Improved Approximation for Node-Disjoint Paths in Grids with Sources on the Boundary

TL;DR

This paper presents a new randomized approximation algorithm for the Node-Disjoint Paths problem on grid graphs with sources on the boundary, improving previous bounds and extending to sources near the boundary.

Contribution

It introduces an efficient randomized approximation algorithm for NDP-Grid with boundary sources, surpassing prior approximation factors and generalizing to sources within a certain distance from the boundary.

Findings

Achieves a $2^{O( oot{ ext{log} n} imes ext{loglog} n)}$-approximation for NDP-Grid with boundary sources.

Extends the approximation to sources within a prescribed distance from the boundary.

Uses a novel approach that departs from the traditional multicommodity flow relaxation.

Abstract

We study the classical Node-Disjoint Paths (NDP) problem: given an undirected $n$-vertex graph G, together with a set {(s_1,t_1),...,(s_k,t_k)} of pairs of its vertices, called source-destination, or demand pairs, find a maximum-cardinality set of mutually node-disjoint paths that connect the demand pairs. The best current approximation for the problem is achieved by a simple greedy $O(\sqrt{n})$-approximation algorithm. A special case of the problem called NDP-Grid, where the underlying graph is a grid, has been studied extensively. The best current approximation algorithm for NDP-Grid achieves an $\tilde{O}(n^{1/4})$-approximation factor. On the negative side, a recent result by the authors shows that NDP is hard to approximate to within factor $2^{\Omega(\sqrt{\log n})}$, even if the underlying graph is a sub-graph of a grid, and all source vertices lie on the grid boundary. In a follow-up work, the authors further show that NDP-Grid is hard to approximate to within factor $\Omega(2^{\log^{1-\epsilon}n})$ for any constant $\epsilon$ under standard complexity assumptions, and to within factor $n^{\Omega(1/(\log\log n)^2)}$ under randomized ETH. In this paper we study NDP-Grid, where all source vertices {s_1,...,s_k} appear on the grid boundary. Our main result is an efficient randomized $2^{O(\sqrt{\log n} \cdot \log\log n)}$-approximation algorithm for this problem. We generalize this result to instances where the source vertices lie within a prescribed distance from the grid boundary. Much of the work on approximation algorithms for NDP relies on the multicommodity flow relaxation of the problem, which is known to have an $\Omega(\sqrt n)$ integrality gap, even in grid graphs. Our work departs from this paradigm, and uses a (completely different) linear program only to select the pairs to be routed, while the routing itself is computed by other methods.