The Even-Path Problem in Directed Single-Crossing-Minor-Free Graphs

TL;DR

This paper extends polynomial-time solutions for the EvenPath problem to H-minor-free directed graphs where H is single-crossing, introduces parity-mimicking networks, and provides algorithms for parity-constrained disjoint paths in planar and bounded treewidth graphs.

Contribution

It presents the first polynomial-time algorithm for the EvenPath problem in H-minor-free directed graphs with single-crossing minors, and introduces parity-mimicking networks and algorithms for parity-constrained disjoint paths.

Findings

Polynomial-time algorithm for EvenPath in H-minor-free directed graphs.

Construction of small, planar, parity-mimicking networks.

Polynomial-time solution for 3-disjoint paths with parity constraints in planar and bounded treewidth graphs.

Abstract

Finding a simple path of even length between two designated vertices in a directed graph is a fundamental NP-complete problem known as the EvenPath problem. Nedev proved in 1999, that for directed planar graphs, the problem can be solved in polynomial time. More than two decades since then, we make the first progress in extending the tractable classes of graphs for this problem. We give a polynomial time algorithm to solve the EvenPath problem for classes of H-minor-free directed graphs,1 where H is a single-crossing graph. We make two new technical contributions along the way, that might be of independent interest. The first, and perhaps our main, contribution is the construction of small, planar, parity-mimicking networks. These are graphs that mimic parities of all possible paths between a designated set of terminals of the original graph. Finding vertex disjoint paths between given source-destination pairs of vertices is another fundamental problem, known to be NP-complete in directed graphs, though known to be tractable in planar directed graphs. We encounter a natural variant of this problem, that of finding disjoint paths between given pairs of vertices, but with constraints on parity of the total length of paths. The other significant contribution of our paper is to give a polynomial time algorithm for the 3-disjoint paths with total parity problem, in directed planar graphs with some restrictions (and also in directed graphs of bounded treewidth).