Derandomizing Isolation In Catalytic Logspace
V. Arvind, Srijan Chakraborty, Samir Datta
TL;DR
This work advances catalytic logspace by derandomizing isolation-based search-to-decision reductions, yielding CL algorithms for several search problems and establishing the position of CL^{NP}_{2-round} among BPP, MA, and ZPP^{NP[1]}. It also shows that NL and LogCFL can be contained within refined catalytic-unambiguous frameworks, with precise space-workload tradeoffs, and extends these ideas to broader oracle-assisted settings. The results collectively suggest CL behaves more like ZPP-like classes than P in several regimes and provide a toolkit for derandomization and upper-bound evidence in space-bounded computation. Overall, the paper broadens the applicability of catalytic computation to fundamental reachability and circuit-evaluation problems while clarifying the landscape of catalytic complexity 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.