MIAO video seminar: KRW composition theorems via lifting
Title: KRW composition theorems via lifting
Speaker: Or Meir, University of Haifa
Host: Jacob Nordström, Department of Computer Science, Lund University
Where: Online - link by registration
One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., separating P from NC^0). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested approaching this problem by proving that depth complexity behaves "as expected" with respect to composition of functions. They showed that the validity of this conjecture would separate P from NC^0. Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function, but only for few inner functions. Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions.
In this work, we extend significantly the range of inner functions that can be handled. First, we consider the monotone version of the KRW conjecture. We prove it for every monotone inner function whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the s-t-connectivity, clique, and generation functions.
In order to carry this progress back to the non-monotone setting,we introduce a new notion of semi-monotone composition, which combines the non-monotone complexity of the outer function with the monotone complexity of the inner function. In this setting, we prove the KRW conjecture for a similar selection of inner functions, but only for a specific choice of the outer function.
Please register at lth.se/digitalth/events/register-miao in order to get an access link to the zoom platform.
The MIAO video seminars are arranged by the Mathematical Insights into Algorithms for Optimization (MIAO) research group at the University of Copenhagen and Lund University. Most of our seminars consist of two parts: first a 50-55-minute regular talk, and then after a break an optional ca-1-hour in-depth technical presentation with (hopefully) a lot of interaction. The intention is that the first part of the seminar will give all listeners a self-contained overview of some exciting research results. For listeners who are particularly interested, there is then an opportunity to return for a second part where we get into more technical details.