Bidimensionality and EPTAS

• Authors: Fedor V. Fomin;Daniel Lokshtanov;Venkatesh Raman;Saket Saurabh
• Venue: Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
• Year: 2011

Bidimensionality theory appears to be a powerful framework for the development of meta-algorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain sub-exponential time parameterized algorithms for problems on H-minor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems, and subsequently improved these results to EPTASs. Fomin et. al [SODA 2010] established a third meta-algorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In this paper we revisit bidimensionality theory from the perspective of approximation algorithms and redesign the framework for obtaining EPTASs to be more powerful, easier to apply and easier to understand. One of the important conditions required in the framework developed by Demaine and Hajiaghayi [SODA 2005] is that to obtain an EPTAS for a graph optimization problem Π, we have to know a constant-factor approximation algorithm for Π. Our approach eliminates this strong requirement, which makes it amenable to more problems. At the heart of our framework is a decomposition lemma which states that for "most" bidimensional problems, there is a polynomial time algorithm which given an H-minor-free graph G as input and an ∈ 0 outputs a vertex set X of size ∈ · OPT such that the treewidth of G\X is O(1/∈). Here, OPT is the objective function value of the problem in question This allows us to obtain EPTASs on (apex)-minor-free graphs for all problems covered by the previous framework, as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors, nor contractions. To the best of our knowledge for many of these problems including Cycle Packing, Vertex-H-Packing, Maximum Leaf Spanning Tree, and Partial r-Dominating Set no EPTASs on planar graphs were previously known.