Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

Abstract

The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. The problem has important applications in numerical algebra, in particular in sparse matrix computations. We develop kernelization algorithms for the problem on several classes of sparse graphs. We obtain linear kernels on planar graphs, and kernels of size O(k 3/2 ) in graphs excluding some fixed graph as a minor and in graphs of bounded degeneracy. As a byproduct of our results, we obtain approximation algorithms with approximation ratios O(log k) on planar graphs and O( √ k log k) on H-minor-free graphs. These results significantly improve the previously known kernelization and approximation results for Minimum Fill-in on sparse graphs. ACM Subject Classification G.2.2 Graph Theory -Graph Algorithms Keywords and phrases IntroductionA graph is chordal (or triangulated) if every cycle of length at least four has a chord, i.e. an edge between nonadjacent vertices of the cycle. In the Minimum Fill-in problem (also known as Minimum Triangulation and Chordal Graph Completion) the task is to check if at most k edges can be added to a graph such that the resulting graph is chordal. That isMinimum Fill-in Input: A graph G = (V, E) and a non-negative integer k.This is a classical computational problem motivated by, and named after, a fundamental issue arising in sparse matrix computations. During Gaussian eliminations of large sparse matrices, new non-zero elements -called fill -can replace original zeros, thus increasing storage requirements, the time needed for the elimination, and the time needed to solve the system after the elimination. The problem of finding the right elimination ordering * The research of Fedor V.

A Benders Approach to the Minimum Chordal Completion Problem

“…Initial algorithmic aspects for directed graphs were investigated by Rose et al (1976) and Rose and Tarjan (1978), motivated by problems arising in Gaussian elimination of sparse linear equality systems. The most general version of the MCCP was only proven NP-Hard later by Yannakakis (1981), and since then minimum graph chordalization methods have been applied in database management (Beeri et al 1983, Tarjan andYannakakis 1984), sparse matrix computation (Grone et al 1984, Fomin et al 2013, artificial intelligence (Lauritzen and Spiegelhalter 1990), computer vision (Chung and Mumford 1994), and in several other contexts; see, e.g., the survey by Heggernes (2006). Most recently, solution methods for the MCCP have gained a central role in semidefinite and nonlinear optimization, in particular for exploiting sparsity of linear and nonlinear constraint matrices (Nakata et al 2003, Kim et al 2011, Vandenberghe and Andersen 2015.…”