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

On a class of strong valid inequalities for the connected matching polytope

We identify a family of $O(|E(G)|^2)$ nontrivial facets of the connected matching polytope of a graph $G$, that is, the convex hull of incidence vectors of matchings in $G$ whose covered vertices induce a connected subgraph. Accompanying software to further inspect the polytope of an input graph is available.Comment: 8 pages, 1 figure. Submitted for publicatio

The matching relaxation for a class of generalized set partitioning problems

This paper introduces a discrete relaxation for the class of combinatorial optimization problems which can be described by a set partitioning formulation under packing constraints. We present two combinatorial relaxations based on computing maximum weighted matchings in suitable graphs. Besides providing dual bounds, the relaxations are also used on a variable reduction technique and a matheuristic. We show how that general method can be tailored to sample applications, and also perform a successful computational evaluation with benchmark instances of a problem in maritime logistics.Comment: 33 pages. A preliminary (4-page) version of this paper was presented at CTW 2016 (Cologne-Twente Workshop on Graphs and Combinatorial Optimization), with proceedings on Electronic Notes in Discrete Mathematic

Fixed cardinality stable sets

Given an undirected graph G=(V,E) and a positive integer k in {1, ..., |V|}, we initiate the combinatorial study of stable sets of cardinality exactly k in G. Our aim is to instigate the polyhedral investigation of the convex hull of fixed cardinality stable sets, inspired by the rich theory on the classical structure of stable sets. We introduce a large class of valid inequalities to the natural integer programming formulation of the problem. We also present simple combinatorial relaxations based on computing maximum weighted matchings, which yield dual bounds towards finding minimum-weight fixed cardinality stable sets, and particular cases which are solvable in polynomial time.publishedVersio

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

Polyhedral results and 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 {ei,ej}∈C, at most one of the edges ei and ej 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 exploit a new relaxation of this problem: fixed cardinality stable sets in the underlying conflict graph H=(E,C). We find interesting properties of the corresponding polytope, and determine stronger dual bounds in a Lagrangean decomposition framework, optimizing over the spanning tree polytope of G and the fixed cardinality stable set polytope of H in the subproblems. This is equivalent to dualizing exponentially many subtour elimination constraints, while limiting the number of multipliers in the dual problem to |E|. It is also a proof of concept for combining Lagrangean relaxation with the power of integer programming solvers over strongly NP-hard subproblems. We present encouraging computational results using a dual method that comprises the Volume Algorithm, initialized with multipliers determined by Lagrangean dual-ascent. In particular, the bound is within 5.5% of the optimum in 146 out of 200 benchmark instances; it actually matches the optimum in 75 cases. All of the implementation is made available in a free, open-source repository.publishedVersio

Fixed cardinality stable sets

Given an undirected graph G=(V,E) and a positive integer k in {1, ..., |V|}, we initiate the combinatorial study of stable sets of cardinality exactly k in G. Our aim is to instigate the polyhedral investigation of the convex hull of fixed cardinality stable sets, inspired by the rich theory on the classical structure of stable sets. We introduce a large class of valid inequalities to the natural integer programming formulation of the problem. We also present simple combinatorial relaxations based on computing maximum weighted matchings, which yield dual bounds towards finding minimum-weight fixed cardinality stable sets, and particular cases which are solvable in polynomial time

The unsuitable neighbourhood inequalities for the fixed cardinality stable set polytope

Given an undirected graph G = (V, E) and an integer k∈{1,…,|V|} , we initiate the combinatorial study of stable sets of cardinality exactly k in G. Our aim is to instigate the polyhedral investigation of the convex hull of fixed cardinality stable sets, and we begin by introducing a large class of valid inequalities to the natural integer programming formulation of the problem