Realizability Makes a Difference: A Complexity Gap for Sink-Finding in USOs

Abstract: Algorithms for finding the sink in Unique Sink Orientations (USOs) of the hypercube can be used to solve many algebraic and geometric problems, most importantly including the P-Matrix Linear Complementarity Problem and Linear Programming. The realizable USOs are those that arise from the reductions of these problems to the USO sink-finding problem. Finding the sink of realizable USOs is thus highly practically relevant, yet it is unknown whether realizability can be exploited algorithmically to find the sink more quickly. However, all (non-trivial) known unconditional lower bounds for sink-finding make use of USOs that are provably not realizable. This indicates that the sink-finding problem might indeed be strictly easier on realizable USOs. In this paper we show that this is true for a subclass of all USOs. We consider the class of Matou\v{s}ek-type USOs, which are a translation of Matou\v{s}ek's LP-type problems into the language of USOs. We show a query complexity gap between sink-finding in all, and sink-finding in only the realizable $n$-dimensional Matou\v{s}ek-type USOs. We provide concrete deterministic algorithms and lower bounds for both cases, and show that in the realizable case $O(log^2 n)$ vertex evaluation queries suffice, while in general exactly $n$ queries are needed. The Matou\v{s}ek-type USOs are the first USO class found to admit such a gap.