Topic
Polynomial-time approximation scheme
About: Polynomial-time approximation scheme is a research topic. Over the lifetime, 1865 publications have been published within this topic receiving 51635 citations. The topic is also known as: PTAS.
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01 Nov 1990TL;DR: This paper focuses on the part of the knapsack problem where the problem of bin packing is concerned and investigates the role of computer codes in the solution of this problem.
Abstract: Introduction knapsack problem bounded knapsack problem subset-sum problem change-making problem multiple knapsack problem generalized assignment problem bin packing problem. Appendix: computer codes.
3,694 citations
TL;DR: It follows that such a complete problem has a polynomial-time approximation scheme iff the whole class does, and that a number of common optimization problems are complete for MAX SNP under a kind of careful transformation that preserves approximability.
1,919 citations
TL;DR: An O(kn) approximation algorithm that guarantees solutions with an objective function value within two times the optimal solution value is presented and it is shown that this approximation algorithm succeeds as long as the set of points satisfies the triangular inequality.
1,784 citations
TL;DR: For P- complete problems such as traveling salesperson, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, etc., it is shown that the approximation problem is also P-complete.
Abstract: For P-complete problems such as traveling salesperson, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, etc., it is shown that the approximation problem is also P-complete. In contrast with these results, a linear time approximation algorithm for the clustering problem is presented.
1,718 citations
TL;DR: The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-aproximation in polynomial time.
Abstract: We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c > 1 and given any n nodes in ℛ2, a randomized version of the scheme finds a (1 + 1/c)-approximation to the optimum traveling salesman tour in O(n(log n)O(c)) time. When the nodes are in ℛd, the running time increases to O(n(log n)(O(√c))d-1). For every fixed c, d the running time is n · poly(logn), that is nearly linear in n. The algorithmm can be derandomized, but this increases the running time by a factor O(nd). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-aproximation in polynomial time.We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, k-TSP, and k-MST. (The running times of the algorithm for k-TSP and k-MST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. We also give efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time.All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as lp for p ≥ 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.
1,113 citations