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CLIQUE as an AND of Polynomial-Sized Monotone Constant-Depth Circuits

Levente Bodnar

TL;DR

This work addresses the problem of representing $k$-CLIQUE on $n$-vertex graphs using monotone, depth-bounded, polynomial-sized circuits arranged as an AND. It introduces a new monotone one-sided switching lemma tailored to cliques and leverages CNF–DNF and DNF–CNF switchings under monotone restrictions to bound how well such circuits can approximate CLIQUE. The main result shows that any such AND must comprise at least $2^{n/4k}$ components to compute $k$-CLIQUE for sufficiently large $k$, while also establishing a strong one-sided correlation bound against random graphs. The findings reveal intrinsic limits of monotone, constant-depth circuit constructions for CLIQUE, contrasting with the relative efficiency of OR-based representations, and open questions about tight correlation bounds and logical expressiveness in the monotone setting.

Abstract

This paper shows that calculating $k$-CLIQUE on $n$ vertex graphs, requires the AND of at least $2^{n/4k}$ monotone, constant-depth, and polynomial-sized circuits, for sufficiently large values of $k$. The proof relies on a new, monotone, one-sided switching lemma, designed for cliques.

CLIQUE as an AND of Polynomial-Sized Monotone Constant-Depth Circuits

TL;DR

This work addresses the problem of representing -CLIQUE on -vertex graphs using monotone, depth-bounded, polynomial-sized circuits arranged as an AND. It introduces a new monotone one-sided switching lemma tailored to cliques and leverages CNF–DNF and DNF–CNF switchings under monotone restrictions to bound how well such circuits can approximate CLIQUE. The main result shows that any such AND must comprise at least components to compute -CLIQUE for sufficiently large , while also establishing a strong one-sided correlation bound against random graphs. The findings reveal intrinsic limits of monotone, constant-depth circuit constructions for CLIQUE, contrasting with the relative efficiency of OR-based representations, and open questions about tight correlation bounds and logical expressiveness in the monotone setting.

Abstract

This paper shows that calculating -CLIQUE on vertex graphs, requires the AND of at least monotone, constant-depth, and polynomial-sized circuits, for sufficiently large values of . The proof relies on a new, monotone, one-sided switching lemma, designed for cliques.
Paper Structure (5 sections, 4 theorems, 16 equations, 1 figure)

This paper contains 5 sections, 4 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Outline
  4. Proofs
  5. Concluding remarks

Key Result

Theorem 1

For any $c_{d}, c_{s}$ constants, large enough $n$ and $\log(n)^{c_{d}+6} < k,$ one cannot have less than $2^{n/4k}$ many monotone circuits $\{f_j\}$ on $\binom{n}{2}$ input bits, each with depth and size bounded by $c_{d}$ and $n^{c_{s}}$ respectively, such that $\bigwedge_j f_j$ computes $k$-CLIQU

Figures (1)

  • Figure 1: The trees on formula $f= q_1 \wedge q_2 \wedge q_3 = (12 \vee 13) \wedge (12 \vee 34) \wedge (14 \vee 25 \vee 45)$ with clauses numbered accordingly.

Theorems & Definitions (13)

  • Theorem 1
  • Proposition 2
  • Definition 3
  • Lemma 4: CNF to DNF switching with small clique error
  • Lemma 5: DNF to CNF switching
  • proof : Proof of Lemma \ref{['hard_lemma']}
  • Claim 7: Relations involving the $\mathcal{G}_d, \mathcal{A}_d, \mathcal{B}_d$ sets
  • proof : Proof of Claim \ref{['hard_lemma_claims']}
  • proof : Proof of Lemma \ref{['easy_lemma']}
  • proof : Proof of Proposition \ref{['correl_prop']}
  • ...and 3 more