Abstract not found

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.

Recent work in network coding shows that, it is necessary to consider both the routing and coding strategies to achieve optimal throughput of information transmission in data networks. So far, most research on network coding has focused on the model of directed networks, where each communication link has a fixed direction. In this paper, we study the benefits of network coding in undirected networks, where each communication link is bidirectional. Our theoretical results show that, for a single unicast or broadcast session, there are no improvements with respect to throughput due to network coding. In the case of a single multicast session, such an improvement is bounded by a factor of two, as long as half integer routing is permitted. This is dramatically different from previous results obtained in directed networks. We also show that multicast throughput in an undirected network is independent of the selection of the sender within the multicast group. We finally show that, rather than improving the optimal achievable throughput, the benefit of network coding is to significantly facilitate the design of efficient algorithms to compute and achieve such optimal throughput. I.

We present a branch-and-cut algorithm to solve capacitated network design problems. Given a capacitated network and point-to-point traffic demands, the objective is to install more capacity on the edges of the network and route traffic simultaneously, so that the overall cost is minimized. We study a mixed-integer programming formulation of the problem and identify some new facet defining inequalities. These inequalities, together with other known combinatorial and mixed-integer rounding inequalities, are used as cutting planes. To choose the branching variable, we use a new rule called "knapsack branching". We also report on our computational experience using real-life data.

Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph together with its ends. Our approach

By applying the matroid partition theorem of J. Edmonds [1] to a hypergraphic generalization of graphic matroids, due to M. Lorea [3], we obtain a generalization of Tutte's disjoint trees theorem for hypergraphs. As a corollary, we prove for positive integers k and q that every (kq)-edge-connected hypergraph of rank q can be decomposed into k connected sub-hypergraphs, a well-known result for q=2. Another by-product is a connectivity-type sufficient condition for the existence of k edge-disjoint Steiner trees in a bipartite graph.

Submodular functions and related polyhedra play an increasing role in combinatorial optimization. The present survey-type paper is intended to provide a brief account of this theory along with several applications in graph theory and combinatorial optimization.

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.

The transmission of information within a data network is constrained by network topology and link capacities. In this paper, we study the fundamental upper bound of information multicast rates with these constraints, given the unique replicable and encodable property of information flows. Based on recent information theory advances in coded multicast rates, we are able to formulate the maximum multicast rate problem as a linear network optimization problem, assuming the general undirected network model. We then proceed to apply Lagrangian relaxation techniques to obtain (1) a necessary and sufficient condition for multicast rate feasibility, and (2) a subgradient solution for computing the maximum rate and the optimal routing strategy to achieve it. The condition we give is a generalization of the well-known conditions for the unicast and broadcast cases. Our subgradient solution takes advantage of the underlying network flow structure of the problem, and therefore outperforms general linear programming solving techniques. It also admits a natural intuitive interpretation, and is amenable to fully distributed implementations.

In this paper, we examine the problem of large-volume data dissemination via overlay networks. A natural way to maximize the throughput of an overlay multicast session is to split the traffic and feed them into multiple trees. While in single-tree solutions, bandwidth of leaf nodes may remain largely under-utilized, multi-tree solutions increase the chances for a node to contribute its bandwidth by being a relaying node in at least one of the trees. We study the following problems: (1) What is the maximum capacity multi-tree solutions can exploit from overlay networks? (2) When multiple sessions compete within the same network, what is the relationship of two contradictory goals: achieving fairness and maximizing overall throughput? (3) What is the impact of IP routing in achieving at constraining the optimal performance of overlay multicast? We extend the multicommodity flow model to the case of overlay data dissemination, where each commodity is associated with an overlay session, rather than the traditional source-destination pair. We first prove that the problem is solvable in polynomial time, then propose an #-approximation algorithm, assuming that each commodity can be split in arbitrary ways. The solution to this problem establishes the theoretical upper bound of overall throughput that any multi-tree solution could reach. We then study the same problem with the restriction that each commodity can only be split and fed into a limited number of trees. A randomized rounding algorithm and an online tree-construction algorithm are presented. All these algorithms are evaluated by extensive simulations.