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 $\mathcal {O}(k^{3/2})$ 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 $\mathcal {O}(\log{k})$ on planar graphs and $\mathcal {O}(\sqrt{k}\log{k})$ on H-minor-free graphs. These results significantly improve the previously known kernelization and approximation results for Minimum Fill-in on sparse graphs.