Graph Complexity and Slice Functions

Abstract. A graph-theoretic approach to study the complexity of Boolean functions was initiated by Pudlák, Rödl, and Savický [PRS] by defining models of computation on graphs. These models generalize well-known models of Boolean complexity such as circuits, branching programs, and two-party communication complexity.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42364/1/30360071.pd

CC-circuits and the expressive power of nilpotent algebras

We show that CC-circuits of bounded depth have the same expressive power as polynomials over finite nilpotent algebras from congruence modular varieties. We use this result to phrase and discuss an algebraic version of Barrington, Straubing and Th\'erien's conjecture, which states that CC-circuits of bounded depth need exponential size to compute AND. Furthermore we investigate the complexity of deciding identities and solving equations in a fixed nilpotent algebra. Under the assumption that the conjecture is true, we obtain quasipolynomial algorithms for both problems. On the other hand, if AND is computable by uniform CC-circuits of bounded depth and polynomial size, we can construct a nilpotent algebra with coNP-complete, respectively NP-complete problem.Comment: 14 page

Multiple Product Modulo Arbitrary Numbers

AbstractLetnbinary numbers of lengthnbe given. The Boolean function “Multiple Product”MPnasks for (some binary representation of ) the value of their product. It has been shown (K.-Y. Siu and V. Roychowdhury, On optimal depth threshold circuits for multiplication and related problems,SIAM J. Discrete Math.7, 285–292 (1994)) that this function can be computed in polynomial-size threshold circuits of depth 4. For many other arithmetic functions, circuits of depth 3 are known. They are mostly based on the fact that the value of the considered function modulo some prime numbers p can be computed easily in threshold circuits of depth 2. In this paper, we investigate the complexity of computingMPnmodulomby depth-2 threshold circuits. It turns out that for all but a few integersm, exponential size is required. In particular, it is shown that form∈{2, 4, 8}, polynomial-size circuits exist, form∈{3, 6, 12, 24}, the question remains open and in all other cases, exponential-size circuits are required. The result still holds if we allowmto grow withn

Almost Optimal Pseudorandom Generators for Spherical Caps

Halfspaces or linear threshold functions are widely studied in complexity theory, learning theory and algorithm design. In this work we study the natural problem of constructing pseudorandom generators (PRGs) for halfspaces over the sphere, aka spherical caps, which besides being interesting and basic geometric objects, also arise frequently in the analysis of various randomized algorithms (e.g., randomized rounding). We give an explicit PRG which fools spherical caps within error $\epsilon$ and has an almost optimal seed-length of $O(\log n + \log(1/\epsilon) \cdot \log\log(1/\epsilon))$. For an inverse-polynomially growing error $\epsilon$, our generator has a seed-length optimal up to a factor of $O( \log \log {(n)})$. The most efficient PRG previously known (due to Kane, 2012) requires a seed-length of $\Omega(\log^{3/2}{(n)})$ in this setting. We also obtain similar constructions to fool halfspaces with respect to the Gaussian distribution. Our construction and analysis are significantly different from previous works on PRGs for halfspaces and build on the iterative dimension reduction ideas of Kane et. al. (2011) and Celis et. al. (2013), the \emph{classical moment problem} from probability theory and explicit constructions of \emph{orthogonal designs} based on the seminal work of Bourgain and Gamburd (2011) on expansion in Lie groups.Comment: 28 Pages (including the title page

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum