Searching for better fill-in

Authors Fedor V. Fomin, Yngve Villanger

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Fedor V. Fomin
Yngve Villanger

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Fedor V. Fomin and Yngve Villanger. Searching for better fill-in. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 8-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


Minimum Fill-in is a fundamental and classical problem arising in sparse matrix computations. In terms of graphs it can be formulated as a problem of finding a triangulation of a given graph with the minimum number of edges. By the classical result of Rose, Tarjan, Lueker, and Ohtsuki from 1976, an inclusion minimal triangulation of a graph can be found in polynomial time but, as it was shown by Yannakakis in 1981, finding a triangulation with the minimum number of edges is NP-hard. In this paper, we study the parameterized complexity of local search for the Minimum Fill-in problem in the following form: Given a triangulation H of a graph G, is there a better triangulation, i.e. triangulation with less edges than H, within a given distance from H? We prove that this problem is fixed-parameter tractable (FPT) being parameterized by the distance from the initial triangulation by providing an algorithm that in time O(f(k) |G|^{O(1)}) decides if a better triangulation of G can be obtained by swapping at most k edges of H. Our result adds Minimum Fill-in to the list of very few problems for which local search is known to be FPT.
  • Local Search
  • Parameterized Complexity
  • Fill-in
  • Triangulation
  • Chordal graph


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