Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm

TL;DR

This paper presents a highly efficient NC algorithm for counting and finding the shortest two disjoint paths in cubic planar graphs, a problem previously unsolved with deterministic polynomial time methods.

Contribution

It introduces the first deterministic NC algorithm for S2DP in cubic planar graphs that also counts the number of solutions, improving over prior randomized and slower algorithms.

Findings

Developed an NC algorithm for S2DP in cubic planar graphs

The algorithm computes the minimum total length of disjoint paths

It counts the number of such shortest path pairs

Abstract

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 Bj\"orklund 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.