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

TL;DR

This work addresses deterministic routing of many demand pairs via edge-disjoint paths in expander-like graphs, achieving almost-linear-time results under weaker expansion assumptions. The core strategy is a reduction to hypergraph perfect matching under a strong Haxell-type condition, augmented by a novel half-layer framework that obviates explicit hypergraph construction. The authors prove that demand-path hypergraphs of expanders satisfy a strong Haxell condition and implement two half-layer oracles—one maximal (via BFS/Dinitz-style methods) and one approximate (via low-step multicommodity flow)—to obtain two running-time trade-offs: almost-quadratic time under polylog factors and almost-linear time under milder conditions. This framework yields the first near-linear deterministic algorithms for disjoint paths on expanders and enables applications such as splitting expanders into sparser expanders, with broader implications for multicommodity routing and fair allocation problems.

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 $φ$ and minimum degree $δ$, 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 $18 \log (n)/φ$ and in $mn^{1+o(1)}\min\{k, φ^{-1}\}$ total time, assuming $φ^3δ\ge (35\log n)^3 k$, or (2) each of length at most $n^{o(1)}/φ$ and in total $m^{1+o(1)}$ time, assuming $φ^3 δ\ge n^{o(1)} k$. 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).

Theorems & Definitions (57)

• theorem 1.1
• corollary 1.2
• definition 1.3: Bipartite Hypergraph Matching
• definition 1.4: Strong Haxell’s condition
• theorem 1.5
• definition 3.1
• proposition 3.2
• proof
• lemma 3.2
• proof : Proof of \ref{['thm: almost linear time hypergraph perfect matching']}