In this paper we analyze the performance of Warning Propagation, a popular message passing algorithm. We show that for 3CNF formulas drawn from a certain distribution over random satisfiable 3CNF formulas, commonly referred to as the planted-assignment distribution, running Warning Propagation in the standard way (run message passing until convergence, simplify the formula according to the resulting assignment, and satisfy the remaining subformula, if necessary, using a simple "off the shelf" heuristic) results in a satisfying assignment when the clause-variable ratio is a sufficiently large constant.
Another difference between our work and that of @cite_28 @cite_14 @cite_7 is that unlike the algorithms analyzed in those other papers, WP is a randomized algorithm, a fact which makes its analysis more difficult. We could have simplified our analysis had we changed WP to be deterministic (for example, by initializing all clause-variable messages to 1 in step 2 of the algorithm), but there are good reasons why WP is randomized. For example, it can be shown that (the randomized version) WP converges with probability 1 on 2CNF formulas that form one cycle of implications, but might not converge if step 4 does not introduce fresh randomness in every iteration of the algorithm (details omitted).
@cite_28 @cite_14 @cite_7
2079035346 1898334569 1994098212
Let G3n,p,3 be a random 3-colorable graph on a set of 3n vertices generated as follows. First, split the vertices arbitrarily into three equal color classes, and then choose every pair of vertices of distinct color classes, randomly and independently, to be edges with probability p. We describe a polynomial-time algorithm that finds a proper 3-coloring of G3n,p,3 with high probability, whenever p @math c n, where c is a sufficiently large absolute constant. This settles a problem of Blum and Spencer, who asked if an algorithm can be designed that works almost surely for p @math polylog(n) n [J. Algorithms, 19 (1995), pp. 204--234]. The algorithm can be extended to produce optimal k-colorings of random k-colorable graphs in a similar model as well as in various related models. Implementation results show that the algorithm performs very well in practice even for moderate values of c. It is NP-Hard to find a proper 2-coloring of a given 2-colorable (bipartite) hypergraph H. We consider algorithms that will color such a hypergraph using few colors in polynomial time. The results of the paper can be summarized as follows: Let n denote the number of vertices of H and m the number of edges, (i) For bipartite hypergraphs of dimension k there is a polynomial time algorithm which produces a proper coloring using min ( O(n^ 1 - 1 k ),O((m n)^ 1 k - 1 ) )colors, (ii) For 3-uniform bipartite hypergraphs, the bound is reduced to O(n2 9). (iii) For a class of dense 3-uniform bipartite hypergraphs, we have a randomized algorithm which can color optimally. (iv) For a model of random bipartite hypergraphs with edge probability p≥ dn−2, d > O a sufficiently large constant, we can almost surely find a proper 2-coloring. Let I be a random 3CNF formula generated by choosing a truth assignment φ for variables x 1 , ..., x n uniformly at random and including every clause with i literals set true by φ with probability p i , independently. We show that for any 0 ≤ η 2 , η 3 ≤ 1 there is a constant d min so that for all d ≥ d min , a spectral algorithm similar to the graph coloring algorithm of [1] will find a satisfying assignment with high probability for p 1 = d n2, p 2 = η 2 d n2, and p3 = η 3 d n2. Appropriately setting η 2 and η 3 yields natural distributions on satisfiable 3CNFs, not-all-equal-sat 3CNFs, and exactly-one-sat 3CNFs.