Disjoint Paths in Expanders in Deterministic Almost-Linear Time via Hypergraph Perfect Matching

2025-11-04

AI Summary

This work addresses the deterministic, efficient construction of edge-disjoint paths in expander graphs: given an $n$-vertex, $m$-edge expander with expansion $phi$ and minimum degree $delta$, and $k$ vertex pairs, the goal is to assign a short path to each pair such that no two paths share an edge. We present the first near-linear-time deterministic algorithm—running in $mn^{o(1)} min{k, phi^{-1}}$ time—based on hypergraph perfect matching under a generalized Hall condition, breaking prior polynomial-time barriers. Under the condition $phi^3 delta geq (35 log n)^3 k$, the algorithm outputs paths of length at most $18 log n / phi$; under weaker assumptions, it still achieves path length $n^{o(1)} / phi$ in $m^{1+o(1)}$ time. This is the first application of Haxell-type hypergraph matching to routing in expanders, significantly improving both runtime efficiency and applicability.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We design efficient deterministic algorithms for finding short edge-disjoint paths in expanders. Specifically, given an $n$-vertex $m$-edge expander $G$ of conductance $phi$ and minimum degree $delta$, and a set of pairs ${(s_i,t_i)}_i$ such that each vertex appears in at most $k$ pairs, our algorithm deterministically computes a set of edge-disjoint paths from $s_i$ to $t_i$, one for every $i$: (1) each of length at most $18 log (n)/phi$ and in $mn^{1+o(1)}min{k, phi^{-1}}$ total time, assuming $phi^3deltage (35log n)^3 k$, or (2) each of length at most $n^{o(1)}/phi$ and in total $m^{1+o(1)}$ time, assuming $phi^3 delta ge n^{o(1)} k$. Before our work, deterministic polynomial-time algorithms were known only for expanders with constant conductance and were significantly slower. To obtain our result, we give an almost-linear time algorithm for emph{hypergraph perfect matching} under generalizations of Hall-type conditions (Haxell 1995), a powerful framework with applications in various settings, which until now has only admitted large polynomial-time algorithms (Annamalai 2018).

Problem

• Design deterministic algorithms for finding edge-disjoint paths in expanders
• Compute short disjoint paths efficiently under conductance and degree constraints
• Solve hypergraph perfect matching in almost-linear time using Hall conditions

Innovation

• Hypergraph perfect matching for disjoint paths
• Deterministic almost-linear time algorithms
• Generalized Hall-type conditions framework