Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Monotone Circuit Complexity of Matching

Bruno Cavalar, Mika Göös, Artur Riazanov, Anastasia Sofronova, Dmitry Sokolov

TL;DR

It is shown that the perfect matching function on $n$-vertex graphs requires monotone circuits of size $n^{\Omega(1)}}} and this improves on the lower bound of Razborov (1985).

Abstract

We show that the perfect matching function on $n$-vertex graphs requires monotone circuits of size $\smash{2^{n^{Ω(1)}}}$. This improves on the $n^{Ω(\log n)}$ lower bound of Razborov (1985). Our proof uses the standard approximation method together with a new sunflower lemma for matchings.

Monotone Circuit Complexity of Matching

TL;DR

It is shown that the perfect matching function on -vertex graphs requires monotone circuits of size $n^{\Omega(1)}}} and this improves on the lower bound of Razborov (1985).

Abstract

We show that the perfect matching function on -vertex graphs requires monotone circuits of size . This improves on the lower bound of Razborov (1985). Our proof uses the standard approximation method together with a new sunflower lemma for matchings.

Paper Structure

This paper contains 15 sections, 7 theorems, 12 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Technique: Matching sunflowers
  3. Other related work
  4. Matching Sunflower Lemma
  5. Reduction to blocky families.
  6. Finding a "vertex" sunflower.
  7. Finding a matching sunflower.
  8. Approximation method
  9. Proof of Lemma \ref{['lem:dnfs-are-dumb-intro']}
  10. Proof of Lemma \ref{['lem:dnfs-approximate-intro']}
  11. Notation.
  12. Proof overview (via plucking).
  13. Approximation.
  14. Proofs of Theorems \ref{['thm:l']} and \ref{['thm:ac0']}
  15. Discussion

Key Result

Theorem 1

$\textsc{Match}$ requires monotone circuits of size at least $2^{n^{1/3-o(1)}}$.

Figures (2)

  • Figure 1: Matching sunflower ${\mathcal{M}}' \subseteq {\mathcal{M}}$ with core $D = \{ab, de\}$ constructed out of a vertex sunflower ${\mathcal{V}}' \subseteq {\mathcal{V}}$ with core $K = \bigcap {\mathcal{V}}' = K_1 \sqcup K_2$ where $K_1 = \{c, f\}$ and $K_2= \{a, b, d, e\}$. The shaded regions indicate where non-core edges of matchings in ${\mathcal{M}}'$ can occur.
  • Figure 2: Example of an input $x\in\mathop{\mathrm{supp}}\nolimits({\mathcal{D}}_0)$ (given by the black/white vertex colouring) that causes an error when plucking a sunflower ${\mathcal{M}}'$ (from \ref{['fig:m-sunflower']}). Edges $ab$ and $de$ in the core $K=\bigcap{\mathcal{M}}'$ are monochromatic, which means $t_K(x)=1$. However, the dashed edges represent a petal $M\setminus K$, $M\in{\mathcal{M}}'$, that contains a bichromatic edge $cc'$, which means $t_M(x)=0$.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1: Robust Sunflower Lemma bcw21
  • Definition 1: Odd cut distribution ${\mathcal{D}}_0$
  • Lemma 2: Matching Sunflower Lemma
  • Claim 1
  • proof
  • Lemma 3
  • Lemma 4
  • ...and 4 more