A Tight Lower Bound for Edge-Disjoint Paths on Planar DAGs

TL;DR

This paper proves that the Edge-Disjoint Paths problem remains computationally hard on planar DAGs, establishing tight lower bounds and answering longstanding open questions about its complexity.

Contribution

It shows W[1]-hardness and ETH-based lower bounds for Edge-Disjoint Paths on planar DAGs, extending known hardness results and closing the complexity landscape.

Findings

W[1]-hardness of Edge-Disjoint Paths on planar DAGs

No f(k)·n^{o(k)} algorithm under ETH

Tightness of the n^{O(k)} algorithm for DAGs

Abstract

(see paper for full abstract) We show that the Edge-Disjoint Paths problem is W[1]-hard parameterized by the number $k$ of terminal pairs, even when the input graph is a planar directed acyclic graph (DAG). This answers a question of Slivkins (ESA '03, SIDMA '10). Moreover, under the Exponential Time Hypothesis (ETH), we show that there is no $f(k)\cdot n^{o(k)}$ algorithm for Edge-Disjoint Paths on planar DAGs, where $k$ is the number of terminal pairs, $n$ is the number of vertices and $f$ is any computable function. Our hardness holds even if both the maximum in-degree and maximum out-degree of the graph are at most $2$. Our result shows that the $n^{O(k)}$ algorithm of Fortune et al. (TCS '80) for Edge-Disjoint Paths on DAGs is asymptotically tight, even if we add an extra restriction of planarity. As a special case of our result, we obtain that Edge-Disjoint Paths on planar directed graphs is W[1]-hard parameterized by the number $k$ of terminal pairs. This answers a question of Cygan et al. (FOCS '13) and Schrijver (pp. 417-444, Building Bridges II, '19), and completes the landscape of the parameterized complexity status of edge and vertex versions of the Disjoint Paths problem on planar directed and planar undirected graphs.