Half-integral linkages in highly connected directed graphs

TL;DR

This paper investigates the half-integral k-Directed Disjoint Paths Problem in highly connected directed graphs, demonstrating its polynomial-time solvability under certain connectivity conditions and establishing NP-completeness in more general cases.

Contribution

It introduces the half-integral variant of the k-Directed Disjoint Paths Problem, showing it is efficiently solvable in highly connected digraphs, unlike the integral version which is NP-complete.

Findings

Half-integral kDDPP is solvable in polynomial time for sufficiently highly connected graphs.

The problem remains NP-complete when the connectivity is proportional to k and k is part of the input.

The paper provides bounds on the connectivity needed for polynomial-time solvability.

Abstract

We study the half-integral $k$-Directed Disjoint Paths Problem ($\tfrac12$kDDPP) in highly strongly connected digraphs. The integral kDDPP is NP-complete even when restricted to instances where $k=2$, and the input graph is $L$-strongly connected, for any $L\geq 1$. We show that when the integrality condition is relaxed to allow each vertex to be used in two paths, the problem becomes efficiently solvable in highly connected digraphs (even with $k$ as part of the input). Specifically, we show that there is an absolute constant $c$ such that for each $k\geq 2$ there exists $L(k)$ such that $\tfrac12$kDDPP is solvable in time $O(|V(G)|^c)$ for a $L(k)$-strongly connected directed graph $G$. As the function $L(k)$ grows rather quickly, we also show that $\tfrac12$kDDPP is solvable in time $O(|V(G)|^{f(k)})$ in $(36k^3+2k)$-strongly connected directed graphs. We also show that for each $\epsilon<1$ deciding half-integral feasibility of kDDPP instances is NP-complete when $k$ is given as part of the input, even when restricted to graphs with strong connectivity $\epsilon k$.