156 research outputs found

    A Linear-Time Algorithm for Integral Multiterminal Flows in Trees

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    In this paper, we study the problem of finding an integral multiflow which maximizes the sum of flow values between every two terminals in an undirected tree with a nonnegative integer edge capacity and a set of terminals. In general, it is known that the flow value of an integral multiflow is bounded by the cut value of a cut-system which consists of disjoint subsets each of which contains exactly one terminal or has an odd cut value, and there exists a pair of an integral multiflow and a cut-system whose flow value and cut value are equal; i.e., a pair of a maximum integral multiflow and a minimum cut. In this paper, we propose an O(n)-time algorithm that finds such a pair of an integral multiflow and a cut-system in a given tree instance with n vertices. This improves the best previous results by a factor of Omega(n). Regarding a given tree in an instance as a rooted tree, we define O(n) rooted tree instances taking each vertex as a root, and establish a recursive formula on maximum integral multiflow values of these instances to design a dynamic programming that computes the maximum integral multiflow values of all O(n) rooted instances in linear time. We can prove that the algorithm implicitly maintains a cut-system so that not only a maximum integral multiflow but also a minimum cut-system can be constructed in linear time for any rooted instance whenever it is necessary. The resulting algorithm is rather compact and succinct

    Parametric min-cuts analysis in a network

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    AbstractThe all pairs minimum cuts problem in a capacitated undirected network is well known. Gomory and Hu showed that the all pairs minimum cuts are revealed by a min-cut tree that can be obtained by solving exactly (n−1) maximum flow problems, where n is the number of nodes in the network.In this paper we consider first the problem of finding parametric min-cuts for a specified pair of nodes when the capacity of an arc i is given by min{bi,λ}, where λ is the parameter, ranging from 0 to ∞. Next we seek the parametric min-cuts for all pairs of nodes, and achieve this by constructing min-cut trees for at most 2m different values of λ, where m is the number of edges in the network

    Two-stage stochastic minimum s − t cut problems: Formulations, complexity and decomposition algorithms

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    We introduce the two‐stage stochastic minimum s − t cut problem. Based on a classical linear 0‐1 programming model for the deterministic minimum s − t cut problem, we provide a mathematical programming formulation for the proposed stochastic extension. We show that its constraint matrix loses the total unimodularity property, however, preserves it if the considered graph is a tree. This fact turns out to be not surprising as we prove that the considered problem is NP-hard in general, but admits a linear time solution algorithm when the graph is a tree. We exploit the special structure of the problem and propose a tailored Benders decomposition algorithm. We evaluate the computational efficiency of this algorithm by solving the Benders dual subproblems as max-flow problems. For many tested instances, we outperform a standard Benders decomposition by two orders of magnitude with the Benders decomposition exploiting the max-flow structure of the subproblems

    A Constant Bound on Throughput Improvement of Multicast Network Coding in Undirected Networks

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    Algorithms for network expansion

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    Maximum Skew-Symmetric Flows and Matchings

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    The maximum integer skew-symmetric flow problem (MSFP) generalizes both the maximum flow and maximum matching problems. It was introduced by Tutte in terms of self-conjugate flows in antisymmetrical digraphs. He showed that for these objects there are natural analogs of classical theoretical results on usual network flows, such as the flow decomposition, augmenting path, and max-flow min-cut theorems. We give unified and shorter proofs for those theoretical results. We then extend to MSFP the shortest augmenting path method of Edmonds and Karp and the blocking flow method of Dinits, obtaining algorithms with similar time bounds in general case. Moreover, in the cases of unit arc capacities and unit ``node capacities'' the blocking skew-symmetric flow algorithm has time bounds similar to those established in Even and Tarjan (1975) and Karzanov (1973) for Dinits' algorithm. In particular, this implies an algorithm for finding a maximum matching in a nonbipartite graph in O(nm)O(\sqrt{n}m) time, which matches the time bound for the algorithm of Micali and Vazirani. Finally, extending a clique compression technique of Feder and Motwani to particular skew-symmetric graphs, we speed up the implied maximum matching algorithm to run in O(nmlog(n2/m)/logn)O(\sqrt{n}m\log(n^2/m)/\log{n}) time, improving the best known bound for dense nonbipartite graphs. Also other theoretical and algorithmic results on skew-symmetric flows and their applications are presented.Comment: 35 pages, 3 figures, to appear in Mathematical Programming, minor stylistic corrections and shortenings to the original versio

    Nonlinear Formations and Improved Randomized Approximation Algorithms for Multiway and Multicut Problems

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    We introduce nonlinear formulations of the multiway cut and multicut problems. By simple linearizations of these formulations we derive several well known formulations and valid inequalities as well as several new ones. Through these formulations we establish a connection between the multiway cut and the maximum weighted independent set problem that leads to the study of the tightness of several LP formulations for the multiway cut problem through the theory of perfect graphs. We also introduce a new randomized rounding argument to study the worst case bound of these formulations, obtaining a new bound of 2a(H)(1 - ) for the multicut problem, where ac(H) is the size of a maximum independent set in the demand graph H
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