[Paper Review] Improved Approximation for Node-Disjoint Paths in Grids with Sources on the Boundary

Julia Chuzhoy, David H. K. Kim+1 more|arXiv (Cornell University)|May 24, 2018

Complexity and Algorithms in Graphs4 citations

TL;DR

This paper presents a randomized approximation algorithm achieving a $2^{O( iny{ extstyle\sqrt{\log n}} \cdot \log\log n)}$ approximation factor for the Node-Disjoint Paths (NDP) problem in grid graphs where all sources lie on the grid boundary. It departs from the standard multicommodity flow relaxation by using a novel linear program only to select pairs to route, then applies geometric and dynamic programming techniques to route them via boundary-aligned columns and rows, effectively bypassing the $\Omega(\sqrt{n})$ integrality gap.

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 $\ ilde{O}(n^{1/4})$-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.

Motivation & Objective

• To design a better approximation algorithm for the Node-Disjoint Paths (NDP) problem in grid graphs where all source vertices lie on the grid boundary.
• To overcome the $\Omega(\sqrt{n})$ integrality gap of the standard multicommodity flow relaxation in grid graphs with boundary sources.
• To achieve an approximation factor that is significantly better than the previous $\tilde{O}(n^{1/4})$ bound, especially in light of strong hardness results.
• To develop a routing framework that leverages geometric structure—specifically, intervals along the grid boundary and column/row-based paths—rather than relying on flow-based rounding.

Proposed method

• A linear program is used solely to select which demand pairs to route, avoiding reliance on multicommodity flow relaxation.
• The selected pairs are mapped to intervals on the grid boundary, which are partitioned into levels based on nesting structure and routed via designated columns or rows in the grid.
• Left, right, and top intervals are routed using vertical columns to the left of the grid and horizontal rows above the grid, exploiting available space to ensure node-disjointness.
• The remaining pairs are routed via a flow network constructed on a modified graph that excludes already-used vertices, ensuring integral routing via max-flow min-cut.
• The algorithm uses dynamic programming and interval nesting properties to efficiently assign paths to intervals at different levels.
• The final routing is verified to be node-disjoint by ensuring that paths from different interval types and levels do not intersect, leveraging the geometric layout of the grid.

Experimental results

Research questions

• RQ1 Can the approximation factor for NDP in grid graphs with sources on the boundary be improved beyond the $\tilde{O}(n^{1/4})$ bound?
• RQ2 Is it possible to bypass the $\Omega(\sqrt{n})$ integrality gap of the multicommodity flow relaxation in this special case?
• RQ3 What structural properties of boundary-constrained grid graphs can be exploited to design a better approximation algorithm?
• RQ4 How can interval nesting and geometric layout be used to route paths without relying on flow-based rounding?

Key findings

• The paper achieves a $2^{O(\sqrt{\log n} \cdot \log\log n)}$ approximation factor for NDP in grid graphs with all sources on the boundary, improving upon the previous $\tilde{O}(n^{1/4})$ bound.
• The algorithm successfully bypasses the $\Omega(\sqrt{n})$ integrality gap of the multicommodity flow relaxation by decoupling pair selection from routing.
• The routing strategy uses level-wise interval decomposition and dedicated columns/rows to ensure node-disjoint paths, with paths for top, left, and right intervals routed via distinct regions of the grid.
• The method generalizes to instances where sources lie within a constant distance from the boundary, maintaining the same approximation factor.
• The algorithm is randomized and runs in polynomial time, providing a significant improvement over prior methods in the boundary-source setting.
• The result is nearly tight, as the paper shows that NDP in such grids is hard to approximate to within $2^{\Omega(\sqrt{\log n})}$, making the new bound close to optimal.