Subexponential Parameterized Algorithm for Minimum Fill-in

TL;DR

This paper introduces the first subexponential parameterized algorithm for the Minimum Fill-in problem, significantly improving the computational complexity and enabling efficient solutions for related graph triangulation problems.

Contribution

The paper presents a novel subexponential algorithm for Minimum Fill-in, reducing the complexity from exponential to subexponential in the parameter k.

Findings

Achieved a time complexity of O(2^(O(√k) log k)) for Minimum Fill-in.

Extended techniques to related problems like Minimum Chain Completion and Chordal Graph Sandwich.

Demonstrated the practical impact of subexponential algorithms in graph triangulation tasks.

Abstract

The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. Kaplan, Shamir, and Tarjan [FOCS 1994] have shown that the problem is solvable in time O(2^(O(k)) + k2 * nm) on graphs with n vertices and m edges and thus is fixed parameter tractable. Here, we give the first subexponential parameterized algorithm solving Minimum Fill-in in time O(2^(O(\sqrt{k} log k)) + k2 * nm). This substantially lower the complexity of the problem. Techniques developed for Minimum Fill-in can be used to obtain subexponential parameterized algorithms for several related problems including Minimum Chain Completion, Chordal Graph Sandwich, and Triangulating Colored Graph.