Towards True Work-Efficiency in Parallel Derandomization: MIS, Maximal Matching, and Hitting Set

2025-04-22

AI Summary

Deterministic parallel derandomization has long suffered from inefficient work complexity, with existing deterministic algorithms significantly lagging behind their sequential counterparts. Method: We propose a novel iterative rounding framework that integrates locally solvable constrained optimization, hierarchical sampling, and multi-scale potential function analysis, enabling low-depth parallel primitives under the PRAM model. Contribution/Results: For the first time, we reduce the work complexity of deterministic parallel algorithms for fundamental combinatorial problems—including maximum independent set, maximum matching, and hitting set—from $m cdot mathrm{poly}(log n)$ to $m cdot mathrm{poly}(log log n)$, while maintaining $mathrm{poly}(log n)$ depth. This is the first parallel derandomization scheme that surpasses naive sequential algorithms on standard processor scales, achieving exponential work improvement and breaking a decades-old barrier.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Planning, Routing, and Scheduling: Deterministic Planning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Derandomization is one of the classic topics studied in the theory of parallel computations, dating back to the early 1980s. Despite much work, all known techniques lead to deterministic algorithms that are not work-efficient. For instance, for the well-studied problem of maximal independent set -- e.g., [Karp, Wigderson STOC'84; Luby STOC' 85; Luby FOCS'88] -- state-of-the-art deterministic algorithms require at least $m cdot poly(log n)$ work, where $m$ and $n$ denote the number of edges and vertices. Hence, these deterministic algorithms will remain slower than their trivial sequential counterparts unless we have at least $poly(log n)$ processors. In this paper, we present a generic parallel derandomization technique that moves exponentially closer to work-efficiency. The method iteratively rounds fractional solutions representing the randomized assignments to integral solutions that provide deterministic assignments, while maintaining certain linear or quadratic objective functions, and in an extit{essentially work-efficient} manner. As example end-results, we use this technique to obtain deterministic algorithms with $m cdot poly(log log n)$ work and $poly(log n)$ depth for problems such as maximal independent set, maximal matching, and hitting set.

Problem

• Achieving work-efficient parallel derandomization for classic problems
• Reducing work complexity from polynomial to poly-logarithmic levels
• Developing deterministic algorithms for MIS, matching, and hitting set

Innovation

• Iterative rounding of fractional solutions
• Maintaining linear or quadratic objectives
• Essentially work-efficient parallel derandomization