Shortest k-Disjoint Paths via Determinants
The well-known k-disjoint path problem (k-DPP) asks for pairwise vertex-disjoint paths between k specified pairs of vertices (s_i, t_i) in a given graph, if they exist. The decision version of the shortest k-DPP asks for the length of the shortest (in terms of total length) such paths. Similarly, the search and counting versions ask for one such and the number of such shortest set of paths, respectively.
We restrict attention to the shortest k-DPP instances on undirected planar graphs where all sources and sinks lie on a single face or on a pair of faces. We provide efficient sequential and parallel algorithms for the search versions of the problem answering one of the main open questions raised by Colin de Verdière and Schrijver [Éric Colin de Verdière and Alexander Schrijver, 2011] for the general one-face problem. We do so by providing a randomised NC^2 algorithm along with an O(n^{omega/2}) time randomised sequential algorithm, for any fixed k. We also obtain deterministic algorithms with similar resource bounds for the counting and search versions. In contrast, previously, only the sequential complexity of decision and search versions of the "well-ordered" case has been studied. For the one-face case, sequential versions of our routines have better running times for constantly many terminals.
The algorithms are based on a bijection between a shortest k-tuple of disjoint paths in the given graph and cycle covers in a related digraph. This allows us to non-trivially modify established techniques relating counting cycle covers to the determinant. We further need to do a controlled inclusion-exclusion to produce a polynomial sum of determinants such that all "bad" cycle covers cancel out in the sum allowing us to count "pure" cycle covers.
disjoint paths
planar graph
parallel algorithm
cycle cover
determinant
inclusion-exclusion
Theory of computation~Parallel algorithms
19:1-19:21
Regular Paper
https://arxiv.org/abs/1802.01338
Samir
Datta
Samir Datta
Chennai Mathematical Institute and UMI ReLaX, Chennai, India
The author was partially funded by a grant from Infosys foundation and SERB grant MTR/2017/000480.
Siddharth
Iyer
Siddharth Iyer
University of Washington, Seattle, USA
The work was done while the author was a student at Birla Institute of Technology and Science, India.
Raghav
Kulkarni
Raghav Kulkarni
Chennai Mathematical Institute, Chennai, India
Anish
Mukherjee
Anish Mukherjee
Chennai Mathematical Institute, Chennai, India
The author was partially supported by a grant from Infosys foundation and TCS PhD fellowship.
10.4230/LIPIcs.FSTTCS.2018.19
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Samir Datta, Siddharth Iyer, Raghav Kulkarni, and Anish Mukherjee
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