Polynomial-Time Approximation Schemes

Let Π be an NP-hard optimization problem, and let A be an approximation algorithm for Π. For an instance I of Π, let A(I) denote the objective value when running A on I, and let OPT (I) denote the optimal objective value. The approximation ratio of A for the instance I is RA(I) = A(I)/OPT (I), thus, when Π is minimization (maximization) problem RA(I) ≥ 1 (RA(I) ≤ 1). A polynomial time approximation scheme is an algorithm which takes as input an additional parameter, ε, which determines the desired approximation ratio. This ratio can be arbitrarily close to 1, when ε approaches 0. The time complexity of the scheme is polynomial in the input size but may be exponential in 1/ε. This gives a clear trade-off between running time and quality of approximation. Formally,

[1]  C. N. Potts,et al.  Analysis of a linear programming heuristic for scheduling unrelated parallel machines , 1985, Discret. Appl. Math..

[2]  Rajeev Motwani,et al.  Towards a syntactic characterization of PTAS , 1996, STOC '96.

[3]  Claire Mathieu,et al.  A Randomized Approximation Scheme for Metric MAX-CUT , 1998, FOCS.

[4]  Samir Khuller,et al.  Approximation Schemes for Broadcasting in Heterogenous Networks , 2004, APPROX-RANDOM.

[5]  Gerhard J. Woeginger,et al.  There is no Asymptotic PTAS for Two-Dimensional Vector Packing , 1997, Inf. Process. Lett..

[6]  A. Frieze,et al.  Approximation algorithms for the m-dimensional 0–1 knapsack problem: Worst-case and probabilistic analyses , 1984 .

[7]  Wolfgang Maass,et al.  Approximation schemes for covering and packing problems in image processing and VLSI , 1985, JACM.

[8]  Marek Karpinski,et al.  Approximation schemes for clustering problems , 2003, STOC '03.

[9]  Rajeev Motwani,et al.  Lecture notes on approximation algorithms: Volume I , 1993 .

[10]  Sartaj Sahni,et al.  Approximate Algorithms for the 0/1 Knapsack Problem , 1975, JACM.

[11]  Ronald L. Graham,et al.  Bounds for certain multiprocessing anomalies , 1966 .

[12]  Stavros G. Kolliopoulos.  Approximating covering integer programs with multiplicity constraints , 2003, Discret. Appl. Math..

[13]  Sartaj Sahni,et al.  Algorithms for Scheduling Independent Tasks , 1976, J. ACM.

[14]  Klaus Jansen,et al.  Polynomial Time Approximation Schemes for the Multiprocessor Open and Flow Shop Scheduling Problem , 2000, STACS.

[15]  David B. Shmoys,et al.  Using dual approximation algorithms for scheduling problems: practical and theoretical results , 1987 .

[16]  Claire Mathieu,et al.  Polynomial-time approximation scheme for data broadcast , 2000, STOC '00.

[17]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[18]  Guy Kortsarz,et al.  Tools for Multicoloring with Applications to Planar Graphs and Partial k-Trees , 2002, J. Algorithms.

[19]  Klaus Jansen,et al.  Makespan Minimization in Job Shops: A Linear Time Approximation Scheme , 2003, SIAM J. Discret. Math..

[20]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[21]  Hadas Shachnai,et al.  Approximation Schemes for Generalized 2-Dimensional Vector Packing with Application to Data Placement , 2003, RANDOM-APPROX.

[22]  Piotr Indyk.  A sublinear time approximation scheme for clustering in metric spaces , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[23]  David R. Karger,et al.  Scheduling Algorithms , 2004, Algorithms and Theory of Computation Handbook.

[24]  Gerhard J. Woeginger,et al.  Makespan minimization in open shops: A polynomial time approximation scheme , 1998, Math. Program..

[25]  Samir Khuller,et al.  Algorithms for non-uniform size data placement on parallel disks , 2003, J. Algorithms.

[26]  Hadas Shachnai,et al.  Approximation Schemes for Deal Splitting and Covering Integer Programs with Multiplicity Constraints , 2004, WAOA.