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TFNP
In computational complexity theory, the complexity class TFNP is the class of total function problems which can be solved in nondeterministic polynomial time. That is, it is the class of function problems that are guaranteed to have an answer, and this answer can be checked in polynomial time, or equivalently it is the subset of FNP where a solution is guaranteed to exist. The abbreviation TFNP stands for "Total Function Nondeterministic Polynomial". TFNP contains many natural problems that are of interest to computer scientists. These problems include integer factorization, finding a Nash Equilibrium of a game, and searching for local optima. TFNP is widely conjectured to contain problems that are computationally intractable, and several such problems have been shown to be hard under cryptographic assumptions. However, there are no known unconditional intractability results or results showing NP-hardness of TFNP problems. TFNP is not believed to have any complete problems.Goldberg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
TFNP Inclusions
In computational complexity theory, the complexity class TFNP is the class of total function problems which can be solved in nondeterministic polynomial time. That is, it is the class of function problems that are guaranteed to have an answer, and this answer can be checked in polynomial time, or equivalently it is the subset of FNP where a solution is guaranteed to exist. The abbreviation TFNP stands for "Total Function Nondeterministic Polynomial". TFNP contains many natural problems that are of interest to computer scientists. These problems include integer factorization, finding a Nash Equilibrium of a game, and searching for local optima. TFNP is widely conjectured to contain problems that are computationally intractable, and several such problems have been shown to be hard under cryptographic assumptions. However, there are no known unconditional intractability results or results showing NP-hardness of TFNP problems. TFNP is not believed to have any complete problems.Goldberg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PPAD
In computer science, PPAD ("Polynomial Parity Arguments on Directed graphs") is a complexity class introduced by Christos Papadimitriou in 1994. PPAD is a subclass of TFNP based on functions that can be shown to be total by a parity argument. The class attracted significant attention in the field of algorithmic game theory because it contains the problem of computing a Nash equilibrium: this problem was shown to be complete for PPAD by Daskalakis, Goldberg and Papadimitriou with at least 3 players and later extended by Chen and Deng to 2 players.*. Definition PPAD is a subset of the class TFNP, the class of function problems in FNP that are guaranteed to be total. The TFNP formal definition is given as follows: :A binary relation P(''x'',''y'') is in TFNP if and only if there is a deterministic polynomial time algorithm that can determine whether P(''x'',''y'') holds given both ''x'' and ''y'', and for every ''x'', there exists a ''y'' such that P(''x'',''y'') holds. Subclas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PLS (complexity)
In computational complexity theory, Polynomial Local Search (PLS) is a complexity class that models the difficulty of finding a locally optimal solution to an optimization problem. The main characteristics of problems that lie in PLS are that the cost of a solution can be calculated in polynomial time and the neighborhood of a solution can be searched in polynomial time. Therefore it is possible to verify whether or not a solution is a local optimum in polynomial time. Furthermore, depending on the problem and the algorithm that is used for solving the problem, it might be faster to find a local optimum instead of a global optimum. Description When searching for a local optimum, there are two interesting issues to deal with: First how to find a local optimum, and second how long it takes to find a local optimum. For many local search algorithms, it is not known, whether they can find a local optimum in polynomial time or not. So to answer the question of how long it takes to find a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PPA (complexity)
In computational complexity theory, PPA is a complexity class, standing for "Polynomial Parity Argument" (on a graph). Introduced by Christos Papadimitriou in 1994 (page 528), PPA is a subclass of TFNP. It is a class of search problems that can be shown to be total by an application of the handshaking lemma: ''any undirected graph that has a vertex whose degree is an odd number must have some other vertex whose degree is an odd number''. This observation means that if we are given a graph and an odd-degree vertex, and we are asked to find some other odd-degree vertex, then we are searching for something that is guaranteed to exist (so, we have a total search problem). Definition PPA is defined as follows. Suppose we have a graph on whose vertices are n-bit binary strings, and the graph is represented by a polynomial-sized circuit that takes a vertex as input and outputs its neighbors. (Note that this allows us to represent an exponentially-large graph on which we can efficiently pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FNP (complexity)
In computational complexity theory, the complexity class FNP is the function problem extension of the decision problem class NP. The name is somewhat of a misnomer, since technically it is a class of binary relations, not functions, as the following formal definition explains: :A binary relation P(''x'',''y''), where ''y'' is at most polynomially longer than ''x'', is in FNP if and only if there is a deterministic polynomial time algorithm that can determine whether P(''x'',''y'') holds given both ''x'' and ''y''. This definition does not involve nondeterminism and is analogous to the verifier definition of NP. There is an NP language directly corresponding to every FNP relation, sometimes called the decision problem ''induced by'' or ''corresponding to'' said FNP relation. It is the language formed by taking all the ''x'' for which P(''x'',''y'') holds given some ''y''; however, there may be more than one FNP relation for a particular decision problem. Many problems in NP, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PPP (complexity)
In computational complexity theory, the complexity class PPP (polynomial pigeonhole principle) is a subclass of TFNP. It is the class of search problems that can be shown to be total by an application of the pigeonhole principle. Christos Papadimitriou introduced it in the same paper that introduced PPAD and PPA. PPP contains both PPAD and PWPP (polynomial weak pigeonhole principle) as subclasses. These complexity classes are of particular interest in cryptography because they are strongly related to cryptographic primitives such as one-way permutations and collision-resistant hash functions. Definition PPP is the set of all function computation problems that admit a polynomial-time reduction to the ''PIGEON'' problem, defined as follows: :Given a Boolean circuit C having the same number n of input bits as output bits, find either an input x that is mapped to the output C(x) = 0^n, or two distinct inputs x \ne y that are mapped to the same output C(x) = C(y). A problem is PPP-co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Complementarity Problem
In mathematical optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It was proposed by Cottle and Dantzig in 1968. Formulation Given a real matrix ''M'' and vector ''q'', the linear complementarity problem LCP(''q'', ''M'') seeks vectors ''z'' and ''w'' which satisfy the following constraints: * w, z \geqslant 0, (that is, each component of these two vectors is non-negative) * z^Tw = 0 or equivalently \sum\nolimits_i w_i z_i = 0. This is the complementarity condition, since it implies that, for all i, at most one of w_i and z_i can be positive. * w = Mz + q A sufficient condition for existence and uniqueness of a solution to this problem is that ''M'' be symmetric positive-definite. If ''M'' is such that has a solution for every ''q'', then ''M'' is a Q-matrix. If ''M'' is such that have a unique solution for every ''q'', then ''M'' is a P-matrix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-matrix
In mathematics, a -matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P_0-matrices, which are the closure of the class of -matrices, with every principal minor \geq 0. Spectra of -matrices By a theorem of Kellogg, the eigenvalues of - and P_0- matrices are bounded away from a wedge about the negative real axis as follows: :If \ are the eigenvalues of an -dimensional -matrix, where n>1, then ::, \arg(u_i), < \pi - \frac,\ i = 1,...,n :If , , are the eigenvalues of an -dimensional -matrix, then :: Remarks The class of nonsingular ''M''-matrices is a subset of the class of -matrices. More precisely, all matrices that are both -matrices and[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Fixed-point Theorem
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. It can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922. Statement ''Definition.'' Let (X, d) be a complete metric space. Then a map T : X \to X is called a contraction mapping on ''X'' if there exists q \in non-empty complete metric space with a contraction mapping T : X \to X. Then ''T'' admits a unique Fixed point (mathematics)">fixed-point x^* in ''X'' (i.e. T(x^*) = x^*). Furthermore, x^* can be found as follows: start with an arbitrary element x_0 \in X and define a sequence (x_n)_ by x_n = T(x_) for n \geq 1. Then \li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karush–Kuhn–Tucker Conditions
In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints. Similar to the Lagrange approach, the constrained maximization (minimization) problem is rewritten as a Lagrange function whose optimal point is a saddle point, i.e. a global maximum (minimum) over the domain of the choice variables and a global minimum (maximum) over the multipliers, which is why the Karush–Kuhn–Tucker theorem is sometimes referred to as the saddle-point theorem. The KKT conditions were originally named after Harold W. Kuhn and Albert W. Tucker, who first published the conditions in 1951. L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Handshaking Lemma
In graph theory, a branch of mathematics, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges is even. In more colloquial terms, in a party of people some of whom shake hands, the number of people who shake an odd number of other people's hands is even. The handshaking lemma is a consequence of the degree sum formula, also sometimes called the handshaking lemma, according to which the sum of the degrees (the numbers of times each vertex is touched) equals twice the number of edges in the graph. Both results were proven by in his famous paper on the Seven Bridges of Königsberg that began the study of graph theory. Beyond the Bridges of Königsberg and their generalization to Euler tours, other applications include proving that for certain combinatorial structures, the number of structures is always even, and assisting with the proofs of Sperner's lemma and the mountain climbing problem. The comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short Integer Solution Problem
Short integer solution (SIS) and ring-SIS problems are two ''average''-case problems that are used in lattice-based cryptography constructions. Lattice-based cryptography began in 1996 from a seminal work by Miklós AjtaiAjtai, Miklós. enerating hard instances of lattice problems.Proceedings of the twenty-eighth annual ACM symposium on Theory of computing. ACM, 1996. who presented a family of one-way functions based on SIS problem. He showed that it is secure in an average case if the shortest vector problem \mathrm_\gamma (where \gamma = n^c for some constant c>0) is hard in a worst-case scenario. Average case problems are the problems that are hard to be solved for some randomly selected instances. For cryptography applications, worst case complexity is not sufficient, and we need to guarantee cryptographic construction are hard based on average case complexity. Lattices A ''full rank lattice'' \mathfrak \subset \R^n is a set of integer linear combinations of n linearly inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
