Search

We prove the following theorem. Given a planar graph $G$ and an integer $k$, it is possible in polynomial time to randomly sample a subset $A$ of vertices of $G$ with the following properties: (i) $A$ induces a subgraph of $G$ of treewidth $\mathcal{O}(\sqrt{k}\log k)$, and (ii) for every connected subgraph $H$ of $G$ on at most $k$ vertices, the probability that $A$ covers the whole vertex set of $H$ is at least $(2^{\mathcal{O}(\sqrt{k}\log^2 k)}\cdot n^{\mathcal{O}(1)})^{-1}$, where $n$ is the number of vertices of $G$. Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound $2^{\mathcal{O}(\sqrt{k} \log^2 k)} n^{\mathcal{O}(1)}$. The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph, examples of such problems include Directed $k$-Path, Weighted $k$-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems can be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface.

12th International Symposium on Parameterized and Exact Computation (IPEC 2017); International audience; Given a graph G, we define bcg(G) as the minimum k for which G can be contracted to the uniformly triangulated grid Γk. A graph class G has the SQGC property if every graph G ∈ G has treewidth O(bcg(G)c) for some 1 ≤ c < 2. The SQGC property is important for algo- rithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a general family of graph classes that satisfy the SQGC property and includes bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for several intersection graph classes of 2-dimensional geometrical objects.