Near-linear Time Algorithm for Approximate Minimum Degree Spanning Trees

Given a graph $G = (V, E)$, we wish to compute a spanning tree whose maximum vertex degree, i.e. tree degree, is as small as possible. Computing the exact optimal solution is known to be NP-hard, since it generalizes the Hamiltonian path problem. For the approximation version of this problem, a $\tilde{O}(mn)$ time algorithm that computes a spanning tree of degree at most $\Delta^* +1$ is previously known [F\"urer \& Raghavachari 1994]; here $\Delta^*$ denotes the minimum tree degree of all the spanning trees. In this paper we give the first near-linear time approximation algorithm for this problem. Specifically speaking, we propose an $\tilde{O}(\frac{1}{\epsilon^7}m)$ time algorithm that computes a spanning tree with tree degree $(1+\epsilon)\Delta^* + O(\frac{1}{\epsilon^2}\log n)$ for any constant $\epsilon \in (0,\frac{1}{6})$. Thus, when $\Delta^*=\omega(\log n)$, we can achieve approximate solutions with constant approximate ratio arbitrarily close to 1 in near-linear time.Comment: 17 page

Approximating the Degree-Bounded Minimum Diameter Spanning Tree Problem

We consider the problem of finding a minimum diameter spanning treewith maximum node degree $B$ in a complete undirected edge-weightedgraph. We provide an $O(sqrt{log_Bn})$-approximation algorithm for theproblem. Our algorithm is purely combinatorial, and relies on acombination of filtering and divide and conquer.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41348/1/453_2004_Article_1121.pd

A differentiated quality of service oriented multimedia multicast protocol

Les systèmes de communication multimédia modernes aspirent à fournir de nouveaux services tels que des communications multipoints. Néanmoins, l'apparition de dispositifs multimédias très diversifiés et le nombre croissant de clients ont révélé de nouveaux besoins pour les mécanismes et les protocoles. Dans une communication multimédia, les flux présentent des contraintes différentes et la QdS requise pour chaque flux n'est pas la même. De plus, dans une communication multipoint, tous les utilisateurs ne peuvent pas ou ne sont pas capables de recevoir la même QdS ; cette contrainte implique que les nouveaux mécanismes de communication doivent prendre en compte les besoins des utilisateurs pour fournir un service adéquat à chaque utilisateur, surtout pour éviter le gaspillage des ressources réseau. Cette thèse propose une architecture multipoint à QdS différentiée appelée M-FPTP. Basée sur des proxies client/serveur, elle relie plusieurs LANs multipoints à travers des liens point-à-point partiellement fiables. Cette architecture fournit une QdS différente à chaque LAN dépendant des besoins des utilisateurs. Pour ce faire, nous proposons un modèle du réseau appelé Arbre Hiérarchisé (AH) qui représente en même temps les performances du réseau et les contraintes de QdS des utilisateurs. Nonobstant, l'application de méthodes standard pour la création d'arbres sur un AH peut conduire à des problèmes de surcharge du degré de sortie dans la source. Pour résoudre ce problème, nous proposons alors un nouvel algorithme appelé Arbre de Plus Courts Chemins à Degré de Sortie Limité. Le déploiement de ce service nécessite, pour gérer les utilisateurs et le déploiement correct des proxies, un nouveau protocole appelé Protocole Simple de Session pour QdS multipoint. L'ensemble des solutions proposées a été modélisé, vérifié, validé et testé en utilisant UML 2.0 et l'outil TAU G2. ABSTRACT : Modern multimedia (MM) communication systems aim to provide new services such as multicast (MC) communication. But the rising of new very different MM capable devices and the growing number of clients drive to new requirements for mechanisms and protocols. In a MM communication, there are some flows that have constraints different from others and the required QoS for each flow is not the same. Furthermore, in MC communications, all the users do not want or are not able to receive the same QoS. These constraints imply that new communication mechanisms have to take into account the user requirements in order to provide an ad hoc service to each user and to avoid wasting the network resources. This dissertation proposes a new differentiated QoS multicast architecture, based on client/server proxies, called M-FPTP, which relays many MC LANs by single partially reliable links. This architecture provides a different QoS to each LAN depending on the users requirements. For doing so, it is also provided a network model called Hierarchized Graph (HG) which represents at the same time the network performances and the users QoS constraints. Nevertheless, the application of standard tree creation methods on an HG can lead to source overloading problems. It is then proposed a new algorithm called Degree-Bounded Shortest-Path-Tree (DgB-SPT) which solves this problem. However, the deployment of such a service needs a new protocol in order to collect users requirements and correctly deploy the proxies. This protocol is called Simple Session Protocol for QoS MC (SSP-QoM). The proposed solutions have been modeled, verified, validated and tested by using UML 2.0 and TAU G2 CASE tool

Minimum Crossing Problems on Graphs

This thesis will address several problems in discrete optimization. These problems are considered hard to solve. However, good approximation algorithms for these problems may be helpful in approximating problems in computational biology and computer science. Given an undirected graph G=(V,E) and a family of subsets of vertices S, the minimum crossing spanning tree is a spanning tree where the maximum number of edges crossing any single set in S is minimized, where an edge crosses a set if it has exactly one endpoint in the set. This thesis will present two algorithms for special cases of minimum crossing spanning trees. The first algorithm is for the case where the sets of S are pairwise disjoint. It gives a spanning tree with the maximum crossing of a set being 2OPT+2, where OPT is the maximum crossing for a minimum crossing spanning tree. The second algorithm is for the case where the sets of S form a laminar family. Let b_i be a bound for each S_i in S. If there exists a spanning tree where each set S_i is crossed at most b_i times, the algorithm finds a spanning tree where each set S_i is crossed O(b_i log n) times. From this algorithm, one can get a spanning tree with maximum crossing O(OPT log n). Given an undirected graph G=(V,E), and a family of subsets of vertices S, the minimum crossing perfect matching is a perfect matching where the maximum number of edges crossing any set in S is minimized. A proof will be presented showing that finding a minimum crossing perfect matching is NP-hard, even when the graph is bipartite and the sets of S are pairwise disjoint

The directed minimum-degree spanning tree problem

Abstract. Consider a directed graph G =(V,E) with n vertices and a root vertex r ∈ V. The DMDST problem for G is one of constructing a spanning tree rooted at r, whose maximal degree is the smallest among all such spanning trees. The problem is known to be NP-hard. A quasipolynomial time approximation algorithm for this problem is presented. The algorithm finds a spanning tree whose maximal degree is at most O( ∆ ∗ + log n) where, ∆ ∗ is the degree of some optimal tree for the problem. The running time of the algorithm is shown to be O(n O(log n)). Experimental results are presented showing that the actual running time of the algorithm is much smaller in practice.