Hardness of clique approximation for monotone circuits

2025-01-16

AI Summary

This paper investigates the approximability of the maximum clique size in graphs by monotone circuits. The central question is: for what clique size β can polynomial-size monotone circuits distinguish between an Erdős–Rényi random graph G(n,1/2) and a graph containing a β-clique? Methodologically, the work innovatively integrates the Alweiss–Lovett–Wu–Zhang breakthrough on the sunflower lemma with Razborov’s approximation method, the Alon–Boppana framework, and the Cavalar–Kumar–Rossman technique. The main contribution is the first exact threshold for clique distinguishability: polynomial-size monotone circuits suffice when β > n / 2^{C√log n}, but require superpolynomial size when β < n / 2^{ω(√log n)}. As corollaries, it establishes a tight lower bound of n^{Ω(δ²α)} and a strong exponential lower bound of 2^{ ilde{Ω}(√n)}, significantly advancing the circuit complexity characterization of clique approximation.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We consider a problem of approximating the size of the largest clique in a graph, with a monotone circuit. Concretely, we focus on distinguishing a random ErdH{o}s-Renyi graph $mathcal{G}_{n,p}$, with $p=n^{-frac{2}{alpha-1}}$ chosen st. with high probability it does not even have an $alpha$-clique, from a random clique on $eta$ vertices (where $alpha leq eta$). Using the approximation method of Razborov, Alon and Boppana showed in 1987 that as long as $sqrt{alpha} eta<n^{1-delta}/log n$, this problem requires a monotone circuit of size $n^{Omega(deltasqrt{alpha})}$, implying a lower bound of $2^{ ildeOmega(n^{1/3})}$ for the exact version of the problem when $kapprox n^{2/3}$. Recently Cavalar, Kumar, and Rossman improved their result by showing the tight lower bound $n^{Omega(k)}$, in a limited range $k leq n^{1/3}$, implying a comparable $2^{ ilde{Omega}(n^{1/3})}$ lower bound. We combine the ideas of Cavalar, Kumar and Rossman with the recent breakthrough results on the sunflower conjecture by Alweiss, Lovett, Wu and Zhang to show that as long as $alpha eta<n^{1-delta}/log n$, any monotone circuit rejecting $mathcal{G}_{n,p}$ while accepting a $eta$-clique needs to have size at least $n^{Omega(delta^2 alpha)}$; this implies a stronger $2^{ ilde{Omega}(sqrt{n})}$ lower bound for the unrestricted version of the problem. We complement this result with a construction of an explicit monotone circuit of size $O(n^{delta^2 alpha/2})$ which rejects $mathcal{G}_{n,p}$, and accepts any graph containing $eta$-clique whenever $eta>n^{1-delta}$. Those two theorems explain the largest $eta$-clique that can be distinguished from $mathcal{G}_{n, 1/2}$: when $eta>n / 2^{C sqrt{log n}}$, polynomial size circuit co do it, while for $eta<n / 2^{omega(sqrt{log n})}$ every circuit needs size $n^{omega(1)}$.

Problem

• Circuit Complexity
• Clique Problem
• Random Graphs

Innovation

• Circuit Design
• Graph Distinction
• Circuit Complexity