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

TL;DR

The paper tackles the long-standing challenge of derandomizing parallel algorithms without sacrificing work-efficiency. It introduces a generic gradual-rounding technique that converts fractional randomized solutions into deterministic integral solutions while maintaining key objectives, enabling them to shrink the problem size by a constant factor at each step. By combining this rounding with a sophisticated hitting-set framework, the authors achieve MIS, maximal matching, and hitting set with $W=(m+n)\operatorname{poly}(\log\log n)$ and $\operatorname{poly}(\log n)$ depth, dramatically improving over prior poly$(\log n)$-overhead methods. This work advances the theoretical frontier of parallel derandomization and moves closer to true work-efficiency, with potential implications for a broader class of parallel problems.

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 \textit{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.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 1.4
• proof : Proof Sketch
• Lemma 2.1: Sorting Lemma
• proof
• Lemma 2.2: Bhatt et al. bhatt1991improved
• Lemma 2.3: Coloring Lemma
• proof