Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
169 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Computing Cliques is Intractable (1305.3218v2)

Published 14 May 2013 in cs.CC

Abstract: The class P is in fact a proper sub-class of NP. We explore topological properties of the Hamming space 2[n] where [n]={1, 2,..., n}. With the developed theory, we show: (i) a theorem that is closely related to Erdos and Rado's sunflower lemma, and claims a stronger statement in most cases, (ii) a new approach to prove the exponential monotone circuit complexity of the clique problem, (iii) NC \ne NP through the impossibility of a Boolean circuit with poly-log depth to compute cliques, based on the construction of (ii), and (iv) P \ne NP through the exponential circuit complexity of the clique problem, based on the construction of (iii). Item (i) leads to the existence of a sunflower with a small core in certain families of sets, which is not an obvious consequence of the sunflower lemma. In (iv), we show that any Boolean circuit computing the clique function CLIQUE_{n,k} (k=n{1/4}) has a size exponential in n. Thus, we will separate P/poly from NP also. Razborov and Rudich showed strong evidence that no natural proof can prove exponential circuit complexity of a Boolean function. We confirm that the proofs for (iii) and (iv) are not natural.

Citations (1)

Summary

We haven't generated a summary for this paper yet.