Related papers: A Distance Amplification Lemma for Monotonicity
In this work, we propose a method for minimizing non-convex functions with Lipschitz continuous $p$th-order derivatives, starting from $p \geq 1$. The method, however, only requires derivative information up to order $(p-1)$, since the…
We present a new technique for efficiently removing almost all short cycles in a graph without unintentionally removing its triangles. Consequently, triangle finding problems do not become easy even in almost $k$-cycle free graphs, for any…
We prove that an $\epsilon$-approximate fixpoint of a map $f:[0,1]^d\rightarrow [0,1]^d$ can be found with $\mathcal{O}(d^2(\log\frac{1}{\epsilon} + \log\frac{1}{1-\lambda}))$ queries to $f$ if $f$ is $\lambda$-contracting with respect to…
We consider the problem of multi-objective maximization of monotone submodular functions subject to cardinality constraint, often formulated as $\max_{|A|=k}\min_{i\in\{1,\dots,m\}}f_i(A)$. While it is widely known that greedy methods work…
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…
Consider an anisotropic independent bond percolation model on the $d$-dimensional hypercubic lattice, $d\geq 2$, with parameter $p$. We show that the two point connectivity function $P_{p}(\{(0,\dots,0)\leftrightarrow (n,0,\dots,0)\})$ is a…
Let $G=(V,E)$ be a weighted undirected graph, with $n$ vertices. A distance oracle is a data structure that can quickly answer distance queries, with some stretch factor. A seminal work of \cite{TZ01}, given an integer $k\ge 1$, provides…
This work designs a framework for boosting the approximation guarantee of maximum matching algorithms. As input, the framework receives a parameter $\epsilon > 0$ and an oracle access to a $\Theta(1)$-approximate maximum matching algorithm…
We prove a Lusin approximation of functions of bounded variation. If $f$ is a function of bounded variation on an open set $\Omega\subset X$, where $X=(X,d,\mu)$ is a given complete doubling metric measure space supporting a $1$-Poincar\'e…
For any $\epsilon \in (0,1)$, a $(1+\epsilon)$-approximate range mode query asks for the position of an element whose frequency in the query range is at most a factor $(1+\epsilon)$ smaller than the true mode. For this problem, we design an…
The Fr\'echet distance is a commonly used distance measure for curves. Computing the Fr\'echet distance between two polygonal curves of $n$ vertices takes roughly quadratic time, and conditional lower bounds suggest that approximating to…
It is shown that, under certain assumptions on the growth and value distribution of a meromorphic function $f(z)$, \begin{equation*} m\left(r,\frac{\Delta_cf - ac}{f' - a}\right)=S(r,f'), \end{equation*} where $\Delta_c f=f(z+c)-f(z)$ and…
We show that for every $\varepsilon>0$ there is an absolute constant $c(\varepsilon)>0$ such that the following is true. The union of any $n$ arithmetic progressions, each of length $n$, with pairwise distinct differences must consist of at…
For each $q\in{\mathbb{N}}_0$, we construct positive linear polynomial approximation operators $M_n$ that simultaneously preserve $k$-monotonicity for all $0\leq k\leq q$ and yield the estimate \[ |f(x)-M_n(f, x)| \leq c…
The randomized query complexity $R(f)$ of a boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is famously characterized (via Yao's minimax) by the least number of queries needed to distinguish a distribution $D_0$ over $0$-inputs from a…
A Boolean $k$-monotone function defined over a finite poset domain ${\cal D}$ alternates between the values $0$ and $1$ at most $k$ times on any ascending chain in ${\cal D}$. Therefore, $k$-monotone functions are natural generalizations of…
We study a family of distance functions on rankings that allow for asymmetric treatments of alternatives and consider the distinct relevance of the top and bottom positions for ordered lists. We provide a full axiomatic characterization of…
We study the query complexity of testing for properties defined by read once formulas, as instances of {\em massively parametrized properties}, and prove several testability and non-testability results. First we prove the testability of any…
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…
In large-data applications, it is desirable to design algorithms with a high degree of parallelization. In the context of submodular optimization, adaptive complexity has become a widely-used measure of an algorithm's "sequentiality".…