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

Eduard Eiben, Robert Ganian, Iyad Kanj, Sebastian Ordyniak, Stefan Szeider

TL;DR

This work analyzes the parameterized complexity of Pow-Hyp-IS-Completion, the data-completion version of Independent Set on induced subgraphs of $r$-th powers of the hypercube. The authors provide a complete complexity picture: a constructive FPT algorithm parameterized by $k+r$ using a sunflower-based pruning framework, together with paraNP-hardness for fixed $r$ and W[1]-hardness in $k$ alone, established via reductions from Independent Set. They also show that FO-model checking on induced subgraphs of hypercubes remains as hard as on general graphs, indicating that FO-tractability does not generalize to all FO-definable problems on this graph class. Moreover, they connect these results to clustering formulations by mapping several FO-definable problems to well-known graph problems on the $igl( ext{induced subgraphs of } Q_digr)^r$ class, providing insights into both algorithmic strategies for incomplete data and fundamental limits of FO-based approaches. The findings have practical implications for designing efficient, parameterized algorithms in incomplete-data clustering and clarifying the boundaries of FO-model-checking tractability in hypercube-derived graph families.

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.

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

TL;DR

This work analyzes the parameterized complexity of Pow-Hyp-IS-Completion, the data-completion version of Independent Set on induced subgraphs of -th powers of the hypercube. The authors provide a complete complexity picture: a constructive FPT algorithm parameterized by using a sunflower-based pruning framework, together with paraNP-hardness for fixed and W[1]-hardness in alone, established via reductions from Independent Set. They also show that FO-model checking on induced subgraphs of hypercubes remains as hard as on general graphs, indicating that FO-tractability does not generalize to all FO-definable problems on this graph class. Moreover, they connect these results to clustering formulations by mapping several FO-definable problems to well-known graph problems on the class, providing insights into both algorithmic strategies for incomplete data and fundamental limits of FO-based approaches. The findings have practical implications for designing efficient, parameterized algorithms in incomplete-data clustering and clarifying the boundaries of FO-model-checking tractability in hypercube-derived graph families.

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.
Paper Structure (7 sections, 10 theorems, 1 equation)

This paper contains 7 sections, 10 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Parameterized Complexity of Pow-Hyp-IS-Completion
  4. Dealing with Unstructured Missing Data
  5. Lower Bounds
  6. On Graph Problems on Induced Subgraphs of the Hypercubes
  7. Conclusion

Key Result

Lemma 1

Let $\mathcal{F}$ be a family of subsets of a universe $U$, each of cardinality exactly $b$, and let $a \in \mathbb{N}$. If $|\mathcal{F}|\geq b!(a-1)^{b}$, then $\mathcal{F}$ contains a sunflower $\mathcal{F}'$ of cardinality at least $a$. Moreover, $\mathcal{F}'$ can be computed in time polynomial

Theorems & Definitions (10)

  • Lemma 1: Erdos60FlumGrohe06
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Corollary 8
  • Lemma 9
  • Theorem 10