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

TL;DR

This paper demonstrates a significant complexity gap in sink-finding algorithms between realizable and non-realizable USOs, specifically within Matoušek-type USOs, showing that realizability can be exploited for more efficient algorithms.

Contribution

The paper establishes the first known complexity gap for sink-finding in USOs by analyzing Matoušek-type USOs, providing algorithms and lower bounds for both cases.

Findings

Realizable USOs allow sink-finding with O(log^2 n) queries.

Non-realizable USOs require exactly n queries.

A proven complexity gap exists for a specific USO subclass.

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.