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Computational Complexity and Statistical Physics$
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Allon Percus, Gabriel Istrate, and Cristopher Moore

Print publication date: 2005

Print ISBN-13: 9780195177374

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195177374.001.0001

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Proving Conditional Randomness using the Principle of Deferred Decisions

Proving Conditional Randomness using the Principle of Deferred Decisions

Chapter 8 Proving Conditional Randomness using the Principle of Deferred Decisions
Computational Complexity and Statistical Physics

Alexis C. Kaporis

Lefteris M. Kirousis

Oxford University Press

In order to prove that a certain property holds asymptotically for a restricted class of objects such as formulas or graphs, one may apply a heuristic on a random element of the class, and then prove by probabilistic analysis that the heuristic succeeds with high probability. This method has been used to establish lower bounds on thresholds for desirable properties such as satisfiability and colorability: lower bounds for the 3-SAT threshold were discussed briefly in the previous chapter. The probabilistic analysis depends on analyzing the mean trajectory of the heuristic—as we have seen in chapter 3—and in parallel, showing that in the asymptotic limit the trajectory’s properties are strongly concentrated about their mean. However, the mean trajectory analysis requires that certain random characteristics of the heuristic’s starting sample are retained throughout the trajectory. We propose a methodology in this chapter to determine the conditional that should be imposed on a random object, such as a conjunctive normal form (CNF) formula or a graph, so that conditional randomness is retained when we run a given algorithm. The methodology is based on the principle of deferred decisions. The essential idea is to consider information about the object as being stored in “small pieces,” in separate registers. The contents of the registers pertaining to the conditional are exposed, while the rest remain unexposed. Having separate registers for different types of information prevents exposing information unnecessarily. We use this methodology to prove various randomness invariance results, one of which answers a question posed by Molloy [402].

Keywords:   Boolean formulas, card model, omniscient intermediary, pure literal, registers

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