Faster Approximation Schemes and Parameterized Algorithms on H-Minor-Free and Odd-Minor-Free Graphs

We improve the running time of the general algorithmic technique known as Baker’s approach (1994) on H-minor-free graphs from O(n) to O(f(|H |)n) showing that it is fixed-parameter tractable w.r.t. the parameter |H |. The numerous applications include e.g. a 2-approximation for coloring and PTASes for various problems such as dominating set and max-cut, where we obtain similar improvements. On classes of odd-minor-free graphs, which have gained significant attention in recent time, we obtain a similar acceleration for a variant of the structural decomposition theorem proved by Demaine et al. (2010) and a Bakerstyle decomposition into 2 graphs of bounded treewidth. We use these algorithms to derive faster 2-approximations; furthermore, we present the first PTASes and subexponential FPT-algorithms for independent set and vertex cover on these graph classes using a novel dynamic programming technique. We also introduce a technique to derive (nearly) subexponential parameterized algorithms on H-minor-free graphs. We provide a uniform algorithm running in time inf0<ǫ≤1 O((1 + ǫ) + n), where n is the size of the input and k is the number of vertices or edges in the solution. Our technique applies, in particular, to problems such as Steiner tree, (directed) subgraph with a property, (directed) longest path, and (connected/independent) dominating set, on some or all proper minor-closed graph classes, many of which were previously not even known to admit an algorithm with running time better than O(2n). We obtain as a corollary that all problems with a minormonotone subexponential kernel and amenable to our technique can be solved in subexponential FPT-time on H-minor free graphs. This results in a general methodology for subexponential parameterized algorithms outside the framework of bidimensionality.