3,381 research outputs found

    Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach

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    In this paper, we study the kk-forest problem in the model of resource augmentation. In the kk-forest problem, given an edge-weighted graph G(V,E)G(V,E), a parameter kk, and a set of mm demand pairs V×V\subseteq V \times V, the objective is to construct a minimum-cost subgraph that connects at least kk demands. The problem is hard to approximate---the best-known approximation ratio is O(min{n,k})O(\min\{\sqrt{n}, \sqrt{k}\}). Furthermore, kk-forest is as hard to approximate as the notoriously-hard densest kk-subgraph problem. While the kk-forest problem is hard to approximate in the worst-case, we show that with the use of resource augmentation, we can efficiently approximate it up to a constant factor. First, we restate the problem in terms of the number of demands that are {\em not} connected. In particular, the objective of the kk-forest problem can be viewed as to remove at most mkm-k demands and find a minimum-cost subgraph that connects the remaining demands. We use this perspective of the problem to explain the performance of our algorithm (in terms of the augmentation) in a more intuitive way. Specifically, we present a polynomial-time algorithm for the kk-forest problem that, for every ϵ>0\epsilon>0, removes at most mkm-k demands and has cost no more than O(1/ϵ2)O(1/\epsilon^{2}) times the cost of an optimal algorithm that removes at most (1ϵ)(mk)(1-\epsilon)(m-k) demands

    The cavity approach for Steiner trees packing problems

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    The Belief Propagation approximation, or cavity method, has been recently applied to several combinatorial optimization problems in its zero-temperature implementation, the max-sum algorithm. In particular, recent developments to solve the edge-disjoint paths problem and the prize-collecting Steiner tree problem on graphs have shown remarkable results for several classes of graphs and for benchmark instances. Here we propose a generalization of these techniques for two variants of the Steiner trees packing problem where multiple "interacting" trees have to be sought within a given graph. Depending on the interaction among trees we distinguish the vertex-disjoint Steiner trees problem, where trees cannot share nodes, from the edge-disjoint Steiner trees problem, where edges cannot be shared by trees but nodes can be members of multiple trees. Several practical problems of huge interest in network design can be mapped into these two variants, for instance, the physical design of Very Large Scale Integration (VLSI) chips. The formalism described here relies on two components edge-variables that allows us to formulate a massage-passing algorithm for the V-DStP and two algorithms for the E-DStP differing in the scaling of the computational time with respect to some relevant parameters. We will show that one of the two formalisms used for the edge-disjoint variant allow us to map the max-sum update equations into a weighted maximum matching problem over proper bipartite graphs. We developed a heuristic procedure based on the max-sum equations that shows excellent performance in synthetic networks (in particular outperforming standard multi-step greedy procedures by large margins) and on large benchmark instances of VLSI for which the optimal solution is known, on which the algorithm found the optimum in two cases and the gap to optimality was never larger than 4 %

    Lagrangian Relaxation and Partial Cover

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    Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover.Comment: 20 pages, extended abstract appeared in STACS 200
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