12,113 research outputs found

    The complexity of finding two disjoint paths with min-max objective function

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    AbstractGiven a network G = (V,E) and two vertices s and t, we consider the problem of finding two disjoint paths from s to t such that the length of the longer path is minimized. The problem has several variants: The paths may be vertex-disjoint or arc-disjoint and the network may be directed or undirected. We show that all four versions as well as some related problems are strongly NP-complete. We also give a pseudo-polynomial-time algorithm for the acyclic directed case

    Robust optimization with incremental recourse

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    In this paper, we consider an adaptive approach to address optimization problems with uncertain cost parameters. Here, the decision maker selects an initial decision, observes the realization of the uncertain cost parameters, and then is permitted to modify the initial decision. We treat the uncertainty using the framework of robust optimization in which uncertain parameters lie within a given set. The decision maker optimizes so as to develop the best cost guarantee in terms of the worst-case analysis. The recourse decision is ``incremental"; that is, the decision maker is permitted to change the initial solution by a small fixed amount. We refer to the resulting problem as the robust incremental problem. We study robust incremental variants of several optimization problems. We show that the robust incremental counterpart of a linear program is itself a linear program if the uncertainty set is polyhedral. Hence, it is solvable in polynomial time. We establish the NP-hardness for robust incremental linear programming for the case of a discrete uncertainty set. We show that the robust incremental shortest path problem is NP-complete when costs are chosen from a polyhedral uncertainty set, even in the case that only one new arc may be added to the initial path. We also address the complexity of several special cases of the robust incremental shortest path problem and the robust incremental minimum spanning tree problem

    Improved Approximation Algorithms for Computing k Disjoint Paths Subject to Two Constraints

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    For a given graph GG with positive integral cost and delay on edges, distinct vertices ss and tt, cost bound CZ+C\in Z^{+} and delay bound DZ+D\in Z^{+}, the kk bi-constraint path (kkBCP) problem is to compute kk disjoint stst-paths subject to CC and DD. This problem is known NP-hard, even when k=1k=1 \cite{garey1979computers}. This paper first gives a simple approximation algorithm with factor-(2,2)(2,2), i.e. the algorithm computes a solution with delay and cost bounded by 2D2*D and 2C2*C respectively. Later, a novel improved approximation algorithm with ratio (1+β,max{2,1+ln1β})(1+\beta,\,\max\{2,\,1+\ln\frac{1}{\beta}\}) is developed by constructing interesting auxiliary graphs and employing the cycle cancellation method. As a consequence, we can obtain a factor-(1.369,2)(1.369,\,2) approximation algorithm by setting 1+ln1β=21+\ln\frac{1}{\beta}=2 and a factor-(1.567,1.567)(1.567,\,1.567) algorithm by setting 1+β=1+ln1β1+\beta=1+\ln\frac{1}{\beta}. Besides, by setting β=0\beta=0, an approximation algorithm with ratio (1,O(lnn))(1,\, O(\ln n)), i.e. an algorithm with only a single factor ratio O(lnn)O(\ln n) on cost, can be immediately obtained. To the best of our knowledge, this is the first non-trivial approximation algorithm for the kkBCP problem that strictly obeys the delay constraint.Comment: 12 page

    Conditional Reliability in Uncertain Graphs

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    Network reliability is a well-studied problem that requires to measure the probability that a target node is reachable from a source node in a probabilistic (or uncertain) graph, i.e., a graph where every edge is assigned a probability of existence. Many approaches and problem variants have been considered in the literature, all assuming that edge-existence probabilities are fixed. Nevertheless, in real-world graphs, edge probabilities typically depend on external conditions. In metabolic networks a protein can be converted into another protein with some probability depending on the presence of certain enzymes. In social influence networks the probability that a tweet of some user will be re-tweeted by her followers depends on whether the tweet contains specific hashtags. In transportation networks the probability that a network segment will work properly or not might depend on external conditions such as weather or time of the day. In this paper we overcome this limitation and focus on conditional reliability, that is assessing reliability when edge-existence probabilities depend on a set of conditions. In particular, we study the problem of determining the k conditions that maximize the reliability between two nodes. We deeply characterize our problem and show that, even employing polynomial-time reliability-estimation methods, it is NP-hard, does not admit any PTAS, and the underlying objective function is non-submodular. We then devise a practical method that targets both accuracy and efficiency. We also study natural generalizations of the problem with multiple source and target nodes. An extensive empirical evaluation on several large, real-life graphs demonstrates effectiveness and scalability of the proposed methods.Comment: 14 pages, 13 figure
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