DIMACS Theoretical Computer Science Seminar

Title: The quantum adversary method and classical formula size lower bounds

Speaker: Mario Szegedy, Rutgers University

Date: September 20, 2005 2:00-3:00pm

Location: DIMACS Center, CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ


We introduce two new complexity measures for Boolean functions, which we name $\spi$ and $\mpi$. The quantity $\spi$ has been emerging through a line of research on quantum query complexity lower bounds via the so-called quantum adversary method culminating in the realization of Spalek and Szegedy that many of the known different formulations are in fact equivalent. Given that $\spi$ turns out to be such a robust invariant of a function, we begin to investigate this quantity in its own right and see that it also has applications to classical complexity theory.

As a surprising application we show that $\spi^2(f)$ is a lower bound on the formula size, and even, up to a constant multiplicative factor, the probabilistic formula size of $f$. We show that several formula size lower bounds in the literature, specifically Khrapchenko and its extensions, including a key lemma of Hastad, are in fact special cases of our method. The second quantity we introduce, $\mpi(f)$, is always at least as large as $\spi(f)$, and is derived from $\spi$ in such a way that $\mpi^2(f)$ remains a lower bound on formula size.

Our main result is proven via a combinatorial lemma which relates the square of the spectral norm of a matrix to the squares of the spectral norms of its submatrices. The generality of this lemma implies that our methods can also be used to lower bound the communication complexity of relations, and a related combinatorial quantity, the rectangle partition number.

To exhibit the strengths and weaknesses of our methods, we look at the $\spi$ and $\mpi$ complexity of a few examples, including the recursive majority of three function, a function defined by Ambainis \cite{A03}, and the collision problem.

(joint with Sophie Laplante and Troy Lee)