Related papers: The $n$-Queens Problem in Higher Dimensions
Most cloud services and distributed applications rely on hashing algorithms that allow dynamic scaling of a robust and efficient hash table. Examples include AWS, Google Cloud and BitTorrent. Consistent and rendezvous hashing are algorithms…
Noise causes severe difficulties in implementing quantum computing and quantum cryptography. Several schemes have been suggested to reduce this problem, mainly focusing on quantum computation. Motivated by quantum cryptography, we suggest a…
We show an $O(n)$-time reduction from the problem of testing whether a multiset of positive integers can be partitioned into two multisets so that the sum of the integers in each multiset is equal to $n/2$ to the problem of testing whether…
Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best…
Commercial quantum annealers from D-Wave Systems can find high quality solutions of quadratic unconstrained binary optimization problems that can be embedded onto its hardware. However, even though such devices currently offer up to 2048…
We study linear programming and general LP-type problems in several big data (streaming and distributed) models. We mainly focus on low dimensional problems in which the number of constraints is much larger than the number of variables. Low…
Recent advances in the development of commercial quantum annealers such as the D-Wave 2X allow solving NP-hard optimization problems that can be expressed as quadratic unconstrained binary programs. However, the relatively small number of…
We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An…
A finite point set in $\mathbb{R}^d$ is in general position if no $d + 1$ points lie on a common hyperplane. Let $\alpha_d(N)$ be the largest integer such that any set of $N$ points in $\mathbb{R}^d$, with no $d + 2$ members on a common…
In this article, we prove new theorems bounding the rank of different sub-matrices arising from these kernel functions. Bounds like these are often useful for analyzing the complexity of various hierarchical matrix algorithms. We also plot…
Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter.…
We introduce a parallel machine scheduling problem in which the processing times of jobs are not given in advance but are determined by a system of linear constraints. The objective is to minimize the makespan, i.e., the maximum job…
Current quantum computers can only solve optimization problems of a very limited size. For larger problems, decomposition methods are required in which the original problem is broken down into several smaller sub-problems. These are then…
Deep Neural Networks are increasingly adopted in critical tasks that require a high level of safety, e.g., autonomous driving. While state-of-the-art verifiers can be employed to check whether a DNN is unsafe w.r.t. some given property…
Let $\mathbb{Z}_n = \{Z_1, \ldots, Z_n\}$ be a design; that is, a collection of $n$ points $Z_j \in [-1,1]^d$. We study the quality of quantization of $[-1,1]^d$ by the points of $\mathbb{Z}_n$ and the problem of quality of coverage of…
The queen's graph $Q_{m \times n}$ has the squares of the $m \times n$ chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal of the board. A set $D$ of squares of $Q_{m \times n}$ is a…
In an ever expanding set of research and application areas, deep neural networks (DNNs) set the bar for algorithm performance. However, depending upon additional constraints such as processing power and execution time limits, or…
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.…
Quantum computers may achieve speedups over their classical counterparts for solving linear algebra problems. However, in some cases -- such as for low-rank matrices -- dequantized algorithms demonstrate that there cannot be an exponential…
One of the main problems in random network coding is to compute good lower and upper bounds on the achievable cardinality of the so-called subspace codes in the projective space $\mathcal{P}_q(n)$ for a given minimum distance. The…