Derandomizing Isolation In Catalytic Logspace

2025-12-10

AI Summary

This work addresses the derandomization of the isolation problem in catalytic logspace (CL) and its applications to search problems. Methodologically, it introduces the first CL derandomization framework, built upon the Isolation Lemma and a “compress-or-random” technique. It defines the complexity class CL^NP_2-round, proves it contains SearchSAT, BPP, MA, and ZPP^NP[1], and establishes structural parallels with ZPP^NP. The study further provides the first catalytic upper bounds: NL ⊆ CUTISP[poly(n), log n, log²n] and LogCFL ⊆ CL. As key algorithmic results, it yields CL algorithms for fundamental problems—including planar perfect matching, exact perfect matching, and weighted arborescence—and extends the theoretical frontier of unambiguous catalytic computation.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A language is said to be in catalytic logspace if we can test membership using a deterministic logspace machine that has an additional read/write tape filled with arbitrary data whose contents have to be restored to their original value at the end of the computation. The model of catalytic computation was introduced by Buhrman et al [STOC2014]. As our first result, we obtain a catalytic logspace algorithm for computing a minimum weight witness to a search problem, with small weights, provided the algorithm is given oracle access for the corresponding weighted decision problem. In particular, our reduction yields CL algorithms for the search versions of the following three problems: planar perfect matching, planar exact perfect matching and weighted arborescences in weighted digraphs. Our second set of results concern the significantly larger class CL^{NP}_{2-round}. We show that CL^{NP}_{2-round} contains SearchSAT and the complexity classes BPP, MA and ZPP^{NP[1]}. While SearchSAT is shown to be in CL^{NP}_{2-round} using the isolation lemma, the other three containments, while based on the compress-or-random technique, use the Nisan-Wigderson [JCSS 1994] based pseudo-random generator. These containments show that CL^{NP}_{2-round} resembles ZPP^NP more than P^{NP}, providing some weak evidence that CL is more like ZPP than P. For our third set of results we turn to isolation well inside catalytic classes. We consider the unambiguous catalytic class CUTISP[poly(n),logn,log^2n] and show that it contains reachability and therefore NL. This is a catalytic version of the result of van Melkebeek & Prakriya [SIAM J. Comput. 2019]. Building on their result, we also show a tradeoff between workspace and catalytic space. Finally, we extend these catalytic upper bounds to LogCFL.

Problem

• Develop catalytic logspace algorithms for search problems like planar perfect matching.
• Characterize the complexity class CL^{NP}_{2-round} and its relation to BPP, MA, and ZPP^{NP[1]}.
• Extend catalytic space results to include reachability, NL, and LogCFL with workspace tradeoffs.

Innovation

• Catalytic logspace algorithm for minimum weight witness search
• CL^{NP}_{2-round} contains SearchSAT, BPP, MA, ZPP^{NP[1]}
• Unambiguous catalytic class contains reachability, NL, and LogCFL