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Why almost all k-colorable graphs are easy
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
A spectral method for MAX2SAT in the planted solution model
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
An Efficient Sparse Regularity Concept
SIAM Journal on Discrete Mathematics
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ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
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Let I be a random 3CNF formula generated by choosing a truth assignment φ for variables x1, ..., xn uniformly at random and including every clause with i literals set true by φ with probability pi, independently. We show that for any 0 ≤ η2, η3 ≤ 1 there is a constant dmin so that for all d ≥ dmin, a spectral algorithm similar to the graph coloring algorithm of [1] will find a satisfying assignment with high probability for p1 = d/n2, p2 = η2d/n2, and p3 = η3d/n2. Appropriately setting η2 and η3 yields natural distributions on satisfiable 3CNFs, not-all-equal-sat 3CNFs, and exactly-one-sat 3CNFs.