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...= F \cap E$. Thus, it is the same as the index of a [[Vapnik–Chervonenkis class]]. It is usually denoted by $\mathrm{VC}(H)$. ...nik–Chervonenkis dimension is $\mathcal{NP}$-hard (cf. also [[NP|$\mathcal{NP}$]]) for many classes of hypergraphs, [[#References|[a1]]], [[#References|[

...entially with the number of variables and turns out to be impractical. The complexity of theoretical and numerical problems that arise in the solution of integer ...le solutions (relative to the "input size" of the problem). The problem "P=NP?" remains open at present (1988).

...99 : ~/encyclopedia/old_files/data/A011/A.0101800 Algorithm, computational complexity of an ...an algorithm to the inputs. A more precise definition of the computational complexity of an algorithm is the concept of a cost function (step-counting function)

...nomial effort coincide for deterministic and nondeterministic functions (P=NP). ...ask processed by a nondeterministic Turing machine may be smaller than the complexity for the corresponding task processed by a (deterministic) Turing machine on

...ng machine|Turing machine]] model of computation. Continuous computational complexity studies continuous problems and tends to use the real number model; see [[# ...al problem can be solved only approximately. The goal of information-based complexity is to compute such an approximation at minimal cost. The error and the cost

...hm]]; [[Algorithm, computational complexity of an|Algorithm, computational complexity of an]]). ...haracteristic function of $S$ in $\Sigma ^ { * }$. In order to speak about complexity, a size measure for a problem instance $x \in \Sigma ^ { * }$ as well as a

...lin graphs. Indeed, the construction just described yields an example of a class of edge-minimal planar $3$-connected graphs. The graph shown below is Halin ...nown that Halin graphs are contained in a so-called $3$-terminal recursive class. As shown in [[#References|[a3]]], this suffices to guarantee linear-time a

...the execution time and the number of the processors matches the sequential complexity of the problem. Examples of such problems include parallel sorting algorith ...ial time (cf. also [[Complexity theory|Complexity theory]]; [[NP|$\mathcal{NP}$]]).

===Complexity Theory of Probabilistic Turing Machines=== ...}[M(x)=1 ]$ denotes the fraction of computations leading to $M(x)=1$. The class of languages decided by PTMs in $O(T(n))$ computation steps is designated a

Computational complexity measures the amount of computational resources, such as time and space, tha Computational complexity studies the inherent difficulty of computational problems. Attention is con

...eory|Complexity theory]]; [[Computational complexity classes|Computational complexity classes]]). Comprehensive introductions to quantum computation and the know ...the classical models of computation. In particular, for formal studies of complexity, many researchers use various versions of quantum Turing machines (cf. also

Traditionally, one has considered the so-called worst-case complexity, which for each input size $n$ determines the maximum amount of resources u ...c worst-case and a $\Theta( n \operatorname { log } n )$ average-case time complexity [[#References|[a22]]]. For a more detailed analysis of its running time bey

$#C+1 = 164 : ~/encyclopedia/old_files/data/C024/C.0204230 Complexity theory ...ticle [[Algorithm, computational complexity of an|Algorithm, computational complexity of an]].

...h the fixed-point property has drawn attention in many ways. The $\mathcal{NP}$-version of the characterization problem is: It is $\mathcal{NP}$-complete when considered in the class of ordered sets of height $5$ [[#References|[a5]]]. The most efficient algo

...h conjecture: Every simple planar graph of maximum degree $6$ or $7$ is of class 1. ...<td valign="top"> V.G. Vizing, "Critical graphs with a given chromatic class" ''Diskret. Anal.'' , '''5''' (1965) pp. 9–17 (In Russian)</td></tr><

...mitation of the number domain may lead to an increase of the computational complexity, however. ====Quantum Complexity====

...ublished at the beginning of the 1970's stimulated numerous studies on the complexity of scheduling problems (see, e.g., [[#References|[5]]]). The results of thi ...n="top">[5]</TD> <TD valign="top"> J.K. Lenstra, A.H.G. Rinnooy Kan, "Complexity of scheduling under precedence constraints" ''Operations Research'' , '''2

..., quantum communication, quantum computation, quantum algorithms and their complexity, and quantum control. The science of quantum information processing is a hi ...cal computer access to qubits as defined above, and then investigating the complexity of algorithms by counting both classical and quantum resources, thus obtain

{| class="wikitable" style="border: 3px solid darkgray;" The elements of an equivalence class can be characterized at a purely

{| class="wikitable" ...over the non-Bayesian methodology is certainly the increased computational complexity. Assume a Gaussian random field model in the form of the form Eq. 1 with kn