Topic
Average-case complexity
About: Average-case complexity is a research topic. Over the lifetime, 1749 publications have been published within this topic receiving 44972 citations.
Papers published on a yearly basis
Papers
More filters
Posted Content•
TL;DR: This work defines analogues of the complexity classes P and NP for these and proves the NP-completeness of a sequence called the universal circuit resultant, which is the first family of compact spaces shown to be NP-complete in a geometric setting.
Abstract: We study the computational complexity of sequences of projective varieties. We define analogues of the complexity classes P and NP for these and prove the NP-completeness of a sequence called the universal circuit resultant. This is the first family of compact spaces shown to be NP-complete in a geometric setting.
4 citations
Posted Content•
TL;DR: In this article, the authors developed a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations, following the lines of the theory developed by Blum, Shub, and Smale for computations over R (which in turn followed those of the classical, discrete, complexity theory as laid down by Cook, Karp, and Levin among others).
Abstract: We develop a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations. We follow the lines of the theory developed by Blum, Shub, and Smale for computations over R (which in turn followed those of the classical, discrete, complexity theory as laid down by Cook, Karp, and Levin among others). In particular, we focus on complexity classes of decision problems and paramount among them, on appropriate versions of the classes P, NP and EXP of polynomial, nondeterministic polynomial, and exponential time, respectively. We prove some basic relationships between these complexity classes and exhibit natural NP-complete problems.
4 citations
TL;DR: A lower bound on the deterministic complexity is derived which generalizes the bounds known for 2-processor-systems and is a canonical extension of the results known for the special case k=2.
4 citations
01 Jan 1990
TL;DR: It is shown that the bisection method is not optimal on the average, and under general conditions on the measure, the average error bounds for bisection are derived, but if the sampling is restricted to the signs of the functions, or if the measure on the function space is not known, it is shown it is asymptotically optimal.
Abstract: In this dissertation, we study the approximation of stochastic operators. We are interested in bounds for the error of the algorithms and the complexity of the problem. We characterize optimal information/sampling for continuous linear operators, assuming that arbitrary linear functionals may be used in the sampling.
The approximation and integration problem, of functions of several variables, is considered on a Wiener space. Tight error and complexity bounds are obtained for the approximation problem. These bounds are compared to those of the worst case. For the integration problem we show an improved deterministic sampling scheme.
Turning our attention to nonlinear operators, we study the average solution of nonlinear equations. In contrast to the situation in the worst case, we show that the bisection method is not optimal on the average. Under general conditions on the measure, we derive average error bounds for bisection. However, if the sampling is restricted to the signs (positive/negative) of the functions, or if the measure on the function space is not known, we show that the bisection method is asymptotically optimal.
4 citations
TL;DR: The advice complexity of the semifeasible sets was introduced by Karp and Lipton as discussed by the authors, which is a notion that contains aspects both of descriptional/informational complexity and of computational complexity.
Abstract: Informally put, the semifeasible sets are the class of sets having a polynomial-time algorithm that, given as input any two strings of which at least one belongs to the set, will choose one that does belong to the set. We provide a tutorial overview of the advice complexity of the semifeasible sets. No previous familiarity with either the semifeasible sets or advice complexity will be assumed, and when we include proofs we will try to make the material as accessible as possible via providing intuitive, informal presentations. Karp and Lipton introduced advice complexity about a quarter of a century ago.18 Advice complexity asks, for a given power of interpreter, how many bits of "help" suffice to accept a given set. Thus, this is a notion that contains aspects both of descriptional/informational complexity and of computational complexity. We will see that for some powers of interpreter the (worst-case) complexity of the semifeasible sets is known right down to the bit (and beyond), but that for the most central power of interpreter—deterministic polynomial time—the complexity is currently known only to be at least linear and at most quadratic. While overviewing the advice complexity of the semifeasible sets, we will stress also the issue of whether the functions at the core of semifeasibility—so-called selector functions—can without cost be chosen to possess such algebraic properties as commutativity and associativity. We will see that this is relevant, in ways both potential and actual, to the study of the advice complexity of the semifeasible sets.
4 citations