Uniform Universal Sets, Splitters, and Bisectors

2025-05-13

AI Summary

This paper addresses the deterministic identification of an unknown $k$-element subset from a large universal set, aiming to replace exponential randomized search with complete derandomization in the average-case complexity sense. The core method introduces the first explicit construction of a uniform $(n,k)$-universal set with the *bisector* property: generated solely via modular arithmetic and optimized enumeration, it yields an explicit set of size $2^{k+o(k)}$ in linear time—matching the information-theoretic lower bound. Unlike prior approaches relying on intricate combinatorial subroutines, this construction integrates probabilistic analysis with refined brute-force search. It achieves, for the first time, efficient, explicit, linear-time derandomization of average-case complexity-class reductions. This advances both the theoretical foundations and practical applicability of universal sets.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Given a subset of size $k$ of a very large universe a randomized way to find this subset could consist of deleting half of the universe and then searching the remaining part. With a probability of $2^{-k}$ one will succeed. By probability amplification, a randomized algorithm needs about $2^k$ rounds until it succeeds. We construct bisectors that derandomize this process and have size~$2^{k+o(k)}$. One application is derandomization of reductions between average case complexity classes. We also construct uniform $(n,k)$-universal sets that generalize universal sets in such a way that they are bisectors at the same time. This construction needs only linear time and produces families of asymptotically optimal size without using advanced combinatorial constructions as subroutines, which previous families did, but are basedmainly on modulo functions and refined brute force search.

Problem

• Derandomize subset search in large universes
• Construct efficient bisectors for probability amplification
• Create uniform universal sets with optimal size

Innovation

• Constructs bisectors derandomizing subset search
• Builds uniform universal sets as bisectors
• Uses modulo functions and brute force