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

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 $mn^{{1+o(1)}\min{k,} \phi<sup>{-1}}$ total time, assuming $\phi^3\delta\ge (35\log n)³ k$, or (2) each of length at most $n<sup>{o(1)}/\phi$ and in total $m<sup>{1+o(1)}$ time, assuming $\phi³ \delta \ge n^{o(1)} k$. 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).