Exact algorithms for spanner problems in graphs

Author(s)

Hugo Vinicius Vaz Braga

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2018-12-14

Examining board members

Yoshiko Wakabayashi; Carlos Eduardo Ferreira; Karla Roberta Pereira Sampaio Lima; Cláudio Nogueira de Meneses; Eduardo Cândido Xavier

• Advisor: Yoshiko Wakabayashi

Abstract

Let (G, w, t) be a triplet consisting of a connected graph G = (V, E) with a nonnegative weight function w defined on E, and a real number t >= 1. A t-spanner of G is a spanning subgraph H of G such that for each pair of vertices u,v, the distance between u and v in H is at most t times the distance between u and v in G. If H is a tree then we call it a tree t-spanner of G. We address the Minimum Weight Tree Spanner Problem (MWTSP), defined as follows. Given a triplet (G, w, t), find a minimum weight tree t-spanner in G. It is known that MWTSP is NP-hard for every fixed t > 1. We propose two ILP formulations for MWTSP, based on arborescences, of polynomial size in the size of G. The formulation that has variables representing distances between pairs of vertices has shown to be better in practice. We also address the Minimum Weight t-Spanner Problem (MWSP) that has the same input as MWTSP and seeks for a minimum weight t-spanner in G. Even for unweighted graphs, it is known that MWSP is NP-hard for every fixed t >= 2. We approach this problem in two ways. We propose an ILP formulation for MWSP that has an an exponential number of restrictions but we show that the separation problem for the corresponding relaxed linear program can be solved in polynomial time in the size of G. We also present a branch-and-price algorithm for MWSP based on an ILP formulation proposed by Sigurd and Zachariasen (2004). We show results on the computational experiments with both formulations for MWTSP and also with the branch-and-price algorithm that we implemented for MWSP. (AU)

• FAPESP's process: 13/22875-9 - Graph Spanners
• Grantee: Hugo Vinicius Vaz Braga
• Support Opportunities: Scholarships in Brazil - Doctorate