When Does a Dynamic Programming Formulation Guarantee the Existence of a Fully Polynomial Time Approximation Scheme (FPTAS)?

We derive results of the following flavor: If a combinatorial optimization problem can be formulated via a dynamic program of a certain structure and if the involved cost and transition functions satisfy certain arithmetical and structural conditions, then the optimization problem automatically possesses a fully polynomial time approximation scheme (FPTAS). Our characterizations provide a natural and uniform approach to fully polynomial time approximation schemes. We illustrate their strength and generality by deducing from them the existence of FPTASs for a multitude of scheduling problems. Many known approximability results follow as corollaries from our main result.

[1]  Chris N. Potts,et al.  Single Machine Scheduling to Minimize Total Weighted Late Work , 1995, INFORMS J. Comput..

[2]  E.L. Lawler,et al.  Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey , 1977 .

[3]  Wieslaw Kubiak,et al.  A Fully Polynomial Approximation Scheme for Minimizing Makespan of Deteriorating Jobs , 1998, J. Heuristics.

[4]  Giorgio Ausiello,et al.  Toward a Unified Approach for the Classification of NP-Complete Optimization Problems , 1979, Theor. Comput. Sci..

[5]  Albert P. M. Wagelmans,et al.  Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems , 2001, Math. Oper. Res..

[6]  Peter Brucker,et al.  Single machine batch scheduling to minimize the weighted number of late jobs , 1996, Math. Methods Oper. Res..

[7]  Samuel Eilon,et al.  Minimising Waiting Time Variance in the Single Machine Problem , 1977 .

[8]  Shlomo Moran,et al.  Non Deterministic Polynomial Optimization Problems and their Approximations , 1977, Theor. Comput. Sci..

[9]  Eugene L. Lawler,et al.  Sequencing and scheduling: algorithms and complexity , 1989 .

[10]  Dorit S. Hochbaum,et al.  Scheduling with batching: minimizing the weighted number of tardy jobs , 1994, Oper. Res. Lett..

[11]  Wieslaw Kubiak,et al.  Completion time variance minimization on a single machine is difficult , 1993, Oper. Res. Lett..

[12]  Eugene L. Lawler,et al.  Fast approximation algorithms for knapsack problems , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

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

[14]  Marc E. Posner,et al.  Earliness-Tardiness Scheduling Problems, I: Weighted Deviation of Completion Times About a Common Due Date , 1991, Oper. Res..

[15]  Gerhard J. Woeginger.  An Approximation Scheme for Minimizing Agreeably Weighted Variance on a Single Machine , 1999, INFORMS J. Comput..

[16]  Chris N. Potts,et al.  Approximation algorithms for scheduling a single machine to minimize total late work , 1992, Oper. Res. Lett..

[17]  J. K. Lenstra,et al.  Deterministic Production Planning: Algorithms and Complexity , 1980 .

[18]  Edward G. Coffman,et al.  Scheduling independent tasks to reduce mean finishing time , 1974, CACM.

[19]  E. L. Lawler,et al.  A fully polynomial approximation scheme for the total tardiness problem , 1982, Oper. Res. Lett..

[20]  Chris N. Potts,et al.  A Fully Polynomial Approximation Scheme for Scheduling a Single Machine to Minimize Total Weighted Late Work , 1994, Math. Oper. Res..

[21]  J. M. Moore,et al.  A Functional Equation and its Application to Resource Allocation and Sequencing Problems , 1969 .

[22]  Giorgio Ausiello,et al.  Theoretical Computer Science Approximate Solution of Np Optimization Problems * , 2022 .

[23]  Stuart E. Dreyfus,et al.  Applied Dynamic Programming , 1965 .

[24]  Giorgio Ausiello,et al.  Structure Preserving Reductions among Convex Optimization Problems , 1980, J. Comput. Syst. Sci..

[25]  Oscar H. Ibarra,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975, JACM.

[26]  Wieslaw Kubiak,et al.  A Fully Polynomial Approximation Scheme for the Weighted Earliness-Tardiness Problem , 1999, Oper. Res..

[27]  Zhi-Long Chen.  Parallel Machine Scheduling with Time Dependent Processing Times , 1996, Discret. Appl. Math..

[28]  Wieslaw Kubiak,et al.  Algorithms for Minclique Scheduling Problems , 1997, Discret. Appl. Math..

[29]  Xiaoqiang Cai,et al.  Minimization of agreeably weighted variance in single machine systems , 1995 .

[30]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[31]  Ellis Horowitz,et al.  Computing Partitions with Applications to the Knapsack Problem , 1974, JACM.

[32]  Ellis Horowitz,et al.  Exact and Approximate Algorithms for Scheduling Nonidentical Processors , 1976, JACM.

[33]  Jan Karel Lenstra,et al.  Complexity of machine scheduling problems , 1975 .

[34]  David S. Johnson,et al.  `` Strong '' NP-Completeness Results: Motivation, Examples, and Implications , 1978, JACM.

[35]  Eugene L. Lawler,et al.  Chapter 9 Sequencing and scheduling: Algorithms and complexity , 1993, Logistics of Production and Inventory.

[36]  Sid Browne,et al.  Scheduling Deteriorating Jobs on a Single Processor , 1990, Oper. Res..

[37]  E. Lawler.  A “Pseudopolynomial” Algorithm for Sequencing Jobs to Minimize Total Tardiness , 1977 .

[38]  Chris N. Potts,et al.  Single Machine Scheduling to Minimize Total Late Work , 1992, Oper. Res..

[39]  Wieslaw Kubiak,et al.  Scheduling deteriorating jobs to minimize makespan , 1998 .

[40]  Eugene Levner,et al.  Fast approximation algorithm for job sequencing with deadlines , 1981, Discret. Appl. Math..