Fast Witness Extraction Using a Decision Oracle

TL;DR

This paper presents a method to efficiently extract witnesses for combinatorial problems using a decision oracle, leveraging group testing techniques to achieve scalable algorithms with practical implementation insights.

Contribution

It introduces a novel approach to convert decision algorithms into witness extraction algorithms with logarithmic query complexity, applicable to algebra-based FPT algorithms.

Findings

Witness extraction achieved with O(k log n) queries.

Practical implementation demonstrated on the k-path problem.

Optimization of finite field arithmetic enhances performance.

Abstract

The gist of many (NP-)hard combinatorial problems is to decide whether a universe of $n$ elements contains a witness consisting of $k$ elements that match some prescribed pattern. For some of these problems there are known advanced algebra-based FPT algorithms which solve the decision problem but do not return the witness. We investigate techniques for turning such a YES/NO-decision oracle into an algorithm for extracting a single witness, with an objective to obtain practical scalability for large values of $n$. By relying on techniques from combinatorial group testing, we demonstrate that a witness may be extracted with $O(k\log n)$ queries to either a deterministic or a randomized set inclusion oracle with one-sided probability of error. Furthermore, we demonstrate through implementation and experiments that the algebra-based FPT algorithms are practical, in particular in the setting of the $k$-path problem. Also discussed are engineering issues such as optimizing finite field arithmetic.