Approximation schemes for NP-hard geometric optimization problems: a survey

Traveling Salesman, Steiner Tree, and many other famous geometric optimization problems are NP-hard. Since we do not expect to design efficient algorithms that solve these problems optimally, researchers have tried to design approximation algorithms, which can compute a provably near-optimal solutio...

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Bibliographic Details
Published inMathematical programming Vol. 97; no. 1-2; pp. 43 - 69
Main Author ARORA, Sanjeev
Format Journal Article
LanguageEnglish
Published Heidelberg Springer 01.07.2003
Springer Nature B.V
Subjects
Algorithms
Applied sciences
Approximation
Digital Object Identifier
Dissection
Exact sciences and technology
Heuristic
Mathematical programming
Operational research and scientific management
Operational research. Management science
Optimization
Traveling salesman problem
NP hard problem
Geometric programming
Mathematical programming
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Summary:Traveling Salesman, Steiner Tree, and many other famous geometric optimization problems are NP-hard. Since we do not expect to design efficient algorithms that solve these problems optimally, researchers have tried to design approximation algorithms, which can compute a provably near-optimal solution in polynomial time. We survey such algorithms, in particular a new technique developed over the past few years that allows us to design approximation schemes for many of these problems. For any fixed constant c> 0, the algorithm can compute a solution whose cost is at most (1 + c) times the optimum. (The running time is polynomial for every fixed c> 0, and in many cases is even nearly linear.) We describe how these schemes are designed, and survey the status of a large number of problems.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-003-0438-y