The Chow Parameters Problem

Abstract. In the 2nd Annual FOCS (1961), Chao-Kong Chow proved that every Boolean threshold function is uniquely determined by its degree-0 and degree-1 Fourier coefficients. These numbers became known as the Chow Parameters. Providing an algorithmic version of Chow’s Theorem—i.e., efficiently constructing a representation of a threshold function given its Chow Parameters—has remained open ever since. This problem has received significant study in the fields of circuit complexity, game theory and the design of voting systems, and learning theory. In this paper we effectively solve the problem, giving a randomized PTAS with the following behavior: Given the Chow Parameters of a Boolean threshold function f over n bits and any constant > 0, the algorithm runs in time O(n2 log2 n) and with high probability outputs a representation of a threshold function f which is -close to f. Along the way we prove several new results of independent interest about Boolean threshold functions...

• Type: Journal
• Year: 2011
• Where: SIAMCOMP
• Authors: Ryan O'Donnell, Rocco A. Servedio
• Keywords: Boolean Threshold Function, Chow Parameters, SIAMCOMP 2011, Software Engineering, Threshold Functions