Randomised enumeration of small witnesses using a decision oracle

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 $n$ elements contains a witness consisting of $k$ 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 $f(k) \cdot poly(n)$, there is a randomised algorithm which enumerates all witnesses in time $e^{k + o(k)} \cdot f(k) \cdot poly(n) \cdot N$, where $N$ 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 which any given witness will be included with high probability. The enumeration algorithm also gives rise to an efficient algorithm to count the total number of witnesses when this number is small.