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Interval completion with few edges

Published: 11 June 2007 Publication History

Abstract

We present an algorithm with runtime O(k(2k)n3 * m) for the following NP-complete problem: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the long-standing open question, first posed by Kaplan, Shamir and Tarjan, of whether this problem could be solved in time f(k) * n(O(1)).The problem has applications in Physical Mapping of DNA and in Profile Minimization for Sparse Matrix Computations. For the first application, our results show tractability for the case of a small number k of false negative errors, and for the second, a small number k of zero elements in the envelope.
Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph, and combines this with a greedy algorithm when graphs of a certain structure are reached by the search. The presented result is surprising, as it was not believed that a bounded search tree algorithm would suffice to answer the open question affirmatively.

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    cover image ACM Conferences
    STOC '07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
    June 2007
    734 pages
    ISBN:9781595936318
    DOI:10.1145/1250790
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    Published: 11 June 2007

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    Author Tags

    1. FPT algorithm
    2. branching
    3. edge completion
    4. interval graphs
    5. physical mapping
    6. profile minimization

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    June 11 - 13, 2007
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    Cited By

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    • (2018)Subexponential Parameterized Algorithm for Interval CompletionACM Transactions on Algorithms10.1145/318689614:3(1-62)Online publication date: 27-Jun-2018
    • (2016)Subexponential parameterized algorithm for interval completionProceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms10.5555/2884435.2884513(1116-1131)Online publication date: 10-Jan-2016
    • (2016)Planar Disjoint-Paths CompletionAlgorithmica10.1007/s00453-015-0046-276:2(401-425)Online publication date: 1-Oct-2016
    • (2015)A Polynomial-Time Algorithm for Outerplanar Diameter ImprovementComputer Science -- Theory and Applications10.1007/978-3-319-20297-6_9(123-142)Online publication date: 23-Jun-2015
    • (2013)Subexponential Parameterized Algorithm for Minimum Fill-InSIAM Journal on Computing10.1137/11085390X42:6(2197-2216)Online publication date: 1-Jan-2013
    • (2013)On the (Non-)Existence of Polynomial Kernels for Pl-Free Edge Modification ProblemsAlgorithmica10.1007/s00453-012-9619-565:4(900-926)Online publication date: 1-Apr-2013
    • (2012)Subexponential parameterized algorithm for minimum fill-inProceedings of the twenty-third annual ACM-SIAM symposium on Discrete algorithms10.5555/2095116.2095254(1737-1746)Online publication date: 17-Jan-2012
    • (2011)Planar disjoint-paths completionProceedings of the 6th international conference on Parameterized and Exact Computation10.1007/978-3-642-28050-4_7(80-93)Online publication date: 6-Sep-2011
    • (2010)Chordal Deletion is Fixed-Parameter TractableAlgorithmica10.5555/3118226.311846357:4(747-768)Online publication date: 1-Aug-2010
    • (2010)Solving MAX-r-SAT above a tight lower boundProceedings of the twenty-first annual ACM-SIAM symposium on Discrete algorithms10.5555/1873601.1873645(511-517)Online publication date: 17-Jan-2010
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