Polynomial Threshold Functions, AC^0 Functions and Spectral Norms

The class of polynomial-threshold functions is studied using harmonic analysis, and the results are used to derive lower bounds related to AC^0 functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). The main result is that polynomial-threshold functions can be characterized by means of their spectral representation. In particular, it is proved that a Boolean function whose L_1 spectral norm is bounded by a polynomial in n is a polynomial-threshold function, and that a Boolean function whose L_∞^(-1) spectral norm is not bounded by a polynomial in n is not a polynomial-threshold function. Some results for AC^0 functions are derived

Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph classes G, an (F, G)-system is a boolean dynamical system such that all local transition functions lie in F and the underlying graph lies in G. Let F be a class of boolean functions which is closed under composition and let G be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If F contains the self-dual functions and G contains the planar graphs then the fixed-point existence problem for (F, G)-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If F contains the self-dual functions and G contains the graphs having vertex covers of size one then the fixed-point existence problem for (F, G)-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24 versio

Comparison of PBO solvers in a dependency solving domain

Linux package managers have to deal with dependencies and conflicts of packages required to be installed by the user. As an NP-complete problem, this is a hard task to solve. In this context, several approaches have been pursued. Apt-pbo is a package manager based on the apt project that encodes the dependency solving problem as a pseudo-Boolean optimization (PBO) problem. This paper compares different PBO solvers and their effectiveness on solving the dependency solving problem.Comment: In Proceedings LoCoCo 2010, arXiv:1007.083