AN EXPONENTIAL TIME PARAMETERIZED ALGORITHM FOR PLANAR DISJOINT PATHS

In the disjoint paths problem‚ the input is an undirected graph G on n vertices and a set of k vertex pairs‚ {si,ti}^k_i=1‚ and the task is to find k pairwise vertex-disjoint paths such that the i’th path connects si to ti. In this paper‚ we give a parameterized algorithm with running time 2^O(^k2)n^O(1) for planar disjoint paths‚ the variant of the problem where the input graph is required to be planar. Our algorithm is based on the unique linkage/treewidth reduction theorem for planar graphs by Adler et al. [J. Combin. Theory Ser. B‚ 122 (2017)‚ pp. 815–843]‚ the algebraic cohomology based technique of Schrijver [SIAM J. Comput., 23 (1994)‚ pp. 780–788]‚ and one of the key combinatorial insights developed by Cygan et al. [Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science‚ 2013‚ pp. 197–206] in their algorithm for disjoint paths on directed planar graphs. To the best of our knowledge‚ our algorithm is the first parameterized algorithm to exploit the fact that the treewidth of the input graph is small‚ and it does so in a way that is completely different from the use of dynamic programming.