Faster parameterized algorithms for minor containment - Adler, Isolde; Dorn, Frederic; Fomin, Fedor V.; Sau, Ignasi; Thilikos, Dimitrios M. (Theoretical computer science, 2011)
The H - Minor containment problem asks whether a graph G contains some fixed graph H as a minor, that is, whether H can be obtained by some subgraph of G after contracting edges. The derivation of a polynomial-time algorithm for H - Minor containment is one of the most important and technical parts of the Graph Minor Theory of Robertson and Seymour and it is a cornerstone for most of the algorithmic applications of this theory. H - Minor containment for graphs of bounded branchwidth is a basic ingredient of this algorithm. The currently fastest solution to this problem, based on the ideas introduced by Robertson and Seymour, was given by Hicks in [I.V. Hicks, Branch decompositions and minor containment, Networks 43 (1) (2004) 1–9], providing an algorithm that in time O ( 3 k 2 ⋅ ( h + k − 1 ) ! ⋅ m ) decides if a graph G with m edges and branchwidth k , contains a fixed graph H on h vertices as a minor. In this work we improve the dependence on k of Hicks’ result by showing that checking if H is a minor of G can be done in time O ( 2 ( 2 k + 1 ) ⋅ log k ⋅ h 2 k ⋅ 2 2 h 2 ⋅ m ) . We set up an approach based on a combinatorial object called rooted packing, which captures the properties of the subgraphs of H that we seek in our dynamic programming algorithm. This formulation with rooted packings allows us to speed up the algorithm when G is embedded in a fixed surface, obtaining the first algorithm for minor containment testing with single-exponential dependence on branchwidth. Namely, it runs in time 2 O ( k ) ⋅ h 2 k ⋅ 2 O ( h ) ⋅ n , with n = ∣ V ( G ) ∣ . Finally, we show that slight modifications of our algorithm permit to solve some related problems within the same time bounds, like induced minor or contraction containment.