22 research outputs found

    A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection

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    Given an undirected graph with edge costs and node weights, the minimum bisection problem asks for a partition of the nodes into two parts of equal weight such that the sum of edge costs between the parts is minimized. We give a polynomial time bicriteria approximation scheme for bisection on planar graphs. Specifically, let WW be the total weight of all nodes in a planar graph GG. For any constant Δ>0\varepsilon > 0, our algorithm outputs a bipartition of the nodes such that each part weighs at most W/2+ΔW/2 + \varepsilon and the total cost of edges crossing the partition is at most (1+Δ)(1+\varepsilon) times the total cost of the optimal bisection. The previously best known approximation for planar minimum bisection, even with unit node weights, was O(log⁥n)O(\log n). Our algorithm actually solves a more general problem where the input may include a target weight for the smaller side of the bipartition.Comment: To appear in STOC 201

    Travelling on Graphs with Small Highway Dimension

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    We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP naturally occur for various applications in logistics. It was previously shown [Feldmann et al. ICALP 2015] that these problems admit a quasi-polynomial time approximation scheme (QPTAS) on graphs of constant highway dimension. We demonstrate that a significant improvement is possible in the special case when the highway dimension is 1, for which we present a fully-polynomial time approximation scheme (FPTAS). We also prove that STP is weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015]

    Shortest paths in linear time on minor-closed graph classes, with an application to Steiner tree approximation

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    AbstractWe generalize the linear-time shortest-paths algorithm for planar graphs with nonnegative edge-weights of Henzinger et al. (1994) to work for any proper minor-closed class of graphs. We argue that their algorithm can not be adapted by standard methods to all proper minor-closed classes. By using recent deep results in graph minor theory, we show how to construct an appropriate recursive division in linear time for any graph excluding a fixed minor and how to transform the graph and its division afterwards, so that it has maximum degree three. Based on such a division, the original framework of Henzinger et al. can be applied. Afterwards, we show that using this algorithm, one can implement Mehlhorn’s (1988) 2-approximation algorithm for the Steiner tree problem in linear time on these graph classes
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