On Routing Disjoint Paths in Bounded Treewidth Graphs

TL;DR

This paper presents improved approximation algorithms for routing disjoint paths in bounded treewidth graphs, achieving polynomial bounds for MaxEDP and MaxNDP problems, advancing previous exponential bounds.

Contribution

It introduces new $O(r^3)$ and matching approximations for MaxEDP and MaxNDP in graphs of bounded treewidth and pathwidth, respectively, improving prior exponential approximations.

Findings

Achieves $O(r^3)$ approximation for MaxEDP in graphs of treewidth $r$.

Provides a matching approximation for MaxNDP in graphs of pathwidth $r$.

Significantly improves upon previous $O(r imes 3^r)$ approximation by Chekuri et al.

Abstract

We study the problem of routing on disjoint paths in bounded treewidth graphs with both edge and node capacities. The input consists of a capacitated graph $G$ and a collection of $k$ source-destination pairs $\mathcal{M} = \{(s_1, t_1), \dots, (s_k, t_k)\}$. The goal is to maximize the number of pairs that can be routed subject to the capacities in the graph. A routing of a subset $\mathcal{M}'$ of the pairs is a collection $\mathcal{P}$ of paths such that, for each pair $(s_i, t_i) \in \mathcal{M}'$, there is a path in $\mathcal{P}$ connecting $s_i$ to $t_i$. In the Maximum Edge Disjoint Paths (MaxEDP) problem, the graph $G$ has capacities $\mathrm{cap}(e)$ on the edges and a routing $\mathcal{P}$ is feasible if each edge $e$ is in at most $\mathrm{cap}(e)$ of the paths of $\mathcal{P}$. The Maximum Node Disjoint Paths (MaxNDP) problem is the node-capacitated counterpart of MaxEDP. In this paper we obtain an $O(r^3)$ approximation for MaxEDP on graphs of treewidth at most $r$ and a matching approximation for MaxNDP on graphs of pathwidth at most $r$. Our results build on and significantly improve the work by Chekuri et al. [ICALP 2013] who obtained an $O(r \cdot 3^r)$ approximation for MaxEDP.