Randomised enumeration of small witnesses using a decision oracle
Kitty Meeks

TL;DR
This paper presents a randomized enumeration algorithm that efficiently lists all small witnesses in combinatorial problems using a decision oracle, with applications to counting and probabilistic witness retrieval.
Contribution
It introduces a novel randomized enumeration method that leverages decision algorithms to find all witnesses with high probability, extending previous decision-to-search transformations.
Findings
Enumeration time is exponential in k but polynomial in n and the number of witnesses.
The method can handle decision algorithms with false negatives, providing high-probability witness lists.
Efficient counting of witnesses is possible when their total number is small.
Abstract
Many combinatorial problems involve determining whether a universe of elements contains a witness consisting of elements which have some specified property. In this paper we investigate the relationship between the decision and enumeration versions of such problems: efficient methods are known for transforming a decision algorithm into a search procedure that finds a single witness, but even finding a second witness is not so straightforward in general. We show that, if the decision version of the problem can be solved in time , there is a randomised algorithm which enumerates all witnesses in time , where is the total number of witnesses. If the decision version of the problem is solved by a randomised algorithm which may return false negatives, then the same method allows us to output a list of witnesses in…
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