Very Large Cliques are Easy to Detect

It is known that, for every constant k 3, the presence of a k-clique (a complete subgraph on k vertices) in an n-vertex graph cannot be detected by a monotone boolean circuit using much fewer than nk gates. We show that, for every constant k, the presence of an (n - k)-clique in an n-vertex graph can be detected by a monotone circuit using only a logarithmic number of fanin-2 OR gates; the total number of gates does not exceed O(n2 log n). Moreover, if we allow unbounded fanin, then a logarithmic number of gates is enough.

• Type: Conference
• Year: 2006
• Where: DAGSTUHL
• Authors: Alexander E. Andreev, Stasys Jukna
• Keywords: DAGSTUHL 2006, DAGSTUHL 2007, Logarithmic Number, Monotone Boolean Circuit, N-vertex Graph