Heavy-Tailed Behavior and Search Algorithms for SAT Tang Yi Based on [1][2][3]

Heavy-Tailed Distribution P[X > x] ~ Cx –α, as x → ∞, 0 0 is a constant independent of x A general class of distributions with power law tails

Cost Distribution of Backtrack Search

Why Heavy-tailed distributions occur in backtrack search Critically constrained variables : Some variables are hard to find good values. If they are set, the values of the other variables are largely determined. Small-world: a small number of global constraints combined with a large number of local connections

Heavy-Tails on the left-hand side of the distribution

Rapid Randomized Restarts(1) Exploit any significant probability mass early on in the distribution. Reduce the variance in runtime and the probability of failure of the search procedure. Suitable cutoff will increase the efficiency of the search.

Rapid Randomized Restarts(2) Run the procedure up to a fixed number of choice points c (the cut off);if the procedure finds a solution or proves that no solution exists,then RRR also found a solution and stops;otherwise restart the backtrack procedure from the beginning for another c decision events, and so on.

RRR speed up backtrack search

Other related results on empirical data No truly heavy-tailed behavior found in randomized systematic search algorithms for SAT(Satz-Rand) Four issues needed to research in these algorithms (1) Occurrence of Stagnation Behavior (2) Detecting Stagnation Behavior (3)Robustness of Performance w.r.t Noise Setting (4)Automatic and Robust Tuning of Parameters

Formal Model: Abstract Tree Search Model Balanced Tree Model: no heavy-tails Imbalanced Tree Model: heavy-tails and restarts Bounded Imbalanced Model: the closest match to heavy-tailed behavior as observed in practice.

Discussion The Heavy-tailed behavior can lead to better search techniques. Not all the available search algorithms can result heavy-tailed phenomena. The study largely based on empirical data, New formal models are needed to help to find new strategy.

Reference [1] Carla Gomes, Bart Selman, Nuno Crato, and Henry Kautz.Heavy- Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems. J. of Automated Reasoning, Vol. 24(1/2), pages , 2000, available at [2] Hubie Chen, Carla Gomes, and Bart Selman Formal Models of Heavy-tailed Behavior in Combinatorial Search.. In Principles and Practices of Constraint Programming (CP-01). Lecture Notes in Computer Science 2239,Springer 2001, available at [3] Holger H. Hoos:Heavy-Tailed Behaviour in Randomised Systematic Search Algorithms for SAT? (Technical Report TR Department of Computer Science, University of British Columbia, 1999) available at