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Bài giảng Phân tích và thiết kế giải thuật: Chương 8 - PGS.TS. Dương Tuấn Anh

Bài giảng chương 8 trang bị cho người học những hiểu biết về thuật toán xấp xỉ. Trong chương này người học có thể tìm hiểu một số bài toán phủ đỉnh và một số vấn đề về phủ đỉnh. để nắm bắt các nội dung chi tiết. | Chapter 8 Approximation Algorithms Outline Why approximation algorithms? The vertex cover problem The set cover problem TSP Why Approximation Algorithms ? Many problems of practical significance are NP-complete but are too important to abandon merely because obtaining an optimal solution is intractable. If a problem is NP-complete, we are unlikely to find a polynomial time algorithm for solving it exactly, but it may still be possible to find near-optimal solution in polynomial time. In practice, near-optimality is often good enough. An algorithm that returns near-optimal solutions is called an approximation algorithm. Performance bounds for approximation algorithms i is an optimization problem instance c(i) be the cost of solution produced by approximate algorithm and c*(i) be the cost of optimal solution. For minimization problem, we want c(i)/c*(i) to be as small as possible. For maximization problem, we want c*(i)/c(i) to be as small as possible. An approximation algorithm for the given problem instance i, has a ratio bound of p(n) if for any input of size n, the cost c of the solution produced by the approximation algorithm is within a factor of p(n) of the cost c* of an optimal solution. That is max(c(i)/c*(i), c*(i)/c(i)) ≤ p(n) Note that p(n) is always greater than or equal to 1. If p(n) = 1 then the approximate algorithm is an optimal algorithm. The larger p(n), the worst algorithm Relative error We define the relative error of the approximate algorithm for any input size as |c(i) - c*(i)|/ c*(i) We say that an approximate algorithm has a relative error bound of ε(n) if |c(i)-c*(i)|/c*(i)≤ ε(n) 1. The Vertex-Cover Problem Vertex cover: given an undirected graph G=(V,E), then a subset V V such that if (u,v) E, then u V or v V (or both). Size of a vertex cover: the number of vertices in it. Vertex-cover problem: find a vertex-cover of minimal size. This problem is NP-hard, since the related decision problem is NP-complete | Chapter 8 Approximation Algorithms Outline Why approximation algorithms? The vertex cover problem The set cover problem TSP Why Approximation Algorithms ? Many problems of practical significance are NP-complete but are too important to abandon merely because obtaining an optimal solution is intractable. If a problem is NP-complete, we are unlikely to find a polynomial time algorithm for solving it exactly, but it may still be possible to find near-optimal solution in polynomial time. In practice, near-optimality is often good enough. An algorithm that returns near-optimal solutions is called an approximation algorithm. Performance bounds for approximation algorithms i is an optimization problem instance c(i) be the cost of solution produced by approximate algorithm and c*(i) be the cost of optimal solution. For minimization problem, we want c(i)/c*(i) to be as small as possible. For maximization problem, we want c*(i)/c(i) to be as small as possible. An approximation algorithm