Finding Two Edge-Disjoint Paths with Length Constraints

TL;DR

This paper studies the fixed-parameter tractability of finding two edge-disjoint paths with specific length constraints in undirected graphs, providing algorithms for most cases and complexity bounds for all.

Contribution

It introduces FPT algorithms for seven out of nine length constraint cases and establishes kernelization lower bounds for the problem.

Findings

Developed FPT algorithms for seven cases of the problem.

Proved no polynomial kernel exists for all cases unless NP ⊆ coNP/poly.

Used structural properties and random partition techniques in algorithms.

Abstract

We consider the problem of finding, for two pairs $(s_1,t_1)$ and $(s_2,t_2)$ of vertices in an undirected graphs, an $(s_1,t_1)$-path $P_1$ and an $(s_2,t_2)$-path $P_2$ such that $P_1$ and $P_2$ share no edges and the length of each $P_i$ satisfies $L_i$, where $L_i \in \{ \le k_i, \; = k_i, \; \ge k_i, \; \le \infty\}$. We regard $k_1$ and $k_2$ as parameters and investigate the parameterized complexity of the above problem when at least one of $P_1$ and $P_2$ has a length constraint (note that $L_i = "\le \infty"$ indicates that $P_i$ has no length constraint). For the nine different cases of $(L_1, L_2)$, we obtain FPT algorithms for seven of them. Our algorithms uses random partition backed by some structural results. On the other hand, we prove that the problem admits no polynomial kernel for all nine cases unless $NP \subseteq coNP/poly$.