP) that have been considered in the literature. The proposed algorithm is augmented with a stabilization procedure to accelerate the convergence of the cut loop and with a primal heuristic to derive high-quality primal solutions. Three different variants of the CTP are considered in a computational study which compares the Benders approach with two compact integer linear programming formulations that are solved with CPLEX. The obtained results show that the proposed algorithm significantly outperforms the two compact models and that it can be used to tackle significantly larger instances than previously considered algorithms based on Lagrangean relaxation.

Optimal Upgrading Schemes for Effective Shortest Paths in Networks

In this paper, a generalization of a recently proposed optimal path problem concerning decisions for improving connectivity is considered [see 6]. Each node in the given network is associated with a connection delay which can be reduced by implementing upgrading actions. For each upgrading action a cost must be paid, and the sum must satisfy a budget constraint. Given a fixed budget, the goal is to choose a set of upgrading actions such that the total delay of establishing paths among predefined node pairs is minimized. This model has applications in areas like multicast communication planning and wildlife reserve design.

Gotta (efficiently) catch them all: Pokémon GO meets Orienteering Problems

Abstract In this paper, a new routing problem, referred to as the Generalized Clustered Orienteering Problem (GCOP) , is studied. The problem is motivated by the mobile phone game Pokemon GO , an augmented reality game for mobile devices holding a record-breaking reception: within the first month of its release, more than 100 million users have installed the game on their devices. The game’s immense popularity has spawned several side businesses, including taxi-tours visiting locations where the game can be played, as well as companies offering to play the game for users during times when they cannot. Further applications arise in typical operative transportation problems that seek for tours that are both time-effective and profitable. Besides the typical traveling distances, in the GCOP we also have prizes or revenues associated with the nodes. Additionally, we are given with K node subsets ( clusters ) and a budget B for the length of the tour. The optimization task is to find a tour that maximizes the total collected prize while ensuring that (i) at least one node of each cluster is visited, and (ii) the total distance of the tour does not exceed the budget B . In order to solve the GCOP to optimality, a polynomial-sized Mixed-Integer Linear Programming (MIP) formulation and an exponential-sized MIP formulation are presented. While the first formulation is tackled by a state-of-the-art branch-and-bound (BB moreover, the proposed B&C is further enhanced with valid inequalities, a lifting procedure for strengthening inequalities, as well as initialization and primal heuristics. The computational performance of the proposed approaches is assessed in an extensive computational study, using real-world instances that combine crowd-sourced data associated with the Pokemon GO game with street maps of three European cities, as well as instances derived from the TSPLIB testbed. The obtained results show that the B&C approach (i) largely outperforms the B&B algorithm, and that (ii) it is very effective for providing optimal or nearly-optimal solutions within reasonable running times for both sets of instances.

Decomposition methods for the two-stage stochastic Steiner tree problem

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.

Solving minimum-cost shared arborescence problems

Abstract In this work, we consider the minimum-cost shared Steiner arborescence problem (SStA). In this problem, the goal is to find a minimum-cost subgraph, which is shared among multiple entities and each entity is able to establish a cost-efficient Steiner arborescence. The SStA has been recently used in the literature to establish shared functional modules in protein-protein interaction networks. We propose a cut-based formulation for the problem, and design two exact algorithmic approaches: one based on the separation of connectivity cut inequalities, and the other corresponding to a Benders decomposition of the former model. Both approaches are enhanced by various techniques, including (i) preprocessing, (ii) stabilized cut generation, (iii) primal heuristics, and (iv) cut management. These two algorithmic alternatives are computationally evaluated and compared with a previously proposed flow-based formulation. We illustrate the effectiveness of the algorithms on two types of instances derived from protein-protein interaction networks (available from the previous literature) and from telecommunication access networks.