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♯P-complete
The #P-complete problems (pronounced "sharp P complete" or "number P complete") form a complexity class in computational complexity theory. The problems in this complexity class are defined by having the following two properties: *The problem is in #P, the class of problems that can be defined as counting the number of accepting paths of a polynomial-time non-deterministic Turing machine. *The problem is #P-hard, meaning that every other problem in #P has a Turing reduction or polynomial-time counting reduction to it. A counting reduction is a pair of polynomial-time transformations from inputs of the other problem to inputs of the given problem and from outputs of the given problem to outputs of the other problem, allowing the other problem to be solved using any subroutine for the given problem. A Turing reduction is an algorithm for the other problem that makes a polynomial number of calls to a subroutine for the given problem and, outside of those calls, uses polynomial t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time Counting Reduction
In the computational complexity theory of counting problems, a polynomial-time counting reduction is a type of reduction (a transformation from one problem to another) used to define the notion of completeness for the complexity class ♯P. These reductions may also be called polynomial many-one counting reductions or weakly parsimonious reductions; they are analogous to many-one reductions for decision problems and they generalize the parsimonious reductions. Definition A polynomial-time counting reduction is usually used to transform instances of a known-hard problem X into instances of another problem Y that is to be proven hard. It consists of two functions f and g, both of which must be computable in polynomial time. The function f transforms inputs for X into inputs for Y, and the function g transforms outputs for Y into outputs for X. These two functions must preserve the correctness of the output. That is, suppose that one transforms an input x for problem X to an input ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
♯P
In computational complexity theory, the complexity class #P (pronounced "sharp P" or, sometimes "number P" or "hash P") is the set of the counting problems associated with the decision problems in the set NP. More formally, #P is the class of function problems of the form "compute ''f''(''x'')", where ''f'' is the number of accepting paths of a nondeterministic Turing machine running in polynomial time. Unlike most well-known complexity classes, it is not a class of decision problems but a class of function problems. The most difficult, representative problems of this class are #P-complete. Relation to decision problems An NP decision problem is often of the form "Are there any solutions that satisfy certain constraints?" For example: * Are there any subsets of a list of integers that add up to zero? (subset sum problem) * Are there any Hamiltonian cycles in a given graph with cost less than 100? (traveling salesman problem) * Are there any variable assignments that satisfy a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parsimonious Reduction
In computational complexity theory and game complexity, a parsimonious reduction is a transformation from one problem to another (a reduction) that preserves the number of solutions. Informally, it is a bijection between the respective sets of solutions of two problems. A general reduction from problem A to problem B is a transformation that guarantees that whenever A has a solution B also has ''at least one'' solution and vice versa. A parsimonious reduction guarantees that for every solution of A, there exists ''a unique solution'' of B and vice versa. Parsimonious reductions are commonly used in computational complexity for proving the hardness of counting problems, for counting complexity classes such as #P. Additionally, they are used in game complexity, as a way to design hard puzzles that have a unique solution, as many types of puzzles require. Formal definition Let x be an instance of problem X. A ''Parsimonious reduction'' R from problem X to problem Y is a reductio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sharp-SAT
In computer science, the Sharp Satisfiability Problem (sometimes called Sharp-SAT or #SAT) is the problem of counting the number of interpretations that satisfies a given Boolean formula, introduced by Valiant in 1979. In other words, it asks in how many ways the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. For example, the formula a\lor \neg b is satisfiable by three distinct boolean value assignments of the variables, namely, for any of the assignments (a = TRUE, b = FALSE), (a = FALSE, b = FALSE), (a = TRUE, b = TRUE), we have a\lor \neg b = TRUE. #SAT is different from Boolean satisfiability problem (SAT), which asks if there exists ''a solution'' of Boolean formula. Instead, #SAT asks to enumerate ''all'' ''the solutions'' to a Boolean Formula. #SAT is harder than SAT in the sense that, once the total number of solutions to a Boolean formula is known, SAT can be decided in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Matching
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph , a perfect matching in is a subset of edge set , such that every vertex in the vertex set is adjacent to exactly one edge in . A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete matching is used. Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs: : In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched. A perfect matching is also a minimum-size edge cover. If there is a perfect matching, then both the matching number and the edge cover number equal . A perfect matching can only occur when the graph has an even n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexity Class
In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. counting problems and function problems) and using other models of computation (e.g. probabilistic Turing machines, interactive proof systems, Boolean circuits, and quantum computers). The study of the relationships between complexity class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Acyclic Graph
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one vertex to another, such that following those directions will never form a closed loop. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge directions. DAGs have numerous scientific and computational applications, ranging from biology (evolution, family trees, epidemiology) to information science (citation networks) to computation (scheduling). Directed acyclic graphs are sometimes instead called acyclic directed graphs or acyclic digraphs. Definitions A graph is formed by vertices and by edges connecting pairs of vertices, where the vertices can be any kind of object that is connected in pairs by edges. In the case of a directed graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order (journal)
''Order'' (subtitled ''A Journal on the Theory of Ordered Sets and its Applications'') is a quarterly peer-reviewed academic journal on order theory and its applications, published by Springer Science+Business Media. It was established in 1984 by Ivan Rival (University of Calgary). From 2010 to 2018, its editor-in-chief was Dwight Duffus (Emory University). He was succeeded in 2019 by Ryan R. Martin (Iowa State University). Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2017 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 0.353. References External links * Order theory Mathematics journals Springer Science+Business Media academic journals Publications established in 1984 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FP (complexity)
In computational complexity theory, the complexity class FP is the set of function problems that can be solved by a deterministic Turing machine in polynomial time. It is the function problem version of the decision problem class P. Roughly speaking, it is the class of functions that can be efficiently computed on classical computers without randomization. The difference between FP and P is that problems in P have one-bit, yes/no answers, while problems in FP can have any output that can be computed in polynomial time. For example, adding two numbers is an FP problem, while determining if their sum is odd is in P. Polynomial-time function problems are fundamental in defining polynomial-time reductions, which are used in turn to define the class of NP-complete problems. Formal definition FP is formally defined as follows: :A binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP (complexity)
In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.''Polynomial time'' refers to how quickly the number of operations needed by an algorithm, relative to the size of the problem, grows. It is therefore a measure of efficiency of an algorithm. An equivalent definition of NP is the set of decision problems ''solvable'' in polynomial time by a nondeterministic Turing machine. This definition is the basis for the abbreviation NP; "nondeterministic, polynomial time". These two definitions are equivalent because the algorithm based on the Turing machine consists of two phases, the first of which consists of a guess ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P (complexity)
In computational complexity theory, P, also known as PTIME or DTIME(''n''O(1)), is a fundamental complexity class. It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. Cobham's thesis holds that P is the class of computational problems that are "efficiently solvable" or " tractable". This is inexact: in practice, some problems not known to be in P have practical solutions, and some that are in P do not, but this is a useful rule of thumb. Definition A language ''L'' is in P if and only if there exists a deterministic Turing machine ''M'', such that * ''M'' runs for polynomial time on all inputs * For all ''x'' in ''L'', ''M'' outputs 1 * For all ''x'' not in ''L'', ''M'' outputs 0 P can also be viewed as a uniform family of boolean circuits. A language ''L'' is in P if and only if there exists a polynomial-time uniform family of boolean circuits \, such that * For all n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
