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

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).