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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] |
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.Goldber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sperner's Lemma
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an simplex contains a cell whose vertices all have different colors. The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain. Sperner colorings have been used for effective computation of fixed points and in root-finding algorithms, and are applied in fair division (cake cutting) algorithms. According to the Soviet ''Mathematical Encyclopaedia'' (ed. I.M. Vinogradov), a related 1929 theorem (of Knaster, Borsuk and Mazurkiewicz) had also become known as the ''Sperner lemma'' – this point is discussed in the English translation (ed. M. Hazewinkel). It is now commonly known as the Knaster–Kuratowski–Mazurkiewicz lemma. Statement One-dimensional case In one dimension, Spern ... [...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- ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithmic Game Theory
Algorithmic game theory (AGT) is an interdisciplinary field at the intersection of game theory and computer science, focused on understanding and designing algorithms for environments where multiple strategic agents interact. This research area combines computational thinking with economic principles to address challenges that emerge when algorithmic inputs come from self-interested participants. In traditional algorithm design, inputs are assumed to be fixed and reliable. However, in many real-world applications—such as online auctions, internet routing, digital advertising, and resource allocation systems—inputs are provided by multiple independent agents who may strategically misreport information to manipulate outcomes in their favor. AGT provides frameworks to analyze and design systems that remain effective despite such strategic behavior. The field can be approached from two complementary perspectives: * ''Analysis'': Evaluating existing algorithms and systems throu ... [...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] |
Fixed-point Computation
Fixed-point computation refers to the process of computing an exact or approximate Fixed point (mathematics), fixed point of a given function. In its most common form, the given function f satisfies the condition to the Brouwer fixed-point theorem: that is, f is continuous and maps the unit N-cube, ''d''-cube to itself. The Brouwer fixed-point theorem guarantees that f has a fixed point, but the proof is not Constructive proof, constructive. Various algorithms have been devised for computing an approximate fixed point. Such algorithms are used in economics for computing a market equilibrium, in game theory for computing a Nash equilibrium, and in dynamic system analysis. Definitions The unit interval is denoted by E := [0, 1], and the unit N-cube, ''d''-dimensional cube is denoted by E^d. A continuous function f is defined on E^d (from E^d to itself)''.'' Often, it is assumed that f is not only continuous but also Lipschitz continuous, that is, for some constant L, , f(x)-f(y), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envy-free Cake-cutting
An envy-free cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the envy-free criterion, namely, that every partner feels that their allocated share is at least as good as any other share, according to their own subjective valuation. When there are only two partners, the problem is easy and was solved in antiquity by the divide and choose protocol. When there are three or more partners, the problem becomes much more challenging. Two major variants of the problem have been studied: * Connected pieces, e.g. if the cake is a 1-dimensional interval then each partner must receive a single sub-interval. If there are n partners, only n-1 cuts are needed. * General pieces, e.g. if the cake is a 1-dimensional interval then each partner can receive a union of disjoint sub-intervals. Short history Modern research into the fair cake-cutting problem started in the 1940s. The first fairness criterion studied was proportional d ... [...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] |
Arrow–Debreu Model
In mathematical economics, the Arrow–Debreu model is a theoretical general equilibrium model. It posits that under certain economic assumptions (convex preferences, perfect competition, and demand independence), there must be a set of prices such that Aggregate supply, aggregate supplies will equal aggregate demands for every commodity in the economy. The model is central to the theory of General equilibrium, general (economic) equilibrium, and it is used as a general reference for other microeconomic models. It was proposed by Kenneth Arrow, Gérard Debreu in 1954, and Lionel W. McKenzie independently in 1954, with later improvements in 1959. The A-D model is one of the most general models of competitive economy and is a crucial part of general equilibrium theory, as it can be used to prove the existence of general equilibrium (or Walrasian equilibrium) of an economy. In general, there may be many equilibria. Arrow (1972) and Debreu (1983) were separately awarded the Nobel Me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Xi Chen
Xi Chen () is a computer scientist. He is a professor of computer science at Columbia University. Chen won the 2021 Gödel Prize and Fulkerson Prize for his co-authored paper "Complexity of Counting CSP with Complex Weights" with Jin-Yi Cai. Biography Chen received his B.S. and Ph.D. from Tsinghua University. He was a postdoctoral fellow at Institute for Advanced Study, Princeton University, University of Southern California, and joined the Columbia faculty in 2011. Chen's research focuses on computational complexity theory. He also received a Presburger Award from the European Association for Theoretical Computer Science in 2015 and a Sloan Research Fellowship The Sloan Research Fellowships are awarded annually by the Alfred P. Sloan Foundation since 1955 to "provide support and recognition to early-career scientists and scholars". This program is one of the oldest of its kind in the United States. ... in 2012. References Living people Tsinghua University a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christos Papadimitriou
Christos Charilaos Papadimitriou (; born August 16, 1949) is a Greek-American theoretical computer scientist and the Donovan Family Professor of Computer Science at Columbia University. Education Papadimitriou studied at the National Technical University of Athens, where in 1972 he received his Bachelor of Arts degree in electrical engineering. He then pursued graduate studies at Princeton University, where he received his Ph.D. in electrical engineering and computer science in 1976 after completing a doctoral dissertation titled "The complexity of combinatorial optimization problems." Career Papadimitriou has taught at Harvard, MIT, the National Technical University of Athens, Stanford, UCSD, University of California, Berkeley and is currently the Donovan Family Professor of Computer Science at Columbia University. Papadimitriou co-authored a paper on pancake sorting with Bill Gates, then a Harvard undergraduate. Papadimitriou recalled "Two years later, I called to tell him ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
