Finding cliques and dense subgraphs using edge queries

TL;DR

This work studies the MCQP and related dense-subgraph problems in $G(n,1/2)$ under edge-query constraints. It introduces an encoding framework that bounds the number of “critical” edges in a queried subgraph by a combinatorial function $\gamma(\ell)$, yielding a fundamental inequality $\alpha^2/2-\alpha-2\gamma(\ell)m^2+(2-\delta)m\le 0$ whose optimal choice of $m$ provides upper bounds on $\alpha_*(\delta,\ell)$. By computing or bounding $\gamma(\ell)$ (with exact values $\gamma'(\infty)=\gamma(\infty)=1/2$, $\gamma'(2)=\gamma(2)=1/4$, $\gamma'(3)=\gamma(3)=3/8$, and $\gamma(\ell)\le 1/2-1/(3\cdot 2^{\ell-1}-4\ell+8)$ for $\ell\ge4$), the paper derives refined upper bounds for all $\delta\in[1,2)$ and $\ell\ge3$, including explicit results like $\alpha_*(1,3) \le 1.577$ (improving the previous $1.62$). It then extends the same methodology to the Maximum Dense Subgraph Query Problem (MDSQP) with density $\eta$, obtaining implicit bounds via an entropy-based expression and showing the bounds are strictly below the trivial density-bound as $\eta$ approaches 1; numerical illustrations and a linear-query lower bound are provided. The work thus clarifies the limitations of adaptive edge-query strategies for detecting large cliques and dense subgraphs in random graphs and connects the finite-round, limited-query regime to a combinatorial optimization framework.

Abstract

We consider the problem of finding a large clique in an Erdős--Rényi random graph where we are allowed unbounded computational time but can only query a limited number of edges. Recall that the largest clique in $G \sim G(n,1/2)$ has size roughly $2\log_{2} n$. Let $α_{\star}(δ,\ell)$ be the supremum over $α$ such that there exists an algorithm that makes $n^δ$ queries in total to the adjacency matrix of $G$, in a constant $\ell$ number of rounds, and outputs a clique of size $α\log_{2} n$ with high probability. We give improved upper bounds on $α_{\star}(δ,\ell)$ for every $δ\in [1,2)$ and $\ell \geq 3$. We also study analogous questions for finding subgraphs with density at least $η$ for a given $η$, and prove corresponding impossibility results.

Figures (5)

• Figure 1: Left: construction for k=6. Right: an alternating path containing the most blue edges.
• Figure 2: A path with many blue edges after deleting $D$.
• Figure 3: Red: trivial upper bound $\frac{2}{1-H(\eta)}$. Blue: $\ell=\infty$. Orange: $\ell=3$. Green: $\ell=2$. ($\delta=1$)
• Figure 4: Blue: $\alpha_1$ for $\delta=1, \ell=3$. Orange: $\alpha_2$ for $\delta=1, \ell=3$.
• Figure 5: Blue: $\alpha_1$ for $\delta=1, \ell=2$. Orange: $\alpha_2$ for $\delta=1, \ell=2$.

Theorems & Definitions (18)

• Theorem 1
• Corollary 2
• Theorem 3
• Remark 6
• Proposition 7
• proof
• Theorem 8
• Lemma 9
• proof
• Lemma 10