Solving the 2-disjoint connected subgraphs problem faster than 2n

Authors:
Marek Cygan;Marcin Pilipczuk;Michał Pilipczuk;Jakub Onufry Wojtaszczyk
Affiliations:
Institute of Informatics, University of Warsaw, Poland;Institute of Informatics, University of Warsaw, Poland;Department of Informatics, University of Bergen, Norway;Google Inc., Warsaw, Poland
Venue:
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Year:
2012

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Abstract

The 2-Disjoint Connected Subgraphs problem, given a graph along with two disjoint sets of terminals Z1, Z2, asks whether it is possible to find disjoint sets A1, A2, such that Z1⊆A1, Z2⊆A2 and A1, A2 induce connected subgraphs. While the naive algorithm runs in O(2nnO(1)) time, solutions with complexity of form O((2−ε)n) have been found only for special graph classes [15, 19]. In this paper we present an O(1.933n) algorithm for 2-Disjoint Connected Subgraphs in general case, thus breaking the 2n barrier. As a counterpoise of this result we show that if we parameterize the problem by the number of non-terminal vertices, it is hard both to speed up the brute-force approach and to find a polynomial kernel.