From Data Completion to Problems on Hypercubes: A Parameterized Analysis of the Independent Set Problem

TL;DR

This paper explores the parameterized complexity of the Independent Set problem on hypercube-induced subgraphs, revealing fixed-parameter tractability under certain parameters and demonstrating the limits of FO model checking complexity.

Contribution

It provides a comprehensive analysis of the Independent Set problem on hypercube graphs and shows the boundaries of fixed-parameter tractability for FO-definable problems.

Findings

Independent Set is fixed-parameter tractable w.r.t. solution size and hypercube power.

FO model checking remains hard on hypercube-induced subgraphs, matching general graph complexity.

The study extends understanding of data completion problems in machine learning and clustering contexts.

Abstract

Several works have recently investigated the parameterized complexity of data completion problems, motivated by their applications in machine learning, and clustering in particular. Interestingly, these problems can be equivalently formulated as classical graph problems on induced subgraphs of powers of partially-defined hypercubes. In this paper, we follow up on this recent direction by investigating the Independent Set problem on this graph class, which has been studied in the data science setting under the name Diversity. We obtain a comprehensive picture of the problem's parameterized complexity and establish its fixed-parameter tractability w.r.t. the solution size plus the power of the hypercube. Given that several such FO-definable problems have been shown to be fixed-parameter tractable on the considered graph class, one may ask whether fixed-parameter tractability could be extended to capture all FO-definable problems. We answer this question in the negative by showing that FO model checking on induced subgraphs of hypercubes is as difficult as FO model checking on general graphs.