The traveling salesman problem with distances one and two
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On paths avoiding forbidden pairs of vertices in a graph
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The hardness of approximate optima in lattices, codes, and systems of linear equations
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
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Improved results for directed multicut
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The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Graph decomposition and a greedy algorithm for edge-disjoint paths
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Approximating Directed Multicuts
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Approximating the k-multicut problem
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ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT
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Improved approximation for directed cut problems
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On Two Problems in the Generation of Program Test Paths
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Finding Paths and Cycles of Superpolylogarithmic Length
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Optimal hierarchical decompositions for congestion minimization in networks
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Polynomial flow-cut gaps and hardness of directed cut problems
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Combination Can Be Hard: Approximability of the Unique Coverage Problem
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Hi-index | 5.23 |
In this paper, we consider the checkpoint problem. The input consists of an undirected graph G, a set of source-destination pairs {(s"1,t"1),(s"2,t"2),...,(s"k,t"k)}, and a collection P of paths connecting the (s"i,t"i) pairs. A feasible solution is a multicut E^', namely, a set of edges whose removal disconnects every source-destination pair. For each p@?P we define cp"E"^"'(p)=|p@?E^'|. In the sum checkpoint (SCP) problem the goal is to minimize @?"p"@?"Pcp"E"^"'(p), while in the maximum checkpoint (MCP) problem the goal is to minimize max"p"@?"Pcp"E"^"'(p). These problems have several natural applications, e.g., in urban transportation and network security. In a sense, they combine the multicut problem and the minimum membership set cover problem. For the sum objective we show that weighted SCP is equivalent, with respect to approximability, to undirected multicut. Thus there exists an O(logn) approximation for SCP in general graphs. Our current approximability results for the max objective have a wide gap: we provide an approximation factor of O(nlogn/opt) for MCP and a hardness of 2 under the assumption PNP. The hardness holds for trees. This solves an open problem of Nelson (2009) [25]. We complement the lower bound by an almost matching upper bound with an asymptotic approximation factor of 2. On trees with all s"i,t"i having an ancestor-descendant relation, we give a combinatorial exact algorithm. Besides the algorithm being combinatorial, its running time improves by many orders of magnitude the LP algorithm that follows from total unimodularity. Finally, we show strong hardness for the well-known problem of finding a path with minimum forbidden pairs, which in a sense can be considered the dual to the checkpoint problem. Despite various works on this problem, hardness of approximation was not known prior to this work. We show that the problem cannot be approximated within cn for some constant c0, unless P=NP. This is the strongest type of hardness possible. It carries over to directed acyclic graphs and is a huge improvement over the plain NP-hardness of Gabow [H.N. Gabow, Finding paths and cycles of superpolylogarithmic length, SIAM J. Comput. 36 (6) (2007) 1648-1671].