Algorithms for finding the sink in Unique Sink Orientations (USOs) of the hypercube can be used to solve many algebraic, geometric, and combinatorial 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 relevant to various theoretical fields, yet it is unknown whether realizability can be exploited algorithmically to find the sink more quickly. However, the best 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 Matoušek-type USOs, which are a translation of Matoušek’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 Matoušek-type USOs. We provide concrete deterministic algorithms and lower bounds for both cases, and show that in the realizable case \(O(\log ^2n)\) vertex evaluation queries suffice, while in general exactly n queries are needed. The Matoušek-type USOs are the first USO class found to admit such a gap.
- Unique Sink Orientation
- Query Complexity
A version of this paper containing the omitted proofs and additional explanations can be accessed at https://arxiv.org/abs/2207.05985.
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A P-matrix is a matrix whose determinants of all principal submatrices are positive.
Branching: a forest of rooted trees, with all edges directed away from the roots.
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We thank Bernd Gärtner for his valuable insights and feedback. This research was supported by the Swiss National Science Foundation under project no. 204320.
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Weber, S., Widmer, J. (2023). Realizability Makes A Difference: A Complexity Gap For Sink-Finding in USOs. In: Morin, P., Suri, S. (eds) Algorithms and Data Structures. WADS 2023. Lecture Notes in Computer Science, vol 14079. Springer, Cham. https://doi.org/10.1007/978-3-031-38906-1_47
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