Derandomizing Isolation In Catalytic Logspace

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.

💡 Research Summary

The paper “Derandomizing Isolation In Catalytic Logspace” investigates how the isolation lemma, a central tool for randomized reductions, can be effectively derandomized within the catalytic logspace (CL) model and explores the consequences of this derandomization for several complexity classes and algorithmic problems.

1. Search‑to‑Weighted‑Decision Reduction in CL
The authors first show that for any NP language L, a minimum‑weight witness can be found in CL given oracle access to the corresponding weighted‑decision problem, provided the weights are polynomially bounded. The reduction follows the classic Mulmuley‑Vazirani‑Vazirani (MVV) isolation approach but replaces the random bits with the contents of the catalytic tape. If a weight assignment isolates a unique minimum‑weight witness, the weighted‑decision oracle can be used to extract it by a sequence of queries. If the assignment fails to isolate, the authors prove that the weight function can be compressed by O(log n) bits, effectively “saving” space on the catalytic tape. By iterating over polynomially many weight assignments, either an isolating assignment is found (yielding the witness) or enough space is saved to run any polynomial‑space algorithm that already knows a witness exists (since L∈NP). This yields a CL‑computable Turing reduction from search to weighted decision and, as concrete applications, gives CL algorithms for the search versions of planar perfect matching, planar exact (red‑blue) matching, and minimum‑weight arborescences in weighted digraphs.

2. The Oracle Class CLⁿᴾ₂‑Round
Motivated by the above reduction, the paper defines a new oracle class, CLⁿᴾ₂‑round, which allows a deterministic logspace machine with a polynomial‑size catalytic tape to ask non‑adaptive NP queries in two parallel rounds. Using the catalytic isolation technique, the authors place SearchSAT (the problem of finding a satisfying assignment) inside this class. They then adapt the Nisan‑Wigderson/Impagliazzo‑Wigderson hardness‑vs‑randomness framework to the catalytic setting: the catalytic tape is viewed as a collection of truth tables; NP queries test whether any of these tables can be computed by small circuits. If a hard table is found, a Nisan‑Wigderson pseudorandom generator (PRG) is built and any BPP algorithm can be derandomized; otherwise the saved space allows a brute‑force simulation in PSPACE. All required NP queries fit into two parallel rounds, establishing the inclusions
 BPP ⊆ CLⁿᴾ₂‑round, MA ⊆ CLⁿᴾ₂‑round, ZPPⁿᴾ