8,315 research outputs found

    Exploring Subexponential Parameterized Complexity of Completion Problems

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    Let F{\cal F} be a family of graphs. In the F{\cal F}-Completion problem, we are given a graph GG and an integer kk as input, and asked whether at most kk edges can be added to GG so that the resulting graph does not contain a graph from F{\cal F} as an induced subgraph. It appeared recently that special cases of F{\cal F}-Completion, the problem of completing into a chordal graph known as Minimum Fill-in, corresponding to the case of F={C4,C5,C6,}{\cal F}=\{C_4,C_5,C_6,\ldots\}, and the problem of completing into a split graph, i.e., the case of F={C4,2K2,C5}{\cal F}=\{C_4, 2K_2, C_5\}, are solvable in parameterized subexponential time 2O(klogk)nO(1)2^{O(\sqrt{k}\log{k})}n^{O(1)}. The exploration of this phenomenon is the main motivation for our research on F{\cal F}-Completion. In this paper we prove that completions into several well studied classes of graphs without long induced cycles also admit parameterized subexponential time algorithms by showing that: - The problem Trivially Perfect Completion is solvable in parameterized subexponential time 2O(klogk)nO(1)2^{O(\sqrt{k}\log{k})}n^{O(1)}, that is F{\cal F}-Completion for F={C4,P4}{\cal F} =\{C_4, P_4\}, a cycle and a path on four vertices. - The problems known in the literature as Pseudosplit Completion, the case where F={2K2,C4}{\cal F} = \{2K_2, C_4\}, and Threshold Completion, where F={2K2,P4,C4}{\cal F} = \{2K_2, P_4, C_4\}, are also solvable in time 2O(klogk)nO(1)2^{O(\sqrt{k}\log{k})} n^{O(1)}. We complement our algorithms for F{\cal F}-Completion with the following lower bounds: - For F={2K2}{\cal F} = \{2K_2\}, F={C4}{\cal F} = \{C_4\}, F={P4}{\cal F} = \{P_4\}, and F={2K2,P4}{\cal F} = \{2K_2, P_4\}, F{\cal F}-Completion cannot be solved in time 2o(k)nO(1)2^{o(k)} n^{O(1)} unless the Exponential Time Hypothesis (ETH) fails. Our upper and lower bounds provide a complete picture of the subexponential parameterized complexity of F{\cal F}-Completion problems for F{2K2,C4,P4}{\cal F}\subseteq\{2K_2, C_4, P_4\}.Comment: 32 pages, 16 figures, A preliminary version of this paper appeared in the proceedings of STACS'1

    The Complexity of Surjective Homomorphism Problems -- a Survey

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    We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of which remain of open complexity

    Letter graphs and geometric grid classes of permutations: characterization and recognition

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    In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a non-constructive way, a polynomial-time recognition of geometric grid classes of permutations and kk-letter graphs for a fixed kk. However, constructive algorithms are available only for k=2k=2. In this paper, we present the first constructive polynomial-time algorithm for the recognition of 33-letter graphs. It is based on a structural characterization of graphs in this class.Comment: arXiv admin note: text overlap with arXiv:1108.6319 by other author

    Subdivision into i-packings and S-packing chromatic number of some lattices

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    An ii-packing in a graph GG is a set of vertices at pairwise distance greater than ii. For a nondecreasing sequence of integers S=(s_1,s_2,)S=(s\_{1},s\_{2},\ldots), the SS-packing chromatic number of a graph GG is the least integer kk such that there exists a coloring of GG into kk colors where each set of vertices colored ii, i=1,,ki=1,\ldots, k, is an s_is\_i-packing. This paper describes various subdivisions of an ii-packing into jj-packings (j\textgreater{}i) for the hexagonal, square and triangular lattices. These results allow us to bound the SS-packing chromatic number for these graphs, with more precise bounds and exact values for sequences S=(s_i,iN)S=(s\_{i}, i\in\mathbb{N}^{*}), s_i=d+(i1)/ns\_{i}=d+ \lfloor (i-1)/n \rfloor
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