Dynamic Programming for H-minor-free Graphs

Abstract

We provide a framework for the design and analysis of dynamic programming algorithms for H-minor-free graphs with branchwidth at most k. Our technique applies to a wide family of problems where standard (deterministic) dynamic programming runs in 2^O(k·logk)·n ^O(1) steps, with n being the number of vertices of the input graph. Extending the approach developed by the same authors for graphs embedded in surfaces, we introduce a new type of branch decomposition for H-minor-free graphs, called an H-minor-free cut decomposition, and we show that they can be constructed in O _h (n ³) steps, where the hidden constant depends exclusively on H. We show that the separators of such decompositions have connected packings whose behavior can be described in terms of a combinatorial object called ℓ-triangulation. Our main result is that when applied on H-minor-free cut decompositions, dynamic programming runs in $2^{O_h(k)}\cdot n^{O(1)}$ steps. This broadens substantially the class of problems that can be solved deterministically in single-exponential time for H-minor-free graphs.

This research was done during a research visit of the first two authors at the Department of Mathematics of the National and Kapodistrian University of Athens. The authors wish to express their gratitude to the decisive support of “Pontios” during that visit. The first author was partially supported by grants JAE-DOC (CSIC), MTM2011-22851, and SEV-2011-0087, the second author was partially supported by project AGAPE (ANR-09-BLAN-0159), and the third author was co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: “Thales. Investing in knowledge society through the European Social Fund”.