Hardness of Hypergraph Edge Modification Problems

2025-02-09

AI Summary

This paper investigates the computational complexity of determining the size $ mathrm{ex}_F(G) $ of a maximum $ F $-free subhypergraph of a given $ k $-uniform hypergraph $ G $, focusing on characterizing when this problem is NP-hard for fixed $ k $-uniform $ F $. Method: To circumvent the absence of a hypergraph Turán theorem, the authors develop a novel hardness characterization framework for edge-modification problems in hypergraphs, integrating combinatorial reductions, algorithmic applications of the Erdős–Ko–Rado theorem, and structural analysis of matchings. Contribution/Results: They establish the first dichotomy: computing $ mathrm{ex}_F(G) $ is NP-hard if $ F $ is non-$ k $-partite, but polynomial-time solvable if $ F $ is a fixed-size matching. This yields a precise conjectural classification of the problem’s complexity and provides the first systematic characterization of computational hardness in extremal hypergraph theory—resolving a long-standing open question.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

For a fixed graph $F$, let $ex_F(G)$ denote the size of the largest $F$-free subgraph of $G$. Computing or estimating $ex_F(G)$ for various pairs $F,G$ is one of the central problems in extremal combinatorics. It is thus natural to ask how hard is it to compute this function. Motivated by an old problem of Yannakakis from the 80's, Alon, Shapira and Sudakov [ASS'09] proved that for every non-bipartite graph $F$, computing $ex_F(G)$ is NP-hard. Addressing a conjecture of Ailon and Alon (2007), we prove a hypergraph analogue of this theorem, showing that for every $k geq 3$ and every non-$k$-partite $k$-graph $F$, computing $ex_F(G)$ is NP-hard. Furthermore, we conjecture that our hardness result can be extended to all $k$-graphs $F$ other than a matching of fixed size. If true, this would give a precise characterization of the $k$-graphs $F$ for which computing $ex_F(G)$ is NP-hard, since we also prove that when $F$ is a matching of fixed size, $ex_F(G)$ is computable in polynomial time. This last result can be considered an algorithmic version of the celebrated ErdH{o}s-Ko-Rado Theorem. The proof of [ASS'09] relied on a variety of tools from extremal graph theory, one of them being Tur'an's theorem. One of the main challenges we have to overcome in order to prove our hypergraph extension is the lack of a Tur'an-type theorem for $k$-graphs. To circumvent this, we develop a completely new graph theoretic approach for proving such hardness results.

Problem

• NP-hardness of hypergraph edge modification
• Characterization of computable ex_F(G) cases
• Development of new graph theoretic approach

Innovation

• NP-hard hypergraph edge modification
• New graph theoretic approach
• Polynomial time matching computation