Markus Leitner

Combining variable neighborhood search with integer linear programming for the generalized minimum spanning tree problem

Abstract We consider the generalized version of the classical Minimum Spanning Tree problem where the nodes of a graph are partitioned into clusters and exactly one node from each cluster must be connected. We present a Variable Neighborhood Search (VNS) approach which uses three different neighborhood types. Two of them work in complementary ways in order to maximize search effectivity. Both are large in the sense that they contain exponentially many candidate solutions, but efficient polynomial-time algorithms are used to identify best neighbors. For the third neighborhood type we apply Mixed Integer Programming to optimize local parts within candidate solution trees. Tests on Euclidean and random instances with up to 1280 nodes indicate especially on instances with many nodes per cluster significant advantages over previously published metaheuristic approaches.

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.

Lagrangian Decomposition, Metaheuristics, and Hybrid Approaches for the Design of the Last Mile in Fiber Optic Networks

We consider a generalization of the (Price Collecting) Steiner Tree Problem on a graph with special redundancy requirements for customer nodes. The problem occurs in the design of the last mile of real-world communication networks. We formulate it as an abstract integer linear program and apply Lagrangian Decomposition to obtain relatively tight lower bounds as well as feasible solutions. Furthermore, a Variable Neighborhood Search and a GRASP approach are described, utilizing a new construction heuristic and special neighborhoods. In particular, hybrids of these methods are also studied and turn out to often perform superior. By comparison to previously published exact methods we show that our approaches are applicable to larger problem instances, while providing high quality solutions together with good lower bounds.

Branch-and-Cut-and-Price for Capacitated Connected Facility Location

We consider a generalization of the Connected Facility Location Problem (ConFL), suitable to model real world network extension scenarios such as fiber-to-the-curb. In addition to choosing a set of facilities and connecting them by a Steiner tree as in ConFL, we aim to maximize the resulting profit by potentially supplying only a subset of all customers. Furthermore, capacity constraints on potential facilities need to be considered. We present two mixed integer programming based approaches which are solved using branch-and-cut and branch-and-cut-and-price, respectively. By studying the corresponding polyhedra we analyze both approaches theoretically and show their advantages over previously presented models. Furthermore, using a computational study we are able to additionally show significant advantages of our models over previously presented ones from a practical point of view.

Overview of Optimization Problems in Electric Car-Sharing System Design and Management

Car-sharing systems are increasingly employing environmentally-friendly electric vehicles. The design and management of Ecar-sharing systems poses several additional challenges with respect to those based on traditional combustion vehicles, mainly related with the limited autonomy allowed by current battery technology. We review the main optimization problems arising in Ecar-sharing systems at strategic, tactical and operational levels, and discuss the existing approaches often developed for similar problems, for example in car-sharing systems with traditional vehicles. We also outline open problems and fruitful research directions.

Self-forming Network Management Topologies in the Madeira Management System

This paper describes a novel approach to network management topologies where multiple customized topologies are self-configured, self-optimized, and maintained automatically by the underlying network of elements. An implementation of these self-forming management topologies as developed in the Celtic European research project Madeira is described. The self-forming topologies use peer-to-peer communication facilities provided by the Madeira platform running on each network element and give a view of the complete network topology, with customization optimised for individual management functionality. Finally, experiences in utilising these topologies are described, highlighting the benefits of this novel approach.

Fault Management based on peer-to-peer paradigms; A case study report from the CELTIC project Madeira

We present an approach to fault management based on an architecture for distributed and collaborative network management as developed in the CELTIC project Madeira. It uses peer-to-peer communication facilities and a logical overlay network facilitating decentralized and iterative alarm processing and correlation. We argue that such an approach might help to overcome key challenges that are posed by NGN scenarios to traditional centralized network management systems. Its feasibility is demonstrated by means of a case study from the area of wireless mesh networks, where an application prototype has been developed.

Stabilized branch-and-price for the rooted delay-constrained steiner tree problem

We consider the rooted delay-constrained Steiner tree problem which arises, e.g., in the design of centralized multicasting networks where quality of service constraints are of concern. We present a mixed integer linear programming formulation based on the concept of feasible paths which has already been considered in the literature for the spanning tree variant. Solving its linear relaxation by column generation has, however, been regarded as computationally not competitive. In this work, we study various possibilities to speed-up the solution of our model by stabilization techniques and embed the column generation procedure in a branch-and-price approach in order to compute proven optimal solutions. Computational results show that the best among the resulting stabilized branch-and-price variants outperforms so-far proposed methods.

Variable Neighborhood Search for a Prize Collecting Capacity Constrained Connected Facility Location Problem

We present a variable neighborhood search approach for a network design problem occurring in real world when the bandwidth of an existing network shall be enhanced. Using two different neighborhood structures we show that a carefully designed combination of a metaheuristic and an exact method based on integer linear programming is able to improve solution quality compared to using heuristic methods only.

Stabilizing Branch-and-Price for Constrained Tree Problems

We consider a rather generic class of network design problems in which a set or subset of given terminal nodes must be connected to a dedicated root node by simple paths and a variety of resource and/or quality of service constraints must be respected. These extensions of the classical Steiner tree problem on a graph can be well modeled by a path formulation in which individual variables are used for all feasible paths. To solve this formulation in practice, branch-and-price is used. It turns out, however, that a naive implementation of column generation suffers strongly from certain degeneracies of the pricing subproblem, leading to excessive running times. After analyzing these computational problems, we propose two methods to accelerate and stabilize column generation by using alternative dual-optimal solutions. The resulting branch-and-price approach is practically tested on the rooted delay-constrained Steiner tree problem and a quota-constrained version of it. Results indicate that the proposed methods in general speed-up the solution process dramatically, far more than a piecewise linear stabilization to which we compare. Furthermore, our branch-and-price approach exhibits on most test instances a better performance than a state-of-the-art branch-and-cut approach based on layered graphs. As the new stabilization technique utilizing alternative dual-optimal solutions is generic in the sense that it easily adapts to the inclusion of a large variety of further constraints and different objective functions, the proposed method is highly promising for a large class of network design problems.