Related papers: Counting independent terms in big-oh notation
Computational problems can be classified according to their algorithmic complexity, which is defined based on how the resources needed to solve the problem, e.g. the execution time, scale with the problem size. Many problems in…
quest for processing speed potential. In fact, we always get a fraction of the technically available computing power (so-called {\em theoretical peak}), and the gap is likely to go hand-to-hand with the hardware complexity of the target…
Computational complexity is a core theory of computer science, which dictates the degree of difficulty of computation. There are many problems with high complexity that we have to deal, which is especially true for AI. This raises a big…
In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like "input", "size" and…
Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
Algorithms for continuous optimization problems have a rich history of design and innovation over the past several decades, in which mathematical analysis of their convergence and complexity properties plays a central role. Besides their…
How best to quantify the information of an object, whether natural or artifact, is a problem of wide interest. A related problem is the computability of an object. We present practical examples of a new way to address this problem. By…
Algorithmic efficiency is essential to reducing energy and time usage for computational problems. Optimizing efficiency is important for tasks involving multiple resources, for example in stochastic calculations where the size of the random…
Benchmarking heuristic algorithms is vital to understand under which conditions and on what kind of problems certain algorithms perform well. In most current research into heuristic optimization algorithms, only a very limited number of…
Two philosophical applications of the concept of program-size complexity are discussed. First, we consider the light program-size complexity sheds on whether mathematics is invented or discovered, i.e., is empirical or is a priori. Second,…
People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at…
Analyzing the computational complexity of evolutionary algorithms for binary search spaces has significantly increased their theoretical understanding. With this paper, we start the computational complexity analysis of genetic programming.…
A general condition determining the optimal performance of a complex system has not yet been found and the possibility of its existence is unknown. To contribute in this direction, an optimization algorithm as a complex system is presented.…
We study output-sensitive algorithms and complexity for multiobjective combinatorial optimization problems. In this computational complexity framework, an algorithm for a general enumeration problem is regarded efficient if it is…
Until now, Computer Scientists have concerned themselves with identifying efficient algorithms for solving the general case of some problem -- that is finding one which performs well when the size of the input tends to infinity. In this…
We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P = NP, no polynomial-time algorithm can give an approximate solution…
Computational feasibility is a widespread concern that guides the framing and modeling of biological and artificial intelligence. The specification of cognitive system capacities is often shaped by unexamined intuitive assumptions about the…
Big data refers to large and complex data sets that, under existing approaches, exceed the capacity and capability of current compute platforms, systems software, analytical tools and human understanding. Numerous lessons on the scalability…
Genetic Programming (GP) has found various applications. Understanding this type of algorithm from a theoretical point of view is a challenging task. The first results on the computational complexity of GP have been obtained for problems…