Abstract

Given a graph G with n vertices and a number s, the decision problem Clique asks whether G contains a fully connected subgraph with s vertices. For this NP-complete problem there exists a non-trivial lower bound: no monotone circuit of a size that is polynomial in n can solve Clique.
This entry provides an Isabelle/HOL formalization of a concrete lower bound (the bound is (n^1/7)^n1/8 for the fixed choice of s = n^1/4), following a proof by Gordeev.

BibTeX

@article{Clique_and_Monotone_Circuits-AFP,
  author  = {René Thiemann},
  title   = {Clique is not solvable by monotone circuits of polynomial size},
  journal = {Archive of Formal Proofs},
  month   = {May},
  year    = {2022},
  note    = {\url{https://isa-afp.org/entries/Clique_and_Monotone_Circuits.html},
            Formal proof development},
  ISSN    = {2150-914x},
}