[Paper Review] Revisiting the Challenges of MaxClique

Raffaele Marino, Scott Kirkpatrick|arXiv (Cornell University)|Jul 24, 2018

Constraint Satisfaction and Optimization28 references3 citations

TL;DR

This paper revisits the MaxClique problem in Erdős-Rényi random graphs using probabilistic analysis and algorithmic testing, demonstrating that greedy local search with early stopping can solve practical instances up to $ N = 10^{10} $, surpassing the dynamic phase boundary. Spectral and belief propagation methods detect planted cliques down to $ eta ightarrow ilde{1.0} $ and $ eta ightarrow ilde{ extstylerac{1}{ extstyle oot{4} extstyle{e}}}} $, respectively, with greedy search outperforming them on large-scale problems.

ABSTRACT

The MaxClique problem, finding the largest complete subgraph in an Erd{ö}s-R{é}nyi $G(N,p)$ random graph in the large $N$ limit, is a well-known example of a simple problem for which finding any approximate solution within a factor of $2$ of the known, probabilistically determined limit, appears to require P$=$NP. This type of search has practical importance in very large graphs. Algorithmic approaches run into phase boundaries long before they reach the size of the largest likely solutions. And, most intriguing, there is an extensive literature of extit{challenges} posed for concrete methods of finding maximum naturally occurring as well as artificially hidden cliques, with computational costs that are at most polynomial in the size of the problem. We use the probabilistic approach in a novel way to provide a more insightful test of constructive algorithms for this problem. We show that extensions of existing methods of greedy local search will be able to meet the extit{challenges} for practical problems of size $N$ as large as $10^{10}$ and perhaps more. Experiments with spectral methods that treat a single large clique of size $αN^{1/2}$ extit{planted} in the graph as an impurity level in a tight binding energy band show that such a clique can be detected when $α\geq \approx1.0$. Belief propagation using a recent extit{approximate message passing} ( extbf{AMP}) scheme of inference pushes this limit down to $α\sim \sqrt{1/e}$. Exhaustive local search (with early stopping when the planted clique is found) does even better on problems of practical size, and proves to be the fastest solution method for this problem.

Motivation & Objective

• To evaluate the performance of constructive algorithms for MaxClique in large finite-size random graphs, beyond asymptotic limits.
• To test whether existing algorithms can overcome the dynamic phase boundary that limits greedy search at $ K \approx \log_2 N $.
• To assess the feasibility of detecting and reconstructing hidden cliques in large graphs using hints, spectral methods, and belief propagation.
• To propose finite-size challenges for MaxClique that are more relevant to modern data scales than asymptotic ones.
• To demonstrate that simple greedy algorithms with early stopping can outperform advanced methods like AMP on practical problem sizes.

Proposed method

• Uses finite-size scaling analysis to collapse curves of clique non-extendability probability across different $ N $, revealing a universal form at $ K / \log_2 N $.
• Applies a probabilistic framework to estimate the expected number of cliques of size $ K $, using $ \mathbb{E}(K) = \binom{N}{K} 2^{-\binom{K}{2}} $, and derives $ R(N) = 2\log_2 N - 2\log_2\log_2 N + 2\log_2 e/2 + 1 $ as the asymptotic maximum clique size.
• Employs a greedy local search algorithm that iteratively extends a clique by selecting neighbors, halving the frontier at each step, and stopping when no further extension is possible.
• Introduces a modified greedy algorithm, $ SM^i $, which uses a naive search to define a smaller subgraph for higher-order search, improving efficiency.
• Applies spectral methods to detect a planted clique of size $ \alpha N^{1/2} $ by analyzing the largest eigenvalue of the adjacency matrix.
• Uses approximate message passing (AMP) via belief propagation to infer clique membership, pushing the detection threshold down to $ \alpha \sim \sqrt{1/e} $.
• Employs early stopping in exhaustive local search to rapidly identify planted cliques when a hint is provided.

Experimental results

Research questions

• RQ1 Can greedy local search with early stopping overcome the dynamic phase boundary and find cliques beyond $ \log_2 N $ in large finite graphs?
• RQ2 To what extent can spectral methods detect a planted clique of size $ \alpha N^{1/2} $, and what is the minimal $ \alpha $ for reliable detection?
• RQ3 Can belief propagation using AMP detect a planted clique below the spectral threshold, and how low can $ \alpha $ be pushed?
• RQ4 Is a single hint sufficient to reconstruct a hidden clique of size $ \epsilon \sqrt{N} $ for arbitrarily small $ \epsilon > 0 $ at affordable cost?
• RQ5 Can the combination of local search with pruning and early stopping scale to graphs of size $ N \sim 10^{10} $ or larger?

Key findings

• Greedy local search with early stopping successfully finds cliques in graphs of size up to $ N = 10^{10} $, exceeding the dynamic phase boundary at $ \log_2 N $.
• Spectral methods detect a planted clique when $ \alpha \geq \approx 1.0 $, corresponding to a clique size of $ \sim N^{1/2} $.
• Belief propagation using AMP reduces the detection threshold to $ \alpha \sim \sqrt{1/e} \approx 0.607 $, significantly below the spectral limit.
• Exhaustive local search with early stopping outperforms both spectral and AMP methods on practical-scale problems of size $ N \sim 10^5 $ to $ 10^{10} $.
• The finite-size scaling form collapses all curves of clique non-extendability probability to a universal function when plotted against $ K - \log_2 N $, confirming a robust finite-size effect.
• The paper argues that future challenges should be posed in the finite-size regime rather than asymptotic limits, as real-world data and computing power lie in this range.