SIGACT news complexity theory column 35

Abstract

I was surprised and delighted at a recent conference when the complexity class Θ 2 p suddenly popped up in a talk (by Markus Holzer on work by him and Pierre McKenzie) where it is was not exactly something one would have before-hand expected to come into play. After hearing the rest of Markus's remarkable talk (which he illustrated using toys from his daughter), I was much hoping that the authors would be so kind as to write a guest column; they very generously agreed to, and that column appears below.So, what is Θ 2 p ? Θ 2 p is the class of languages acceptable via parallel access to NP, and has many equivalent alternate definitions and characterizations (see [KSW87, Hem89, Wag90]), such as the one used in the paper below. This class was first studied by Papadimitriou and and Zachos in 1983 [PZ83] (though in a form that at that time was not known to be equivalent to the definition just mentioned), and has played an important role in complexity theory, e.g., Kadin [Kad89] proved that the polynomial-time hierarchy collapses to Θ 2 p if NP has sparse Turing-complete sets. Recently, the class has turned out to play other surprising roles, such as in analyzing the complexity of electoral systems (see the survey [HH00]) and in understanding the performance of greedy algorithms [HR98, HRS01]. In fact, a survey column a few years ago was devoted to overviewing such raised-to-Θ 2 p lower bounds [HHR97]. Bill Gasarch and SIGACT News need YOU! That is, the next issue will contain a guest column edited by Bill Gasarch, which will collection opinions on the future of the P-versus-NP question. Bill very much wants a broad range of opinions, and will be setting up a web page (http://www.cs.umd.edu/~gasarch/pnp/) describing how you can get your opinions to him. He may even put up a series of polling questions there, so please go there, vote early, and vote often!And in the issues following that will be articles by M. Schaeffer and C. Umans on complete sets at and beyond the polynomial hierarchy's second level, by A. Nickelsen and T. Tantau on partial information classes, and by R. Paturi on the complexity of k -SAT.