Jin-Yi Cai

An optimal lower bound on the number of variables for graph identification

In this paper we show that Ω(n) variables are needed for first-order logic with counting to identify graphs onn vertices. Thek-variable language with counting is equivalent to the (k−1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant becausen variables obviously suffice to identify graphs onn vertices.

The Boolean hierarchy I: structural properties

In this paper, we study the complexity of sets formed by boolean operations (union, intersection, and complement) on NP sets. These are the sets accepted by trees of hardware with NP predicates as leaves, and together these form the boolean hierarchy.We present many results about the structure of the boolean hierarchy: separation and immunity results, natural complete languages, and structural asymmetries between complementary classes.We show that in some relativized worlds the boolean hierarchy is infinite, and that for every k there is a relativized world in which the boolean hierarchy extends exactly k levels. We prove natural languages, variations of VERTEX COVER, complete for the various levels of the boolean hierarchy. We show the following structural asymmetry: though no set in the boolean hierarchy is

The Boolean hierarchy II: applications

{\text{D}}^{\text{P}}

On the power of parity polynomial time

-immune, there is a relativized world in which the boolean hierarchy contains

With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy

{\text{coD}}^{\text{P}}

An improved worst-case to average-case connection for lattice problems

-immune sets.Thus, this paper explores the structural properties of the...

Holographic algorithms: from art to science

The Boolean Hierarchy I: Structural Properties [J. Cai et al., SIAM J. Comput ., 17 (1988), pp. 1232–252] explores the structure of the boolean hierarchy, the closure of NP with respect to boolean ...

Holant problems and counting CSP

This paper proves that the complexity class ⊕P, parity polynomial time [PZ], contains the class of languages accepted byNP machines with few accepting paths. Indeed, ⊕P contains a broad class of languages accepted by path-restricted nondeterministic machines. In particular, ⊕P contains the polynomial accepting path versions ofNP, of the counting hierarchy, and of ModmNP form>1. We further prove that the class of nondeterministic path-restricted languages is closed under bounded truth-table reductions.

The Boolean Hierarchy: Hardware over NP

Abstract We consider how much error a fixed depth Boolean circuit must make in computing the parity function. We show that with an exponential bound of the form exp( n λ ) on the size of the circuits, they make a 50% error on all possible inputs, asymptotically and uniformly. As a consequence, we show that a random oracle set A separates PSPACE from the entire polynomial-time hierarchy with probability one.

Approximating the SVP to within a Factor (1+1/dimε) Is NP-Hard under Randomized Reductions

We improve a connection of the worst-case complexity and the average-case complexity of some well-known lattice problems. This fascinating connection was first discovered by Ajtai (1995). We improve the exponent of this connection from 8 to 3.5+/spl epsiv/.