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LATIN 2012: Theoretical Informatics

Volume 7256 of the series Lecture Notes in Computer Science pp 195-206

Solving the 2-Disjoint Connected Subgraphs Problem Faster Than 2 n

  • Marek CyganAffiliated withInstitute of Informatics, University of Warsaw
  • , Marcin PilipczukAffiliated withInstitute of Informatics, University of Warsaw
  • , Michał PilipczukAffiliated withDepartment of Informatics, University of Bergen
  • , Jakub Onufry WojtaszczykAffiliated withGoogle Inc.

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Abstract

The 2-Disjoint Connected Subgraphs problem, given a graph along with two disjoint sets of terminals Z 1 ,Z 2 , asks whether it is possible to find disjoint sets A 1 ,A 2 , such that Z 1 ⊆ A 1 , Z 2 ⊆ A 2 and A 1 ,A 2 induce connected subgraphs. While the naive algorithm runs in O(2 n n O(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(1.933 n ) algorithm for 2-Disjoint Connected Subgraphs in general case, thus breaking the 2 n 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.