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A note on the computational complexity of weak saturation

Martin Tancer, Mykhaylo Tyomkyn

TL;DR

The paper investigates the computational complexity of the weak saturation problem, proving NP-hardness for the case when the pattern is $K_3$ by constructing a reduction from 3-SAT. The authors establish a novel link between weak saturation and the shellability/collapsibility of simplicial complexes, leveraging a 2D complex $K_\phi$ associated to a 3-CNF formula $\phi$ and its barycentric subdivisions to connect satisfiability with topological properties. By extending to $L_\phi=\mathrm{sd}^2K_\phi$, they translate shellability and collapsibility questions into a weak-saturation decision problem $\mathrm{wsat}(L_\phi^{(1)},K_3)$, yielding a polynomial-time reduction from 3-SAT to wsat. This work highlights fundamental hardness in classifying weak saturation numbers and suggests deep connections between combinatorial graph processes and topological properties of simplicial complexes.

Abstract

We prove that determining the weak saturation number of a host graph $F$ with respect to a pattern graph $H$ is already a computationally hard problem when $H$ is the triangle. As our main tool we establish a connection between weak saturation and shellability of simplicial complexes.

A note on the computational complexity of weak saturation

TL;DR

The paper investigates the computational complexity of the weak saturation problem, proving NP-hardness for the case when the pattern is by constructing a reduction from 3-SAT. The authors establish a novel link between weak saturation and the shellability/collapsibility of simplicial complexes, leveraging a 2D complex associated to a 3-CNF formula and its barycentric subdivisions to connect satisfiability with topological properties. By extending to , they translate shellability and collapsibility questions into a weak-saturation decision problem , yielding a polynomial-time reduction from 3-SAT to wsat. This work highlights fundamental hardness in classifying weak saturation numbers and suggests deep connections between combinatorial graph processes and topological properties of simplicial complexes.

Abstract

We prove that determining the weak saturation number of a host graph with respect to a pattern graph is already a computationally hard problem when is the triangle. As our main tool we establish a connection between weak saturation and shellability of simplicial complexes.
Paper Structure (5 sections, 3 theorems, 1 equation, 1 figure)

This paper contains 5 sections, 3 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Weak saturation.
  3. Simplicial complexes, shellability and collapsibility.
  4. Hardness of shellability.
  5. The proof of Theorem \ref{['t:auxiliary']}

Key Result

Theorem 1

Given on input a graph $F$ with $n$ vertices, it is NP-hard to decide whether $\mathop{\mathrm{wsat}}\nolimits(F, K_3) = n-1$.

Figures (1)

  • Figure 1: The figure displays three options how $\vartheta_i$ may meet $L_\phi[\vartheta_1,\dots,\vartheta_{i-1}]$ according to the number of shared edges. (The third displayed case is not fully realistic globally because the displayed $L_\phi[\vartheta_1,\dots,\vartheta_{i-1}]$ is not shellable. Any realistic example unfortunately contains a nontrivial 2-dimensional homology, which makes it harder to depict.)

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2: Essentially Proposition 8 from goaoc-patak-patakova-tancer-wagner19
  • Theorem 3
  • proof : Proof of Theorem \ref{['t:main']}
  • proof : Proof of (ii) $\Rightarrow$ (iv)
  • proof : Proof of (iv) $\Rightarrow$ (v)
  • proof : Proof of (v) $\Rightarrow$ (iii)