# Algorithms for hub allocation and related problems

## par Shih-shun Kao

Projet de thèse en Informatique

Sous la direction de Ralf Klasing et de Sun-Yuan Hsieh.

Thèses en préparation à Bordeaux en cotutelle avec National Cheng Kung University , dans le cadre de École doctorale de mathématiques et informatique (Talence, Gironde) , en partenariat avec Laboratoire bordelais de recherche en informatique (laboratoire) et de Combinatoire et algorithmiques (equipe de recherche) depuis le 04-10-2019 .

• ## Résumé

Hub allocation problems have various applications in transportation and telecommunication systems. However, many of them are NP-hard, even in metric graphs (which model large classes of networks deployed in many real-life scenarios). For example, the star p-hub center problem is NP-hard in metric graphs. The goal of this thesis is to design efficient algorithms and to study the hardness of important hub allocation and related problems in the class of β-metric graphs (i.e. a well-known class of graphs generalizing metric graphs). More precisely, we say a metric graph G = (V, E, w) is called β-metric graph for some β ≥ 1/2 if the distance function w(·, ·) satisfies the β-triangle inequality, i.e., for all u, v, x ∈ V, w(u, v) ≤ β · (w(u, x) + w(v, x)). For each 1/2 ≤ β < 1, it defines a sub-class of metric graphs. An interesting question is whether the considered problem is still NP-hard when the input graph belongs to a subclass of metric graphs. In our preliminary results on solving the star p-hub center problem, we obtained that if β is small, it is possible to solve the problem optimally in polynomial time. Moreover, if β is larger than some specific constant, we are able to design approximation algorithms to solve it. In this thesis, we want to extend those preliminary fruitful results to other important hub allocation and related problems, including e.g.: (i) the star p-hub center problem with routing cost optimization in β-metric graphs; (ii) the facility location problem in β-metric graphs; (iii) the Steiner tree problem in β-metric graphs. The goal of this thesis is to show that for some β, the above problems can be solved optimally in polynomial time, or to show that when β is greater than some constant, the problem is NP-hard to approximate to within a function of β and to design polynomial time approximation algorithms for them.

Algorithms for hub allocation and related problems

• ## Résumé

Hub allocation problems have various applications in transportation and telecommunication systems. However, many of them are NP-hard, even in metric graphs (which model large classes of networks deployed in many real-life scenarios). For example, the star p-hub center problem is NP-hard in metric graphs. The goal of this thesis is to design efficient algorithms and to study the hardness of important hub allocation and related problems in the class of β-metric graphs (i.e. a well-known class of graphs generalizing metric graphs). More precisely, we say a metric graph G = (V, E, w) is called β-metric graph for some β ≥ 1/2 if the distance function w(·, ·) satisfies the β-triangle inequality, i.e., for all u, v, x ∈ V, w(u, v) ≤ β · (w(u, x) + w(v, x)). For each 1/2 ≤ β < 1, it defines a sub-class of metric graphs. An interesting question is whether the considered problem is still NP-hard when the input graph belongs to a subclass of metric graphs. In our preliminary results on solving the star p-hub center problem, we obtained that if β is small, it is possible to solve the problem optimally in polynomial time. Moreover, if β is larger than some specific constant, we are able to design approximation algorithms to solve it. In this thesis, we want to extend those preliminary fruitful results to other important hub allocation and related problems, including e.g.: (i) the star p-hub center problem with routing cost optimization in β-metric graphs; (ii) the facility location problem in β-metric graphs; (iii) the Steiner tree problem in β-metric graphs. The goal of this thesis is to show that for some β, the above problems can be solved optimally in polynomial time, or to show that when β is greater than some constant, the problem is NP-hard to approximate to within a function of β and to design polynomial time approximation algorithms for them.