Related papers: The Complexity of Computing the Size of an Interva…
We elucidate why an interval algorithm that computes the exact bounds on the amplitude and phase of the discrete Fourier transform can run in polynomial time. We address this question from a formal perspective to provide the mathematical…
Properties of intervals in the lattice of antichains of subsets of a universe of finite size are investigated. New objects and quantities in this lattice are defined. Expressions and numerical values are deduced for the number of connected…
The method of partial derivatives is one of the most successful lower bound methods for arithmetic circuits. It uses as a complexity measure the dimension of the span of the partial derivatives of a polynomial. In this paper, we consider…
In ongoing work to define a principled method for syntenic block discovery and structuring, work based on homology-derived constraints and a generalization of common intervals, we faced a fundamental computational problem: how to determine…
We develop a class of mixed-integer formulations for disjunctive constraints intermediate to the big-M and convex hull formulations in terms of relaxation strength. The main idea is to capture the best of both the big-M and convex hull…
It is proved that, if $(P_n)$ is a sequence of polynomials with complex coefficients having unbounded valences and tending to infinity at sufficiently many points, then there is an infinite dimensional closed subspace of entire functions,…
The paper investigates the problem of performing correlation analysis when the number of observations is very large. In such a case, it is often necessary to combine the random observations to achieve dimensionality reduction of the…
Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process from an initial instance to the final acceptance is motivated because many natural processes have been recognized…
We consider the computational complexity of reconfiguration problems, in which one is given two combinatorial configurations satisfying some constraints, and is asked to transform one into the other using elementary transformations, while…
How hard is it to estimate a discrete-time signal $(x_{1}, ..., x_{n}) \in \mathbb{C}^n$ satisfying an unknown linear recurrence relation of order $s$ and observed in i.i.d. complex Gaussian noise? The class of all such signals is…
The gauge function, closely related to the atomic norm, measures the complexity of a statistical model, and has found broad applications in machine learning and statistical signal processing. In a high-dimensional learning problem, the…
Interval arithmetic is a simple way to compute a mathematical expression to an arbitrary accuracy, widely used for verifying floating-point computations. Yet this simplicity belies challenges. Some inputs violate preconditions or cause…
Let $\cal{N}=\{1,\cdots,n\}$. The entropy function $\bf h$ of a set of $n$ discrete random variables $\{X_i:i\in\cal N\}$ is a $2^n$-dimensional vector whose entries are ${\bf{h}}({\cal{A}})\triangleq H(X_{\cal{A}}),\cal{A}\subset{\cal N}…
Holant problems are a general framework to study the computational complexity of counting problems. It is a more expressive framework than counting constraint satisfaction problems (CSP) which are in turn more expressive than counting graph…
A discrete temporal constraint satisfaction problem is a constraint satisfaction problem (CSP) whose constraint language consists of relations that are first-order definable over $(\Bbb Z,<)$. Our main result says that every distance CSP is…
Using properties of Blum complexity measures and certain complexity class operators, we exhibit a total computable and non-decreasing function $t_{\mathsf{poly}}$ such that for all $k$, $\Sigma_k\mathsf{P} =…
The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such…
Promise CSPs are a relaxation of constraint satisfaction problems where the goal is to find an assignment satisfying a relaxed version of the constraints. Several well-known problems can be cast as promise CSPs including approximate graph…
We propose a new order, the small polynomial path order (sPOP* for short). The order sPOP* provides a characterisation of the class of polynomial time computable function via term rewrite systems. Any polynomial time computable function…
The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of…