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

TL;DR

This paper presents a deterministic, nearly linear time algorithm for finding edge-disjoint paths in expanders, leveraging hypergraph perfect matching techniques under generalized Hall conditions, improving efficiency over prior methods.

Contribution

It introduces the first almost-linear time algorithm for hypergraph perfect matching under generalized Hall conditions, enabling faster deterministic path-finding in expanders.

Findings

Deterministic algorithms for edge-disjoint paths in expanders with near-linear time complexity.

New hypergraph perfect matching algorithm under generalized Hall conditions.

Improved efficiency over previous polynomial-time algorithms for expanders.

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 $\phi^3\delta\ge (35\log n)^3 k$, 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).