The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NP-complete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in high-speed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.

The Steiner packing problem is to find the maximum number of edge-disjoint subgraphs of a given graph G that connect a given set of required points S. This problem is motivated by practical applications in VLSI-layout and broadcasting, as well as theoretical reasons. In this paper, we study this problem and present an algorithm with an asymptotic approximation factor of |S|/4. This gives a sufficient condition for the existence of k edge-disjoint Steiner trees in a graph in terms of the edge-connectivity of the graph. We will show that this condition is the best possible if the number of terminals is 3. At the end, we consider the fractional version of this problem, and observe that it can be reduced to the minimum Steiner tree problem via the ellipsoid algorithm.

Web Usage Mining is the application of data mining techniques to Web clickstream data in order to extract usage patterns. As Web sites continue to grow in size and complexity, the results of Web Usage Mining have become critical for a number of applications such as Web site design, business and marketing decision support, personalization, usability studies, and network trac analysis. The two major challenges involved in Web Usage Mining are preprocessing the raw data to provide an accurate picture of how a site is being used, and ltering the results of the various data mining algorithms in order to present only the rules and patterns that are potentially interesting. This thesis develops and tests an architecture and algorithms for performing Web Usage Mining. An evidence combination framework referred to as the information lter is developed to compare and combine usage, content, and structure information about a Web site. The information lter automatically identi es the discovered ...

The max-flow min-cut theorem of Ford and Fulkerson is based on an even more foundational result, namely Menger's theorem on graph connectivity. Menger's theorem provides a good characterization for the following single-source disjoint paths problem: given a graph G, with a source vertex s and terminals t 1 , ..., t k , decide whether there exist edge-disjoint s-t i paths, for i = 1, ..., k. We consider a natural, NP-hard generalization of this problem, which we call the single-source unsplittable flow problem. We are given a source and terminals as before; but now each terminal t i has a demand ae i 1, and each edge e of G has a capacity c e 1. The problem is to decide whether one can choose a single s-t i path, for each i, so that the resulting set of paths respects the capacity constraints --- the total amount of demand routed across any edge e must be bounded by the capacity c e . The main results of this paper are constant-factor approximation algorithms for three n...

We present an approximation algorithm for solving graph problems in which a low-cost set of edges must be selected that has certain vertex-connectivity properties. In the survivable network design problem, one is given a value r ij for each pair of vertices i and j, and must find a minimum-cost set of edges such that there are r ij vertex-disjoint paths between vertices i and j. In the case for which r ij 2 f0; 1; 2g for all i; j, we can find a solution of cost no more than 3 times the optimal cost in polynomial time. In the case in which r ij = k for all i; j, we can find a solution of cost no more than 2H(k) times optimal, where H(n) = 1 + 1 2 + \Delta \Delta \Delta + 1 n . No approximation algorithms were previously known for these problems. Our algorithms rely on a primal-dual approach which has recently led to approximation algorithms for many edge-connectivity problems. 1 Introduction Let G = (V; E) be an undirected graph with non-negative costs c e 0 on all edges e 2 E. In...

Many NP-hard graph problems can be solved in polynomial time for graphs with bounded treewidth. Equivalent results are known for pathwidth and branchwidth. In recent years, several studies have shown that this result is not only of theoretical interest but can successfully be applied to find (almost) optimal solutions or lower bounds for many optimization problems. To apply a tree decomposition approach, the treewidth of the graph has to be determined, independently of the application at hand. Although for fixed k, linear time algorithms exist to solve the decision problem “treewidth < k”, their practical use is very limited. The computational tractability of treewidth has been rarely studied so far. In this paper, we compare four heuristics and two lower bounds for instances from applications such as the frequency assignment problem and the vertex coloring problem. Three of the heuristics are based on well-known algorithms to recognize triangulated graphs. The fourth heuristic recursively improves a tree decomposition by the computation of minimal separating vertex sets in subgraphs. Lower bounds can be computed from maximal cliques and the minimum degree of induced subgraphs. A computational analysis shows that the treewidth of several graphs can be identified by these methods. For other graphs, however, more sophisticated techniques are necessary.

Many network problems are based on fundamental relationships involving time. Consider, for example, the problems of modeling the flow of information through a distributed network, studying the spread of a disease through a population, or analyzing the reachability properties of an airline timetable. In such settings, a natural model is that of a graph in which each edge is annotated with a time label specifying the time at which its endpoints “communicated. ” We will call such a graph a temporal network. To model the notion that information in such a network “flows ” only on paths whose labels respect the ordering of time, we call a path time-respecting if the time labels on its edges are non-decreasing. The central motivation for our work is the following question: how do the basic combinatorial and algorithmic properties of graphs change when we impose this additional temporal condition? The notion of a path is intrinsic to many of the most fundamental algorithmic problems on graphs; spanning trees, connectivity, flows, and cuts are some examples. When we focus on time-respecting paths in place of arbitrary paths, many of these problems acquire a character that is different from the

Given an undirected multigraph G and a subset of vertices S ` V (G), the STEINER TREE PACKING problem is to find a largest collection of edge-disjoint trees that each connects S. This problem and its generalizations have attracted considerable attention from researchers in different areas because of their wide applicability. This problem was shown to be APX-hard (no polynomial time approximation scheme unless P=NP). In fact, prior to this paper, not evenan approximation algorithm with asymptotic ratio o(n) wasknown despite several attempts. In this work, we close this huge gap by presenting the first polynomial time constant factor approximation algorithm for the STEINER TREE PACKING problem. The maintheorem is an approximate min-max relation between the maximum number of edge-disjoint trees that each connects S (i.e. S-trees) and the minimum size of an edge-cut thatdisconnects some pair of vertices in S (i.e. S-cut). Specifically, we prove that if the minimum S-cut in G has 26k edges, then G has at least k edge-disjoint S-trees; this answers Kriesell's conjecture affirmatively up to a constant multiple. The techniques that we use are purely combinatorial, where matroid theory is the underlying ground work.

Network coding is an elegant and novel technique introduced at the turn of the millennium to improve network throughput and performance. It is expected to be a critical technology for networks of the future. This tutorial addresses the first most natural questions one would ask about this new technique: how network coding works and what are its benefits, how network codes are designed and how much it costs to deploy networks implementing such codes, and finally, whether there are methods to deal with cycles and delay that are present in all real networks. A companion issue deals primarily with applications of network coding. 1

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