Polyhedra and algorithms for problems bridging notions of connectivity and independence

I denne avhandlinga interesserer vi oss for å finne delgrafer som svarer til utvalgte modeller for begrepene sammenheng og uavhengighet. I korthet betyr dette stabile (også kalt uavhengige) mengder med gitt kardinalitet, stabile (også kalt konfliktfrie) spenntrær og pardannelser (eller uavhengige kantmengder) som induserer en sammenhengende delgraf. Dette er kombinatoriske strukturer som kan generaliseres til ulike modeller for nettverksdesign innen telekommunikasjon og forsyningsvirksomhet, plassering av anlegg, fylogenetikk, og mange andre applikasjoner innen operasjonsanalyse og optimering. Vi argumenterer for at de valgte strukturene reiser interessante forskningsspørsmål, og vi bidrar med forbedret matematisk forståelse av selve strukturene, samt forbedrede algoritmer for å takle de tilhørende kombinatoriske optimeringsproblemene. Med det mener vi metoder for å identifisere en optimal struktur, forutsatt at elementene som danner dem (hjørner eller kanter i en gitt graf) er tildelt verdier. Forskninga vår omfatter ulike områder innenfor algoritmer, kombinatorikk og optimering. De fleste resultatene omhandler det å finne bedre beskrivelser av de geometriske strukturene (nemlig 0/1-polytoper) som representerer alle mulige løsninger for hvert av problemene. Slike forbedrede beskrivelser oversettes til lineære ulikheter i heltallsprogrammeringsmodeller, noe som igjen gir mer effektive beregningsresultater når man løser referanseinstanser av hvert problem. Vi påpeker gjentatte ganger betydninga av å dele kildekoden til implementasjonen av alle utviklede algoritmer og verktøy når det foreslås nye modeller og løsningsmetoder for heltallsprogrammering og kombinatorisk optimering. Kodearkivene våre inkluderer fullstendige implementasjoner, utformet med effektivitet og modulær design i tankene, og fremmer dermed gjenbruk, videre forskning og nye anvendelser innen forskning og utvikling.We are interested in finding subgraphs that capture selected models of connectivity and independence. In short: fixed cardinality stable (or independent) sets, stable (or conflict-free) spanning trees, and matchings (or independent edge sets) inducing a connected subgraph. These are combinatorial structures that can be generalized to a number of models across network design in telecommunication and utilities, facility location, phylogenetics, among many other application domains of operations research and optimization. We argue that the selected structures raise appealing research questions, and seek to contribute with improved mathematical understanding of the structures themselves, as well as improved algorithms to face the corresponding combinatorial optimization problems. That is, methods to identify an optimal structure, assuming the elements that form them (vertices or edges in a given graph) have a weight. Our research spans different lines within algorithmics, combinatorics and optimization. Most of the results concern finding better descriptions of the geometric structures (namely, 0/1-polytopes) that represent all feasible solutions to each of the problems. Such improved descriptions translate to linear inequalities in integer programming formulations which, in turn, provide stronger computational results when solving benchmark instances of each problem. We repeatedly remark the importance of sharing an open-source implementation of all algorithms and tools developed when proposing new models and solution methods in integer programming and combinatorial optimization. Our code repositories include full implementations, crafted with efficiency and modular design in mind, thus fostering reuse, further research and new applications in research and development.Doktorgradsavhandlin

Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory. Published July 201

Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

The Minimum Flow Cost Hamiltonian Tour Problem

In this thesis we introduce the minimum flow cost Hamiltonian tour problem(FCHT). Given a graph and positive flow between pairs of vertices, the FCHT consists of �finding a Hamiltonian cycle that minimizes the total cost for sending flows between pairs of vertices thorough the shortest path on the cycle. We prove that the FCHT belongs to the class of NP-hard problems and study the polyhedral structure of its set of feasible solutions. In particular, we present �five di�different MIP formulations which are theoretically and computationally compared. We also develop some approximate and exact solution procedures to solve the FCHT. We present a combinatorial bound and two heuristic procedures: a greedy deterministic method and a greedy randomized adaptive search procedure. Finally, a branch-and-cut algorithm is also proposed to solve the problem exactly

An exact mathematical programming approach to multiple RNA sequence-structure alignment

One of the main tasks in computational biology is the computation of alignments of genomic sequences to reveal their commonalities. In case of DNA or protein sequences, sequence information alone is usually sufficient to compute reliable alignments. RNA molecules, however, build spatial conformations—the secondary structure—that are more conserved than the actual sequence. Hence, computing reliable alignments of RNA molecules has to take into account the secondary structure. We present a novel framework for the computation of exact multiple sequence-structure alignments: We give a graph- theoretic representation of the sequence-structure alignment problem and phrase it as an integer linear program. We identify a class of constraints that make the problem easier to solve and relax the original integer linear program in a Lagrangian manner. Experiments on a recently published benchmark show that our algorithms has a comparable performance than more costly dynamic programming algorithms, and outperforms all other approaches in terms of solution quality with an increasing number of input sequences

Combinatorial optimization with one quadratic term

Diese Arbeit befasst sich mit einer neuen Herangehensweise für binäre kombinatorische Optimierungsprobleme. Die wesentliche Idee hierbei ist, die Anzahl der quadratischen Terme in der Zielfunktion auf einen einzigen zu beschränken, und das durch eine Linearisierung entstehende Polyeder zu analysieren. Für diesen Ansatz gibt es mehrere Motivationsgründe. Im Allgemeinen ist das ursprüngliche Problem mit beliebig vielen quadratischen Termen NP-schwer. Doch obwohl eine gute polyedrische Beschreibung mit schnellen Separierungsroutinen die Optimierung in einem Branch-and-Cut-Verfahren signifikant beschleunigen könnte, gibt es bislang nur wenige Erkenntnisse zur polyedrischen Struktur des binären quadratischen Optimierungsproblems. Betrachtet man das reduzierte Problem mit einem quadratischen Term, dann ist eine effiziente Optimierung möglich, falls die zugrundeliegende lineare Version effizient lösbar ist. Somit können hier auch die facettendefinierenden Ungleichungen effizient separiert werden. Darüberhinaus bleiben alle zulässigen Ungleichungen des reduzierten Problems zulässig für das ursprüngliche Problem. In Kombination bedeutet dies, dass Erkenntnisse zur Facettenstruktur des Problems mit einem quadratischen Term direkt zu einer verbesserten polyedrische Beschreibung des Ursprungsproblems führen. Für eine praktische Anwendung dieses theoretischen Ansatzes betrachten wir verschiedene konkrete Optimierungsprobleme mit einem quadratischen Term und analysieren deren jeweilige polyedrische Struktur, die sich nach der Linearisierung ergibt. Konkret betrachten wir das Minimale Spannwald- und das Minimale Spannbaumproblem, das Minimale Branching- und das Minimale Arboreszenzproblem, das Minimale Assignmentproblem und das Maximale Matchingproblem. Für jedes dieser Optimierungsprobleme leiten wir neue Klassen von facettendefinierenden Ungleichungen her. Außerdem präsentieren wir für das Minimale Spannwald- und das Minimale Spannbaumproblem eine vollständige Beschreibung der jeweiligen Polytope. Für die verwandten gerichteten Probleme, das Minimale Branching- und das Minimale Arboreszenzproblem, zeigen wir zwar einerseits einige Gemeinsamkeiten mit den ungerichteten Problemen, andererseits aber auch, dass sich die polyedrischen Strukturen im gerichteten Fall durch die zusätzlichen Gradbedingungen deutlich verkomplizieren. Bei der Untersuchung des Minimalen Assignmentproblems mit einem quadratischen Term stellen wir nicht nur die Vermutung über die vollständige polyedrische Beschreibung auf, sondern kommen insbesondere zu der überraschenden Erkenntnis, dass bereits ein einziger quadratischer Term genügen kann, um die Anzahl der Facetten von polynomiell auf exponentiell zu erhöhen. Die größte Vielfalt an Facettenklassen leiten wir für das Polyeder des Maximalen Matchingproblems mit einem quadratischen Term her. Wir zeigen jedoch auch, dass diese noch nicht genügen, um die vollständige Beschreibung des Polyeders zu erhalten. Da die meisten der hergeleiteten Facettenklassen von exponentieller Größe sind, leiten wir verschiedene Routinen für eine polynomielle Separierung her. Unsere exemplarischen Rechenergebnisse für das quadratische Minimale Spannwald- und das quadratische Minimale Spannbaumproblem zeigen die praktische Relevanz unseres Ansatzes

Mathematical Modelling for Load Balancing and Minimization of Coordination Losses in Multirobot Stations

The automotive industry is moving from mass production towards an individualized production, in order to improve product quality and reduce costs and material waste. This thesis concerns aspects of load balancing of industrial robots in the automotive manufacturing industry, considering efficient algorithms required by an individualized production. The goal of the load balancing problem is to improve the equipment utilization. Several approaches for solving the load balancing problem are presented along with details on mathematical tools and subroutines employed.Our contributions to the solution of the load balancing problem are manifold. First, to circumvent robot coordination we have constructed disjoint robot programs, which require no coordination schemes, are more flexible, admit competitive cycle times for some industrial instances, and may be preferred in an individualized production. Second, since solving the task assignment problem for generating the disjoint robot programs was found to be unreasonably time-consuming, we modelled it as a generalized unrelated parallel machine problem with set packing constraints and suggested a tighter model formulation, which was proven to be much more tractable for a branch--and--cut solver. Third, within continuous collision detection it needs to be determined whether the sweeps of multiple moving robots are disjoint. Our solution uses the maximum velocity of each robot along with distance computations at certain robot configurations to derive a function that provides lower bounds on the minimum distance between the sweeps. The lower bounding function is iteratively minimized and updated with new distance information; our method is substantially faster than previously developed methods

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM