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The Art of Computer Programming: Satisfiability, Volume 4, Fascicle 6 by Donald E. Knuth

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355. [HM21] In (152), prove that Ej ≤ 1/δ when (p1, . . . , pm) has positive slack δ. Hint: Consider replacing pj by pj + δpj.

Image 356. [M33] (The Clique Local Lemma.) Let G be a graph on {1, . . . , m}, and let G | U1, . . . , G | Ut be cliques that cover all the edges of G. Assign numbers θij ≥ 0 to the vertices of each Uj, such that Σj = ∑i ∈ U j θij < 1. Assume that

Image

a) Prove that (p1, . . . , pm) ∈ R(G). Hint: Letting Image denote , show that

b) Also ...

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