2,693 research outputs found

    A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface

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    Given a graph GG cellularly embedded on a surface Σ\Sigma of genus gg, a cut graph is a subgraph of GG such that cutting Σ\Sigma along GG yields a topological disk. We provide a fixed parameter tractable approximation scheme for the problem of computing the shortest cut graph, that is, for any ε>0\varepsilon >0, we show how to compute a (1+ε)(1+ \varepsilon) approximation of the shortest cut graph in time f(ε,g)n3f(\varepsilon, g)n^3. Our techniques first rely on the computation of a spanner for the problem using the technique of brick decompositions, to reduce the problem to the case of bounded tree-width. Then, to solve the bounded tree-width case, we introduce a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which may be of independent interest

    A PTAS for Bounded-Capacity Vehicle Routing in Planar Graphs

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    The Capacitated Vehicle Routing problem is to find a minimum-cost set of tours that collectively cover clients in a graph, such that each tour starts and ends at a specified depot and is subject to a capacity bound on the number of clients it can serve. In this paper, we present a polynomial-time approximation scheme (PTAS) for instances in which the input graph is planar and the capacity is bounded. Previously, only a quasipolynomial-time approximation scheme was known for these instances. To obtain this result, we show how to embed planar graphs into bounded-treewidth graphs while preserving, in expectation, the client-to-client distances up to a small additive error proportional to client distances to the depot
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