We exhibit an unambiguous$k$-DNF formula that requires CNF width$\tilde\Omega(k^{2})$, which is optimal up to logarithmic factors. As a consequence, we get a near-optimal solution to the Alon–Saks–Seymour problem in graph theory (posed in 1991), which asks: How large a gap can there be between the chromatic number of a graph and its biclique partition number? Our result is also known to imply several other improved separations in query and communication complexity.

We study the quantum query complexity of two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most k. We call this the Dyck_{k,n} problem. We prove a lower bound of Ω(c^k √n), showing that the complexity of this problem increases exponentially in k. Here n is the length of the word. When k is a constant, this is interesting as a representative example of star-free languages for which a surprising Õ(√n) query quantum algorithm was recently constructed by Aaronson et al. [Scott Aaronson et al., 2018]. Their proof does not give rise to a general algorithm. When k is not a constant, Dyck_{k,n} is not context-free. We give an algorithm with O(√n(log n)^{0.5k}) quantum queries for Dyck_{k,n} for all k. This is better than the trival upper bound n for k = o({log(n)}/{log log n}). Second, we consider connectivity problems on grid graphs in 2 dimensions, if some of the edges of the grid may be missing. By embedding the "balanced parentheses" problem into the grid, we show a lower bound of Ω(n^{1.5-ε}) for the directed 2D grid and Ω(n^{2-ε}) for the undirected 2D grid. The directed problem is interesting as a black-box model for a class of classical dynamic programming strategies including the one that is usually used for the well-known edit distance problem. We also show a generalization of this result to more than 2 dimensions.

In this paper we study quantum algorithms for NP-complete problems whose best classical algorithm is an exponential time application of dynamic programming. We introduce the path in the hypercube problem that models many of these dynamic programming algorithms. In this problem we are asked whether there is a path from 0n to 1n in a given subgraph of the Boolean hypercube, where the edges are all directed from smaller to larger Hamming weight. We give a quantum algorithm that solves path in the hypercube in time O*(1.817n). The technique combines Grover's search with computing a partial dynamic programming table. We use this approach to solve a variety of vertex ordering problems on graphs in the same time O*(1.817n), and graph bandwidth in time O*(2.946n). Then we use similar ideas to solve the travelling salesman problem and minimum set cover in time O*(1.728n).

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total Boolean function is given by the function f on n = 2 k bits defined by a complete binary tree of NAND gates of depth k , which achieves R 0 ( f ) = O ( D ( f ) 0.7537… ). We show that this is false by giving an example of a total Boolean function f on n bits whose deterministic query complexity is Ω( n ) while its zero-error randomized query complexity is Õ(√ n ). We further show that the quantum query complexity of the same function is Õ( n 1/4 ), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities. We also construct a total Boolean function g on n variables that has zero-error randomized query complexity Ω( n / log ( n )) and bounded-error randomized query complexity R ( g ) = Õ(√ n ). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is Q E ( g ) = Õ(√ n ). These functions show that the relations D ( f ) = O ( R 1 ( f ) 2 ) and R 0 ( f ) = Õ( R ( f ) 2 ) are optimal, up to polylogarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R 0 , a 3/2-power separation between Q E and R , and a 4th-power separation between approximate degree and bounded-error randomized query complexity. All of these examples are variants of a function recently introduced by Göös, Pitassi, and Watson, which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function f on n=2k bits defined by a complete binary tree of NAND gates of depth k, which achieves R0(f) = O(D(f)0.7537…). We show this is false by giving an example of a total boolean function f on n bits whose deterministic query complexity is Ω(n/log(n)) while its zero-error randomized query complexity is Õ(√n). We further show that the quantum query complexity of the same function is Õ(n1/4), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities.