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

• Published in: Mathematical programming Vol. 97; no. 1-2; pp. 43 - 69
• Main Author: ARORA, Sanjeev
• Format: Journal Article
• Language: English
• 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
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-003-0438-y

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.