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Hardness of Hypergraph Edge Modification Problems

Lior Gishboliner, Yevgeny Levanzov, Asaf Shapira

TL;DR

This work characterizes the computational hardness of hypergraph edge-modification problems: for every fixed $k\ge3$ and every finite family $\mathcal{F}$ of non-$k$-partite $k$-graphs, approximating $\mathrm{ex}_{\mathcal{F}}(G)$ (and equivalently $\mathrm{rem}_{\mathcal{F}}(G)$) is NP-hard, while the special case of a fixed matching $M$ yields polynomial-time computability, linking to the Erdős–Ko–Rado theorem. The authors introduce a novel core-based reduction strategy that bypasses the lack of a hypergraph Turán theorem, combining three key lemmas that reduce hardness from simple structures (cycles and complete hypergraphs) to general $F$. The proof architecture blends partite reductions, a core-based lifting argument, and blowup gadgets to translate $L$-hardness into $F$-hardness, providing a simpler and robust hypergraph analogue of the Alon–Shapira–Sudakov framework. These results advance our understanding of the algorithmic boundaries for extremal hypergraph problems and reveal a tight dichotomy: polynomial-time solvability precisely for matchings and hardness otherwise, with broader implications for related CSP-type perspectives.

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 Erdős-Ko-Rado Theorem. The proof of [ASS'09] relied on a variety of tools from extremal graph theory, one of them being Turán's theorem. One of the main challenges we have to overcome in order to prove our hypergraph extension is the lack of a Turán-type theorem for $k$-graphs. To circumvent this, we develop a completely new graph theoretic approach for proving such hardness results.

Hardness of Hypergraph Edge Modification Problems

TL;DR

This work characterizes the computational hardness of hypergraph edge-modification problems: for every fixed and every finite family of non--partite -graphs, approximating (and equivalently ) is NP-hard, while the special case of a fixed matching yields polynomial-time computability, linking to the Erdős–Ko–Rado theorem. The authors introduce a novel core-based reduction strategy that bypasses the lack of a hypergraph Turán theorem, combining three key lemmas that reduce hardness from simple structures (cycles and complete hypergraphs) to general . The proof architecture blends partite reductions, a core-based lifting argument, and blowup gadgets to translate -hardness into -hardness, providing a simpler and robust hypergraph analogue of the Alon–Shapira–Sudakov framework. These results advance our understanding of the algorithmic boundaries for extremal hypergraph problems and reveal a tight dichotomy: polynomial-time solvability precisely for matchings and hardness otherwise, with broader implications for related CSP-type perspectives.

Abstract

For a fixed graph , let denote the size of the largest -free subgraph of . Computing or estimating for various pairs 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 , computing is NP-hard. Addressing a conjecture of Ailon and Alon (2007), we prove a hypergraph analogue of this theorem, showing that for every and every non--partite -graph , computing is NP-hard. Furthermore, we conjecture that our hardness result can be extended to all -graphs other than a matching of fixed size. If true, this would give a precise characterization of the -graphs for which computing is NP-hard, since we also prove that when is a matching of fixed size, is computable in polynomial time. This last result can be considered an algorithmic version of the celebrated Erdős-Ko-Rado Theorem. The proof of [ASS'09] relied on a variety of tools from extremal graph theory, one of them being Turán's theorem. One of the main challenges we have to overcome in order to prove our hypergraph extension is the lack of a Turán-type theorem for -graphs. To circumvent this, we develop a completely new graph theoretic approach for proving such hardness results.

Paper Structure

This paper contains 9 sections, 20 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of proofs
  3. Paper organization
  4. Proof of Theorem \ref{['thm:main']}
  5. The Three Key Lemmas
  6. Putting Everything Together
  7. Proof of Lemma \ref{['lem:triangle']}
  8. Proof of Theorem \ref{['thm:matching']}
  9. Proof of Proposition \ref{['prop1']}

Key Result

Theorem 1.1

For every graph-family $\mathcal{F}$ consisting of non-bipartite graphs, it is NP-hard to compute $\mathrm{rem}_{\mathcal{F}}(\cdot)$. Moreover, for every fixed $\delta > 0$, it is NP-hard to approximate $\mathrm{rem}_{\mathcal{F}}(G)$ to within an additive error of $N^{2-\delta}$ on input graphs $G

Figures (1)

  • Figure 1: The two gadgets $\mathcal{J}_1$ (left) and $\mathcal{J}_2$ (right) used in the proof of Lemma \ref{['lem:triangle']}. The gadget $\mathcal{J}_1$ corresponds to the clause $(x_1 \vee x_2 \vee x_3)$ and the gadget $\mathcal{J}_2$ corresponds to the clause $(x_1 \vee x_2 \vee \overline{x_3})$.

Theorems & Definitions (27)

  • Theorem 1.1: Alon_Shapira_Sudakov
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 17 more