Searching for better fill-in

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. 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.