PPZ For More Than Two Truth Values - An Algorithm for Constraint Satisfaction Problems

We analyze the so-called ppz algorithm for (d,k)-CSP problems for general values of d (number of values a variable can take) and k (number of literals per constraint). To analyze its success probability, we prove a correlation inequality for submodular functions.

💡 Research Summary

The paper studies the PPZ algorithm, originally designed for Boolean k‑SAT, in the broader context of (d, k)-CSP, where each variable can take one of d values and each constraint is a disjunction of at most 2k literals of the form “x ≠ c”. The authors first describe a straightforward generalization of PPZ: variables are processed in a random permutation π; for each variable x, all unit constraints of the form (x ≠ c) that are currently present are collected, the corresponding values c are marked as forbidden, and x is assigned uniformly at random among the remaining allowed values. This process is repeated until all variables are assigned, and the algorithm succeeds if the resulting assignment satisfies the formula.

The central technical challenge is to bound the success probability of this algorithm. For a fixed satisfying assignment α (assumed without loss of generality to be the all‑1 vector), the probability that PPZ returns α given a permutation π is the product over all variables x of 1/|S(x, π, α)|, where S(x, π, α) denotes the set of values not forbidden for x after processing all variables that precede x in π. The overall success probability is the sum of these probabilities over all satisfying assignments.

To analyze the expectation of the product, the authors first apply Jensen’s inequality in a standard way, obtaining a lower bound of 2^{‑∑_x E