## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Wednesday, March 27, 2013, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Dominik Scheder (Aarhus University)

## After Traxler and Amano: The relationship between average sensitivity, clause size, and solution density

A boolean function f: {0,1}^n -> {0,1} is sensitive to a coordinate i at point x if flipping the i-th coordinate of x changes the value of f. That is, if f(x) != f(x + e_i). The sensitivity of a boolean function f at point x, written S(f,x), is the number of coordinates to which f is sensitive at x. This is an integer between 0 and n. The average sensitivity of a function, S(f), is, well, the average of S(f,x) over all x.

In 2009, Patrick Traxler used the PPZ k-SAT algorithm by Paturi, Pudlak, and Zane to upper bound the average sensitivity of k-CNF formulas. His upper bound is k * (a function depending on mu), where mu is the density of satisfying assignments of f. This function peaks at around 1.06. In 2011, Kazuyuki Amano proved an upper bound of k, which is tight. This improves Patrick's upper bound for the worst-case density, but does not say too much about the average sensitivity of k-CNF formulas of density 0.1, say.

We determine the exact relationship between these three parameters: Clause width k, solution density mu, and average sensitivity. That is, we give a function s: [0,1] -> [0,1] such that

Upper Bound: S(f) ≤ k s(mu) if f is a k-CNF formula of solution density mu;
Lower Bound: We construct k-CNF formulas that come arbitrarily close to this upper bound.

If you think the number of clauses is more relevant than the clause width, we have the answer for you, too:

S(f) ≤ log(m) s(mu) (1 + o(1)), where m is the number of clauses and mu is the solution density, and o(1) goes to 0 as m grows. The function s is the same as above.

This is joint work with Li-Yang Tan.

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