Related papers: Average-case deterministic query complexity of boo…
The state complexity of basic operations on finite languages (considering complete DFAs) has been in studied the literature. In this paper we study the incomplete (deterministic) state and transition complexity on finite languages of…
We study the sample complexity of learning a uniform approximation of an $n$-dimensional cumulative distribution function (CDF) within an error $\epsilon > 0$, when observations are restricted to a minimal one-bit feedback. This serves as a…
We prove a lower bound on the communication complexity of computing the $n$-fold xor of an arbitrary function $f$, in terms of the communication complexity and rank of $f$. We prove that $D(f^{\oplus n}) \geq n \cdot…
Given an $n$-point metric space $(M,d)$, {\sc metric $1$-median} asks for a point $p\in M$ minimizing $\sum_{x\in M}\,d(p,x)$. We show that for each computable function $f\colon \mathbb{Z}^+\to\mathbb{Z}^+$ satisfying $f(n)=\omega(1)$, {\sc…
Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function $f$, the deterministic query complexity, $D(f)$, is at most quartic in the quantum query complexity, $Q(f)$: $D(f) = O(Q(f)^4)$. This matches the…
We introduce a new structural graph parameter called \emph{partial matching width}. For each (sufficiently large) integer $k \geq 1$, we introduce a class $\mathcal{G}_k$ of graphs of treewidth at most $k$ and max-degree $7$ such that for…
A method to calculate the average size of Davis-Putnam-Loveland-Logemann (DPLL) search trees for random computational problems is introduced, and applied to the satisfiability of random CNF formulas (SAT) and the coloring of random graph…
Consider a search from the root of an ordered tree with $n$ edges to some target node at a fixed distance $\ell$ from that root. We compare the average time complexity of the breadth-first search (BFS) and depth-first search (DFS)…
This note is an attempt to unconditionally prove the existence of weak one way functions (OWF). Starting from a provably intractable decision problem $L_D$ (whose existence is nonconstructively assured from the well-known discrete…
We study the quantum query complexity of the Boolean hidden shift problem. Given oracle access to f(x+s) for a known Boolean function f, the task is to determine the n-bit string s. The quantum query complexity of this problem depends…
The negative weight adversary method, $\mathrm{ADV}^\pm(g)$, is known to characterize the bounded-error quantum query complexity of any Boolean function $g$, and also obeys a perfect composition theorem $\mathrm{ADV}^\pm(f \circ g^n) =…
We introduce Tree Decision Diagrams (TDD) as a model for Boolean functions that generalizes OBDD. They can be seen as a restriction of structured d-DNNF; that is, d-DNNF that respect a vtree $T$. We show that TDDs enjoy the same…
The problem "Given a Boolean function $f$ of $n$ variables by its truth table vector. Find (if exists) a vector $\alpha \in \{0,1\}^n$ of maximal (or minimal) weight, such that $f(\alpha)= 1$." is considered here. It is closely related to…
The approximate non-deterministic degree of a Boolean function $f$, denoted $\mathsf{ndeg}_\epsilon(f)$ (written $\mathsf{N}_\epsilon(f)$ for brevity), is the minimum degree of a real polynomial $p$ such that $0 \le |p(x)| \le \epsilon$…
The theorem states that: Every Boolean function can be $\epsilon -approximated$ by a Disjunctive Normal Form (DNF) of size $O_{\epsilon}(2^{n}/\log{n})$. This paper will demonstrate this theorem in detail by showing how this theorem is…
We improve both upper and lower bounds for the distribution-free testing of monotone conjunctions. Given oracle access to an unknown Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ and sampling oracle access to an unknown distribution…
The approximate degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that approximates $f$ pointwise: $|f(x)-p(x)|\leq1/3$ for all $x\in\{0,1\}^n.$ For every $\delta>0,$ we construct CNF…
We show that any quantum algorithm deciding whether an input function $f$ from $[n]$ to $[n]$ is 2-to-1 or almost 2-to-1 requires $\Theta(n)$ queries to $f$. The same lower bound holds for determining whether or not a function $f$ from…
We establish two results regarding the query complexity of bounded-error randomized algorithms. * Bounded-error separation theorem. There exists a total function $f : \{0,1\}^n \to \{0,1\}$ whose $\epsilon$-error randomized query complexity…
We revisit the problem of computing with noisy information considered in Feige et al. 1994, which includes computing the OR function from noisy queries, and computing the MAX, SEARCH and SORT functions from noisy pairwise comparisons. For…