Randomised Enumeration of Small Witnesses Using a Decision Oracle

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) · poly(n), there is a randomised algorithm which enumerates all witnesses in time e k+o(k) · f (k) · poly(n) · 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.

Approximately counting and sampling small witnesses using a colourful decision oracle

“…Thus we can consider this to be a "fine-grained FPT overhead". Theorems 1 and 2 can therefore be applied immediately to any self-contained k-witness problem (see [39]); that is, any problem with integer parameter k in which we are interested in the existence of witnesses consisting of k-element subsets of some given universe, and we have the ability to quickly test whether any given k-element set is such a witness. Examples include weight-k solutions to CSPs, size-k solutions to database queries, and sets of k vertices in a (weighted) graph or hypergraph which induce a sub(hyper)graph with specific properties.…”

Efficiently enumerating hitting sets of hypergraphs arising in data profiling

“…Creignou et al give a general overview of the field of parameterised enumeration [16]. The parameterised complexity of extension problems in particular has been treated by Meeks [46] and, very recently, by Casel et al [11].…”

Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling

“…It is a common pattern in the design of enumeration algorithms to base them on an extension oracle [6,26,38,39,42,45,46,51,52]. The oracle decides, for a given collection of vertices, whether there is a solution using these vertices (possibly avoiding some other set).…”

“…Thus, modifications are necessary for practical use, cf. [46,52]. For transversals, this space reduction can achieved via a decision tree pruned by the extension oracle.…”

“…Creignou et al give a general overview of the field of parameterised enumeration [16]. The parameterised complexity of extension problems in particular has been treated by Meeks [46] and, very recently, by Casel et al [11].…”