A Cut-Matching Game for Constant-Hop Expanders

This paper extends and generalizes the well-known cut-matching game framework and provides a novel cut-strategy that produces constant-hop expanders. Constant-hop expanders are a significant strengthening of regular expanders with the additional guarantee that any demand can be (obliviously) routed along constant-hop flow-paths - in contrast to the $\Omega(\log n)$-hop paths in expanders. Cut-matching games for expanders are key tools for obtaining linear-time approximation algorithms for many hard problems, including finding (balanced or approximately-largest) sparse cuts, certifying the expansion of a graph by embedding an (explicit) expander, as well as computing expander decompositions, hierarchical cut decompositions, oblivious routings, multi-cuts, and multi-commodity flows. The cut-matching game of this paper is crucial in extending this versatile and powerful machinery to constant-hop and length-constrained expanders and has been already been extensively used. For example, as a key ingredient in several recent breakthroughs, including, computing constant-approximate $k$-commodity (min-cost) flows in $(m+k)^{1+\epsilon}$ time as well as the optimal constant-approximate deterministic worst-case fully-dynamic APSP-distance oracle - in all applications the constant-approximation factor directly traces to and crucially relies on the expanders from a cut-matching game guaranteeing constant-hop routing paths.