Approximation Schemes for Planar Graph Problems

Many NP-hard graph problems become easier to approximate on planar graphs and their generalizations. (A graph is planar if it can be drawn in the plane (or the sphere) without crossings. For definitions of other related graph classes, see the entry on Bidimensionality (2004; Demaine, Fomin, Hajiaghayi, Thilikos).) For example, maximum independent set asks to find a maximum subset of vertices in a graph that induce no edges. This problem is inapproximable in general graphs within a factor of n1− for any > 0 unless NP = ZPP (and inapproximable within n1/2− unless P = NP), while for planar graphs there is a 4-approximation (or simple 5-approximation) by taking the largest color class in a vertex 4-coloring (or 5-coloring). Another is minimum dominating set, where the goal is to find a minimum subset of vertices such that every vertex is either in or adjacent to the subset. This problem is inapproximable in general graphs within log n for some > 0 unless P = NP, but as we will see, for planar graphs the problem admits a polynomial-time approximation scheme (PTAS): a collection of (1 + )-approximation algorithms for all > 0. There are two main general approaches for designing PTASs for problems on planar graphs and their generalizations: the separator approach and the Baker approach. Lipton and Tarjan [15, 16] introduced the first approach, which is based on planar separators. The first step in this approach is to find a separator of O( √ n) vertices or edges, where n is the size of the graph, whose removal splits the graph into two or more pieces each of which is a constant fraction smaller than the original graph. Then recurse in each piece, building a recursion tree of separators, and stop when the pieces have some constant size such as 1/ . The problem can be solved on these pieces by brute force, and then it remains to combine the solutions up the recursion tree. The induced error can often be bounded in terms of the total size of all separators, which in turn can be bounded by n. If the optimal solution is at least some constant factor times n, this approach often leads to a PTAS.