Faster parameterized algorithms for minor containment

• Authors: Isolde Adler;Frederic Dorn;Fedor V. Fomin;Ignasi Sau;Dimitrios M. Thilikos
• Venue: Theoretical Computer Science
• Year: 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^)^@?^l^o^g^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.