Finding Hidden Cliques in Linear Time with High Probability

We are given a graph $G$ with $n$ vertices, where a random subset of $k$ vertices has been made into a clique, and the remaining edges are chosen independently with probability $\tfrac12$. This random graph model is denoted $G(n,\tfrac12,k)$. The hidden clique problem is to design an algorithm that finds the $k$-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov uses spectral techniques to find the hidden clique with high probability when $k = c \sqrt{n}$ for a sufficiently large constant $c > 0$. Recently, an algorithm that solves the same problem was proposed by Feige and Ron. It has the advantages of being simpler and more intuitive, and of an improved running time of $O(n^2)$. However, the analysis in the paper gives success probability of only $2/3$. In this paper we present a new algorithm for finding hidden cliques that both runs in time $O(n^2)$, and has a failure probability that is less than polynomially small.

💡 Research Summary

The paper addresses the hidden‑clique problem in the random graph model $G(n,\tfrac12,k)$, where a set of $k$ vertices forms a clique and all other edges appear independently with probability $1/2$. The goal is to recover the planted clique in polynomial time with high probability. Earlier work by Alon, Krivelevich, and Sudakov showed that spectral methods succeed when $k=c\sqrt{n}$ for a sufficiently large constant $c$, but these techniques are relatively involved. Feige and Ron later introduced a much simpler algorithm based on degree thresholds and random sampling; it runs in $O(n^{2})$ time but the analysis guarantees only a $2/3$ success probability.

The contribution of this paper is a new $O(n^{2})$‑time algorithm whose failure probability is polynomially small (e.g., $n^{-c}$ for some constant $c$). The improvement comes from two technical ideas. First, the algorithm uses a finely tuned degree cutoff
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