Min-max Relations in Combinatorial Optimization

Author(s)

Marcel Kenji de Carli Silva

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2007-04-04

Examining board members

Yoshiko Wakabayashi; Paulo Feofiloff; Cláudio Leonardo Lucchesi

• Advisor: Yoshiko Wakabayashi

Abstract

Min-max relations are central objects in combinatorial optimization. They basically state that, in a given structure, the optimum value of a certain minimization problem equals the optimum value of a different, maximization problem. Relations of this kind provide good characterizations and polyhedral descriptions to several important problems and, moreover, they often come with efficient algorithms for the corresponding problems. Usually, such efficient algorithms are obtained naturally from the constructive proofs involved; even when that is not the case, these relations reveal enough of the combinatorial structure of the problem, leading to the development of efficient algorithms. The main focus of this dissertation is the study of these relations in graphs. Our emphasis is on directed graphs. We present Edmonds and Giles\' powerful polyhedral framework concerning submodular flows, as well as Frank\'s algorithm for a special case of this framework: the Lucchesi-Younger Theorem. We also derive several min-max relations about packing connectors, starting with Edmonds\' Disjoint Branchings Theorem and ending with Feofiloff-Younger and Schrijver\'s Disjoint Dijoins Theorem. We further derive some classical min-max relations on matchings, T-joins and S-paths. To this end, we use a theorem due to Frank, Tardos, and Sebö and a general framework due to Chudnovsky, Geelen, Gerards, Goddyn, Lohman, and Seymour. Throughout the text, we illustrate several recurrent themes, such as the use of tools from polyhedral combinatorics, the uncrossing technique, the use of submodular functions, matroids and exchange properties, as well as some results involving forbidden substructures. (AU)