Searching for better fill-in

Abstract

Abstract 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. 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 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.