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N COMPUTER applications we are used to live with approximation. Var­ I ious notions of approximation appear, in fact, in many circumstances. One notable example is the type of approximation that arises in numer­ ical analysis or in computational geometry from the fact that we cannot perform computations with arbitrary precision and we have to truncate the representation of real numbers. In other cases, we use to approximate com­ plex mathematical objects by simpler ones: for example, we sometimes represent non-linear functions by means of piecewise linear ones. The need to solve difficult optimization problems is another reason that forces us to deal with approximation. In particular, when a problem is computationally hard (i. e. , the only way we know to solve it is by making use of an algorithm that runs in exponential time), it may be practically unfeasible to try to compute the exact solution, because it might require months or years of machine time, even with the help of powerful parallel computers. In such cases, we may decide to restrict ourselves to compute a solution that, though not being an optimal one, nevertheless is close to the optimum and may be determined in polynomial time. We call this type of solution an approximate solution and the corresponding algorithm a polynomial-time approximation algorithm. Most combinatorial optimization problems of great practical relevance are, indeed, computationally intractable in the above sense. In formal terms, they are classified as Np-hard optimization problems.

Programming --- Computer science --- Mathematical statistics --- algoritmen --- 681.3*G16 Optimization: constrained optimization; gradient methods; integer programming; least squares methods; linear programming; nonlinear programming (Numericalanalysis) --- Optimization: constrained optimization; gradient methods; integer programming; least squares methods; linear programming; nonlinear programming (Numericalanalysis) --- 681.3*G12 Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- 681.3*F2 Analysis of algorithms and problem complexity--See also {681.3*B6}; {681.3*B7}; {681.3*F13} --- Analysis of algorithms and problem complexity--See also {681.3*B6}; {681.3*B7}; {681.3*F13} --- 681.3*G3 Probability and statistics: probabilistic algorithms (including Monte Carlo);random number generation; statistical computing; statistical software (Mathematics of computing) --- Probability and statistics: probabilistic algorithms (including Monte Carlo);random number generation; statistical computing; statistical software (Mathematics of computing) --- 681.3*G3 --- Combinatorial optimization --- Computational complexity --- Computer algorithms --- 681.3*F2 --- 681.3*G12 --- 681.3*G16 --- 681.3*G2 --- Complexity, Computational --- Optimization, Combinatorial --- 681.3*G2 Discrete mathematics (Mathematics of computing) --- Discrete mathematics (Mathematics of computing) --- Algorithms --- Electronic data processing --- Machine theory --- Combinatorial analysis --- Mathematical optimization --- Computational Complexity --- Optimisation combinatoire --- Complexité de calcul (Informatique) --- Algorithmes --- Computer programming. --- Algorithms. --- Computer mathematics. --- Computer science—Mathematics. --- Operations research. --- Decision making. --- Information technology. --- Business—Data processing. --- Programming Techniques. --- Algorithm Analysis and Problem Complexity. --- Computational Mathematics and Numerical Analysis. --- Discrete Mathematics in Computer Science. --- Operations Research/Decision Theory. --- IT in Business. --- IT (Information technology) --- Technology --- Telematics --- Information superhighway --- Knowledge management --- Deciding --- Decision (Psychology) --- Decision analysis --- Decision processes --- Making decisions --- Management --- Management decisions --- Choice (Psychology) --- Problem solving --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory --- Computer mathematics --- Mathematics --- Algorism --- Algebra --- Arithmetic --- Computers --- Electronic computer programming --- Electronic digital computers --- Programming (Electronic computers) --- Coding theory --- Decision making --- Foundations

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Computer games --- Microcomputers --- 681.3*G --- 681.3*F2 --- 681.3*I21 --- 681.3*I26 --- 681.3*I28 --- 681.3*G Mathematics of computing --- Mathematics of computing --- Home computers --- Micro computers --- Micros (Microcomputers) --- PCs (Microcomputers) --- Personal computers --- Small computers --- Minicomputers --- Application software --- Electronic games --- 681.3*I28 Problem solving, control methods and search: backtracking dynamic program- ming graph and tree search strategies heuristics plan execution, formationand generation (Artificial intelligence)--See also {681.3*F22} --- Problem solving, control methods and search: backtracking dynamic program- ming graph and tree search strategies heuristics plan execution, formationand generation (Artificial intelligence)--See also {681.3*F22} --- 681.3*I26 Learning: analogies concept learning induction knowledge acquisition language acquisition parameter learning (Artificial intelligence)--See also {681.3*K32} --- Learning: analogies concept learning induction knowledge acquisition language acquisition parameter learning (Artificial intelligence)--See also {681.3*K32} --- 681.3*I21 Applications and expert systems (Artificial intelligence). Cartography. Games. Industrial automation. Law. Medicine and science. Natural language interfaces. Office automation--See also {681.3*H4} {681.3*J} --- Applications and expert systems (Artificial intelligence). Cartography. Games. Industrial automation. Law. Medicine and science. Natural language interfaces. Office automation--See also {681.3*H4} {681.3*J} --- Analysis of algorithms and problem complexity --- Computer science. --- Information technology. --- Business --- Algorithms. --- Computer science --- Artificial intelligence. --- Calculus of variations. --- Computer Science. --- Algorithm Analysis and Problem Complexity. --- Artificial Intelligence (incl. Robotics). --- Mathematics of Computing. --- Calculus of Variations and Optimal Control; Optimization. --- IT in Business. --- Data processing. --- Mathematics. --- 681.3*I28 Problem solving, control methods and search: backtracking; dynamic program- ming; graph and tree search strategies; heuristics; plan execution, formationand generation (Artificial intelligence)--See also {681.3*F22} --- Problem solving, control methods and search: backtracking; dynamic program- ming; graph and tree search strategies; heuristics; plan execution, formationand generation (Artificial intelligence)--See also {681.3*F22} --- 681.3*I26 Learning: analogies; concept learning; induction; knowledge acquisition; language acquisition; parameter learning (Artificial intelligence)--See also {681.3*K32} --- Learning: analogies; concept learning; induction; knowledge acquisition; language acquisition; parameter learning (Artificial intelligence)--See also {681.3*K32} --- 681.3*I21 Applications and expert systems (Artificial intelligence). Cartography. Games. Industrial automation. Law. Medicine and science. Natural language interfaces. Office automation--See also {681.3*H4}; {681.3*J} --- Applications and expert systems (Artificial intelligence). Cartography. Games. Industrial automation. Law. Medicine and science. Natural language interfaces. Office automation--See also {681.3*H4}; {681.3*J} --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Computer mathematics --- Discrete mathematics --- Algorism --- Algebra --- Arithmetic --- IT (Information technology) --- Technology --- Telematics --- Information superhighway --- Knowledge management --- Informatics --- Science --- Mathematics --- Foundations --- Computer software. --- Mathematical optimization. --- Artificial Intelligence. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Operations research --- System analysis --- Software, Computer --- Computer systems --- Computer science—Mathematics. --- Business—Data processing. --- Internet games --- Television games --- Videogames --- Games --- Video games