Derandomizing Isolation In Catalytic Logspace

TL;DR

This paper explores derandomization techniques in catalytic logspace, providing algorithms for search problems, analyzing the class CL^{NP}_{2-round}, and establishing containment results for reachability and LogCFL within catalytic classes.

Contribution

It introduces a catalytic logspace algorithm for minimum weight witnesses, analyzes the structure of CL^{NP}_{2-round}, and extends catalytic bounds to reachability and LogCFL.

Findings

Catalytic logspace algorithms for planar perfect matching and weighted arborescences.

CL^{NP}_{2-round} contains SearchSAT, BPP, MA, and ZPP^{NP[1]} classes.

Reachability and LogCFL are contained in certain catalytic classes.

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 CTISP[poly(n),logn,log^2n]^UL 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 the workspace of the oracle and the catalytic space of the base machine. Finally, we extend these catalytic upper bounds to LogCFL.