A Simple and Fast Approach for Solving Problems on Planar Graphs

Abstract

AbstractIt is well known that the celebrated Lipton-Tarjan planar separation theorem, in a combination with a divide-and-conquer strategy leads to many complexity results for planar graph problems. For example, by using this approach, many planar graph problems can be solved in time 2\(^{O(\sqrt{n})}\), where n is the number of vertices. However, the constants hidden in big-Oh, usually are too large to claim the algorithms to be practical even on graphs of moderate size. Here we introduce a new algorithm design paradigm for solving problems on planar graphs. The paradigm is so simple that it can be explained in any textbook on graph algorithms: Compute tree or branch decomposition of a planar graph and do dynamic programming. Surprisingly such a simple approach provides faster algorithms for many problems. For example, Independent Set on planar graphs can be solved in time \(O(2^{3.182\sqrt{n}}n + n^4)\) and Dominating Set in time \(O(2^{5.043\sqrt{n}}n + n^4)\). In addition, significantly broader class of problems can be attacked by this method. Thus with our approach, Longest cycle on planar graphs is solved in time \(O(2^{2.29\sqrt{n}(ln n + 0.94)}n^{5/4} + n^4)\) and Bisection is solved in time \(O(2^{3.182\sqrt{n}}n + n^4)\). The proof of these results is based on complicated combinatorial arguments that make strong use of results derived by the Graph Minors Theory. In particular we prove that branch-width of a planar graph is at most 2.122\(\sqrt{n}\). In addition we observe how a similar approach can be used for solving different fixed parameter problems on planar graphs. We prove that our method provides the best so far exponential speed-up for fundamental problems on planar graphs like Vertex Cover, (Weighted) Dominating Set, and many others.KeywordsPlanar GraphVertex CoverPerfect CodeKernel GraphRadial GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.