0000000000461966
AUTHOR
Jevg��nijs Vihrovs
Sensitivity versus Certificate Complexity of Boolean Functions
Sensitivity, block sensitivity and certificate complexity are basic complexity measures of Boolean functions. The famous sensitivity conjecture claims that sensitivity is polynomially related to block sensitivity. However, it has been notoriously hard to obtain even exponential bounds. Since block sensitivity is known to be polynomially related to certificate complexity, an equivalent of proving this conjecture would be showing that certificate complexity is polynomially related to sensitivity. Previously, it has been shown that $bs(f) \leq C(f) \leq 2^{s(f)-1} s(f) - (s(f)-1)$. In this work, we give a better upper bound of $bs(f) \leq C(f) \leq \max\left(2^{s(f)-1}\left(s(f)-\frac 1 3\righ…
Quantum Speedups for Exponential-Time Dynamic Programming Algorithms
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 $0^n$ to $1^n$ 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 $O^*(1.817^n)$. 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 o…
On the Inner Product Predicate and a Generalization of Matching Vector Families
Motivated by cryptographic applications such as predicate encryption, we consider the problem of representing an arbitrary predicate as the inner product predicate on two vectors. Concretely, fix a Boolean function $P$ and some modulus $q$. We are interested in encoding $x$ to $\vec x$ and $y$ to $\vec y$ so that $$P(x,y) = 1 \Longleftrightarrow \langle\vec x,\vec y\rangle= 0 \bmod q,$$ where the vectors should be as short as possible. This problem can also be viewed as a generalization of matching vector families, which corresponds to the equality predicate. Matching vector families have been used in the constructions of Ramsey graphs, private information retrieval (PIR) protocols, and mor…
Size of Sets with Small Sensitivity: a Generalization of Simon's Lemma
We study the structure of sets $S\subseteq\{0, 1\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\subseteq\{0, 1\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving lower bounds on sensitivity of Boolean functions, with applications to the theory of parallel computing and the "sensitivity vs. block sensitivity" conjecture. In this paper, we take a deeper look at the size of such sets and their structure. We show an unexpected "gap theorem": if $S\subseteq\{0, 1\}^n$ has sensitivity $s$, then we either have $|S|=2^{n-s}$ or $|S|\geq \frac{3}{2} 2^{n-s}$. This is shown via classifying such sets into sets that can be con…