Towards stronger Lagrangean bounds for stable spanning trees

Given a graph G=(V,E) and a set C of unordered pairs of edges regarded as being in conflict, a stable spanning tree in G is a set of edges T inducing a spanning tree in G, such that for each {e_i, e_j} in C, at most one of the edges e_i and e_j is in T. The existing work on Lagrangean algorithms to the NP-hard problem of finding minimum weight stable spanning trees is limited to relaxations with the integrality property. We have recently initiated the combinatorial and polyhedral study of fixed cardinality stable sets [see https://doi.org/10.1016/j.dam.2021.01. 019], which motivates a new formulation for stable spanning trees based on Lagrangean Decomposition. By optimizing over the spanning tree polytope of G and the fixed cardinality stable set polytope of the conflict graph H=(E,C) in the subproblems, we are able to determine stronger Lagrangean bounds (equivalent to dualizing exponentially-many subtour elimination constraints), while limiting the number of multipliers in the dual problem to |E|. This naturally asks for more sophisticated dual algorithms, requiring the fewest iterations possible, and we derive a collection of Lagrangean dual ascent directions to this end.publishedVersio

A genetic-inspired joint multicast routing and channel assignment algorithm in wireless mesh networks

Copyright @ 2008 IEEEThis paper proposes a genetic algorithm (GA) based optimization approach to search a minimum-interference multicast tree which satis¯es the end-to-end delay constraint and optimizes the usage of the scarce radio network resource in wireless mesh networks. The path-oriented en- coding method is used and each chromosome is represented by a tree data structure (i.e., a set of paths). Since we expect the multicast trees on which the minimum-interference channel assignment can be produced, a fitness function that returns the total channel conflict is devised. Crossover and mutation are well designed to adapt to the tree structure. A simple yet effective channel assignment algorithm is proposed to reduce the channel conflict. Simulation results show that the proposed GA based multicast algorithm achieves better performance in terms of both the total channel conflict and the tree cost than that of a well known algorithm

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

Joint QoS multicast routing and channel assignment in multiradio multichannel wireless mesh networks using intelligent computational methods

Copyright @ 2010 Elsevier B.V. All rights reserved.In this paper, the quality of service multicast routing and channel assignment (QoS-MRCA) problem is investigated. It is proved to be a NP-hard problem. Previous work separates the multicast tree construction from the channel assignment. Therefore they bear severe drawback, that is, channel assignment cannot work well with the determined multicast tree. In this paper, we integrate them together and solve it by intelligent computational methods. First, we develop a unified framework which consists of the problem formulation, the solution representation, the fitness function, and the channel assignment algorithm. Then, we propose three separate algorithms based on three representative intelligent computational methods (i.e., genetic algorithm, simulated annealing, and tabu search). These three algorithms aim to search minimum-interference multicast trees which also satisfy the end-to-end delay constraint and optimize the usage of the scarce radio network resource in wireless mesh networks. To achieve this goal, the optimization techniques based on state of the art genetic algorithm and the techniques to control the annealing process and the tabu search procedure are well developed separately. Simulation results show that the proposed three intelligent computational methods based multicast algorithms all achieve better performance in terms of both the total channel conflict and the tree cost than those comparative references.This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) of UK under Grant EP/E060722/1