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Finding hidden cliques in linear time

Abstract : In the hidden clique problem, one needs to find the maximum clique in an $n$-vertex graph that has a clique of size $k$ but is otherwise random. An algorithm of Alon, Krivelevich and Sudakov that is based on spectral techniques is known to solve this problem (with high probability over the random choice of input graph) when $k \geq c \sqrt{n}$ for a sufficiently large constant $c$. In this manuscript we present a new algorithm for finding hidden cliques. It too provably works when $k > c \sqrt{n}$ for a sufficiently large constant $c$. However, our algorithm has the advantage of being much simpler (no use of spectral techniques), running faster (linear time), and experiments show that the leading constant $c$ is smaller than in the spectral approach. We also present linear time algorithms that experimentally find even smaller hidden cliques, though it remains open whether any of these algorithms finds hidden cliques of size $o(\sqrt{n})$.
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Uriel Feige, Dorit Ron. Finding hidden cliques in linear time. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. pp.189-204, ⟨10.46298/dmtcs.2802⟩. ⟨hal-01185601⟩



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