Kruskal-Based Approximation Algorithm for the Multi-Level Steiner Tree Problem

We study the multi-level Steiner tree problem: a generalization of the Steiner tree problem in graphs where terminals $T$ require varying priority, level, or quality of service. In this problem, we seek to find a minimum cost tree containing edges of varying rates such that any two terminals $u$, $v$ with priorities $P(u)$, $P(v)$ are connected using edges of rate $\min\{P(u),P(v)\}$ or better. The case where edge costs are proportional to their rate is approximable to within a constant factor of the optimal solution. For the more general case of non-proportional costs, this problem is hard to approximate with ratio $c \log \log n$, where $n$ is the number of vertices in the graph. A simple greedy algorithm by Charikar et al., however, provides a $\min\{2(\ln |T|+1), \ell \rho\}$-approximation in this setting, where $\rho$ is an approximation ratio for a heuristic solver for the Steiner tree problem and $\ell$ is the number of priorities or levels (Byrka et al. give a Steiner tree algorithm with $\rho\approx 1.39$, for example). In this paper, we describe a natural generalization to the multi-level case of the classical (single-level) Steiner tree approximation algorithm based on Kruskal's minimum spanning tree algorithm. We prove that this algorithm achieves an approximation ratio at least as good as Charikar et al., and experimentally performs better with respect to the optimum solution. We develop an integer linear programming formulation to compute an exact solution for the multi-level Steiner tree problem with non-proportional edge costs and use it to evaluate the performance of our algorithm on both random graphs and multi-level instances derived from SteinLib

Clustering with shallow trees

We propose a new method for hierarchical clustering based on the optimisation of a cost function over trees of limited depth, and we derive a message--passing method that allows to solve it efficiently. The method and algorithm can be interpreted as a natural interpolation between two well-known approaches, namely single linkage and the recently presented Affinity Propagation. We analyze with this general scheme three biological/medical structured datasets (human population based on genetic information, proteins based on sequences and verbal autopsies) and show that the interpolation technique provides new insight.Comment: 11 pages, 7 figure

Hypergraphic LP Relaxations for Steiner Trees

We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number of variables could be exponential, the number of positive variables is at most the number of terminals. Relations with other relaxations: We show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Integrality gap upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap of these hypergraph relaxations in general graphs. In the special case of uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~ 1.216. By our equivalence theorem, the latter result implies an improved upper bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010

Performance improvement of an optical network providing services based on multicast

Operators of networks covering large areas are confronted with demands from some of their customers who are virtual service providers. These providers may call for the connectivity service which fulfils the specificity of their services, for instance a multicast transition with allocated bandwidth. On the other hand, network operators want to make profit by trading the connectivity service of requested quality to their customers and to limit their infrastructure investments (or do not invest anything at all). We focus on circuit switching optical networks and work on repetitive multicast demands whose source and destinations are {\em \`a priori} known by an operator. He may therefore have corresponding trees "ready to be allocated" and adapt his network infrastructure according to these recurrent transmissions. This adjustment consists in setting available branching routers in the selected nodes of a predefined tree. The branching nodes are opto-electronic nodes which are able to duplicate data and retransmit it in several directions. These nodes are, however, more expensive and more energy consuming than transparent ones. In this paper we are interested in the choice of nodes of a multicast tree where the limited number of branching routers should be located in order to minimize the amount of required bandwidth. After formally stating the problem we solve it by proposing a polynomial algorithm whose optimality we prove. We perform exhaustive computations to show an operator gain obtained by using our algorithm. These computations are made for different methods of the multicast tree construction. We conclude by giving dimensioning guidelines and outline our further work.Comment: 16 pages, 13 figures, extended version from Conference ISCIS 201

Spanning Trees With Edge Conflicts and Wireless Connectivity

We introduce the problem of finding a spanning tree along with a partition of the tree edges into fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we seek to model the irregularities seen in actual wireless environments. Not all node pairs may be able to communicate, even if geographically close - thus, the available pairs are specified with a link graph {L}=(V,E). Also, signal attenuation need not follow a nice geometric formula - hence, interference is modeled by a conflict (hyper)graph {C}=(E,F) on the links. The objective is to maximize the efficiency of the communication, or equivalently, to minimize the length of a schedule of the tree edges in the form of a coloring. We find that in spite of all this generality, the problem can be approximated linearly in terms of a versatile parameter, the inductive independence of the interference graph. Specifically, we give a simple algorithm that attains a O(rho log n)-approximation, where n is the number of nodes and rho is the inductive independence, and show that near-linear dependence on rho is also necessary. We also treat an extension to Steiner trees, modeling multicasting, and obtain a comparable result. Our results suggest that several canonical assumptions of geometry, regularity and "niceness" in wireless settings can sometimes be relaxed without a significant hit in algorithm performance

A 3/2-approximation algorithm for some minimum-cost graph problems

International audienceWe consider a class of graph problems introduced in a paper of Goemans and Williamson that involve finding forests of minimum edge cost. This class includes a number of location/routing problems; it also includes a problem in which we are given as input a parameter k, and want to find a forest such that each component has at least k vertices. Goemans and Williamson gave a 2-approximation algorithm for this class of problems. We give an improved 3/2-approximation algorithm