Reconstructing Sets of Strings from Their k-way Projections: Algorithms & Complexity

TL;DR

This paper introduces the String Set Reconstruction problem, exploring its computational complexity and proposing a new algorithm based on overlap graphs to reconstruct string sets from k-way projections, with experimental validation.

Contribution

It presents the first algorithm for reconstructing string sets from k-way projections using modified overlap graphs, addressing non-contiguous k-mers and analyzing complexity.

Findings

The problem is computationally hard with inapproximability results.

The proposed algorithm is efficient and scalable based on experimental results.

Analytic approximations explain the observed complexity scaling.

Abstract

Graphs are a powerful tool for analyzing large data sets, but many real-world phenomena involve interactions that go beyond the simple pairwise relationships captured by a graph. In this paper we introduce and study a simple combinatorial model to capture higher order dependencies from an algorithms and computational complexity perspective. Specifically, we introduce the String Set Reconstruction problem, which asks when a set of strings can be reconstructed from seeing only the k-way projections of strings in the set. This problem is distinguished from genetic reconstruction problems in that we allow projections from any k indices and we maintain knowledge of those indices, but not which k-mer came from which string. We give several results on the complexity of this problem, including hardness results, inapproximability, and parametrized complexity. Our main result is the introduction of a new algorithm for this problem using a modified version of overlap graphs from genetic reconstruction algorithms. A key difference we must overcome is that in our setting the k-mers need not be contiguous, unlike the setting of genetic reconstruction. We exhibit our algorithm's efficiency in a variety of experiments, and give high-level explanations for how its complexity is observed to scale with various parameters. We back up these explanation with analytic approximations. We also consider the related problems of: whether a single string can be reconstructed from the k-way projections of a given set of strings, and finding the largest k at which we get no information about the original data set from its k-way projections (i.e., the largest $k$ for which it is "k-wise independent").