eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-12-06
19:1
19:13
10.4230/LIPIcs.ISAAC.2018.19
article
Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm
Björklund, Andreas
1
Husfeldt, Thore
2
Department of Computer Science, Lund University, Sweden
BARC, IT University of Copenhagen, Denmark and Lund University, Sweden
Given an undirected graph and two disjoint vertex pairs s_1,t_1 and s_2,t_2, the Shortest two disjoint paths problem (S2DP) asks for the minimum total length of two vertex disjoint paths connecting s_1 with t_1, and s_2 with t_2, respectively.
We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs.
Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs with arbitrary placement of the terminals. In contrast, the randomized polynomial time algorithm by Björklund and Husfeldt, ICALP 2014, for general graphs is much slower, is serial in nature, and cannot count the solutions.
Our results are built on an approach by Hirai and Namba, Algorithmica 2017, for a generalisation of S2DP, and fast algorithms for counting perfect matchings in planar graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol123-isaac2018/LIPIcs.ISAAC.2018.19/LIPIcs.ISAAC.2018.19.pdf
Shortest disjoint paths
Cubic planar graph