12 research outputs found

    On the fine-grained complexity of rainbow coloring

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    The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in kk colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any k≄2k\ge 2, there is no algorithm for Rainbow k-Coloring running in time 2o(n3/2)2^{o(n^{3/2})}, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set SS of pairs of vertices and we ask if there is a coloring in which all the pairs in SS are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by ∣S∣|S|. We also study Maximum Rainbow k-Coloring problem, where we are additionally given an integer qq and we ask if there is a coloring in which at least qq anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by qq and has a kernel of size O(q)O(q) for every k≄2k\ge 2 (thus proving that the problem is FPT), extending the result of Ananth et al. [FSTTCS 2011]

    Linear kernels for outbranching problems in sparse digraphs

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    In the kk-Leaf Out-Branching and kk-Internal Out-Branching problems we are given a directed graph DD with a designated root rr and a nonnegative integer kk. The question is to determine the existence of an outbranching rooted at rr that has at least kk leaves, or at least kk internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with O(k2)O(k^2) vertices are known on general graphs. In this work we show that kk-Leaf Out-Branching admits a kernel with O(k)O(k) vertices on H\mathcal{H}-minor-free graphs, for any fixed family of graphs H\mathcal{H}, whereas kk-Internal Out-Branching admits a kernel with O(k)O(k) vertices on any graph class of bounded expansion.Comment: Extended abstract accepted for IPEC'15, 27 page

    Approximation and Parameterized Complexity of Minimax Approval Voting

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    We present three results on the complexity of Minimax Approval Voting. First, we study Minimax Approval Voting parameterized by the Hamming distance dd from the solution to the votes. We show Minimax Approval Voting admits no algorithm running in time O⋆(2o(dlog⁥d))\mathcal{O}^\star(2^{o(d\log d)}), unless the Exponential Time Hypothesis (ETH) fails. This means that the O⋆(d2d)\mathcal{O}^\star(d^{2d}) algorithm of Misra et al. [AAMAS 2015] is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time O⋆((3/Ï”)2d)\mathcal{O}^\star(\left({3}/{\epsilon}\right)^{2d}), which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for Minimax Approval Voting, which runs in time nO(1/Ï”2⋅log⁥(1/Ï”))⋅poly(m)n^{\mathcal{O}(1/\epsilon^2 \cdot \log(1/\epsilon))} \cdot \mathrm{poly}(m), almost matching the running time of the fastest known PTAS for Closest String due to Ma and Sun [SIAM J. Comp. 2009].Comment: 14 pages, 3 figures, 2 pseudocode

    On Directed Feedback Vertex Set parameterized by treewidth

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    We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time 2o(tlog⁡t)⋅nO(1)2^{o(t\log t)}\cdot n^{\mathcal{O}(1)} on general directed graphs, where tt is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time 2O(tlog⁡t)⋅nO(1)2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}. On the other hand, we show that if the input digraph is planar, then the running time can be improved to 2O(t)⋅nO(1)2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}

    Tight Lower Bound for the Channel Assignment Problem

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    On Directed Feedback Vertex Set parameterized by treewidth

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    We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time 2o(tlog⁡t)⋅nO(1)2^{o(t\log t)}\cdot n^{\mathcal{O}(1)} on general directed graphs, where tt is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time 2O(tlog⁡t)⋅nO(1)2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}. On the other hand, we show that if the input digraph is planar, then the running time can be improved to 2O(t)⋅nO(1)2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}

    On Directed Feedback Vertex Set Parameterized by Treewidth.

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    We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time 2 o ( t log t )· n O ( 1 ) on general directed graphs, where t is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time 2 O ( t log t )· n O ( 1 ). On the other hand, we show that if the input digraph is planar, then the running time can be improved to 2 O ( t )· n O ( 1 )
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