Approximation schemes for a class of subset selection problems
In this paper we develop an easily applicable algorithmic technique/tool for developing approximation schemes for certain types of combinatorial optimization problems. Special cases that are covered by our result show up in many places in the literature. For every such special case, a particular rou...
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| Published in: | Theoretical computer science Vol. 382; no. 2; pp. 151 - 156 |
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| Main Authors: | , |
| Format: | Journal Article Conference Proceeding |
| Language: | English |
| Published: |
Amsterdam
Elsevier B.V
31.08.2007
Elsevier |
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| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | In this paper we develop an easily applicable algorithmic technique/tool for developing approximation schemes for certain types of combinatorial optimization problems. Special cases that are covered by our result show up in many places in the literature. For every such special case, a particular rounding trick has been implemented in a slightly different way, with slightly different arguments, and with slightly different worst case estimations. Usually, the rounding procedure depended on certain upper or lower bounds on the optimal objective value that have to be justified in a separate argument. Our easily applied result unifies many of these results, and sometimes it even leads to a simpler proof.
We demonstrate how our result can be easily applied to a broad family of combinatorial optimization problems. As a special case, we derive the existence of an FPTAS for the scheduling problem of minimizing the weighted number of late jobs under release dates and preemption on a single machine. The approximability status of this problem has been open for some time. |
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| AbstractList | In this paper we develop an easily applicable algorithmic technique/tool for developing approximation schemes for certain types of combinatorial optimization problems. Special cases that are covered by our result show up in many places in the literature. For every such special case, a particular rounding trick has been implemented in a slightly different way, with slightly different arguments, and with slightly different worst case estimations. Usually, the rounding procedure depended on certain upper or lower bounds on the optimal objective value that have to be justified in a separate argument. Our easily applied result unifies many of these results, and sometimes it even leads to a simpler proof.
We demonstrate how our result can be easily applied to a broad family of combinatorial optimization problems. As a special case, we derive the existence of an FPTAS for the scheduling problem of minimizing the weighted number of late jobs under release dates and preemption on a single machine. The approximability status of this problem has been open for some time. |
| Author | Pruhs, Kirk Woeginger, Gerhard J. |
| Author_xml | – sequence: 1 givenname: Kirk surname: Pruhs fullname: Pruhs, Kirk email: kirk@cs.pitt.edu organization: Department of Computer Science, University of Pittsburgh, Pittsburgh, PA 15260, USA – sequence: 2 givenname: Gerhard J. surname: Woeginger fullname: Woeginger, Gerhard J. email: gwoegi@win.tue.nl organization: Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands |
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| Cites_doi | 10.1016/S0020-0190(02)00205-3 10.1145/321941.321951 10.1287/opre.35.1.70 10.1007/BF02392825 10.1287/moor.4.4.339 10.1016/0166-218X(81)90008-1 10.1016/S0167-5060(08)70356-X 10.1287/moor.17.1.36 10.1145/321921.321934 10.1007/BF02248588 10.1287/mnsc.16.1.77 10.1016/S0167-6377(01)00069-4 10.1145/321906.321909 10.1287/ijoc.12.1.57.11901 10.1145/321812.321823 |
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| Keywords | Pseudo-polynomial algorithm Approximation scheme Approximation algorithm FPTAS Combinatorial optimization Scheduling theory Worst case analysis Polynomial Lower bound Approximation Computer theory Combinatorial problem Estimation Optimization method Algorithmics Scheduling Implementation Selection problem Upper bound Proof Procedure Single machine |
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| References | Ibarra, Kim (b11) 1975; 22 Warburton (b20) 1987; 35 Ergun, Sinha, Zhang (b3) 2002; 83 Karp (b12) 1972 Bellman, Dreyfus (b1) 1962 Håstad (b8) 1999; 182 Horowitz, Sahni (b10) 1976; 23 Lawler (b13) 1979; 4 Martello, Toth (b18) 1990 Cormen, Leiserson, Rivest, Stein (b2) 2001 Lawler, Moore (b16) 1969; 16 Lawler (b14) 1990; 26 Sahni (b19) 1976; 23 Woeginger (b21) 2000; 12 Hassin (b7) 1992; 17 Graham, Lawler, Lenstra, Rinnooy Kan (b6) 1979; 5 Horowitz, Sahni (b9) 1974; 21 Lorenz, Raz (b17) 2001; 28 Garey, Johnson (b4) 1979 Gens, Levner (b5) 1981; 3 Lawler, Lenstra, Rinnooy Kan, Shmoys (b15) 1993; vol. 4 Bellman (10.1016/j.tcs.2007.03.006_b1) 1962 Graham (10.1016/j.tcs.2007.03.006_b6) 1979; 5 Horowitz (10.1016/j.tcs.2007.03.006_b10) 1976; 23 Horowitz (10.1016/j.tcs.2007.03.006_b9) 1974; 21 Ibarra (10.1016/j.tcs.2007.03.006_b11) 1975; 22 Gens (10.1016/j.tcs.2007.03.006_b5) 1981; 3 Woeginger (10.1016/j.tcs.2007.03.006_b21) 2000; 12 Cormen (10.1016/j.tcs.2007.03.006_b2) 2001 Garey (10.1016/j.tcs.2007.03.006_b4) 1979 Lawler (10.1016/j.tcs.2007.03.006_b13) 1979; 4 Warburton (10.1016/j.tcs.2007.03.006_b20) 1987; 35 Martello (10.1016/j.tcs.2007.03.006_b18) 1990 Karp (10.1016/j.tcs.2007.03.006_b12) 1972 Hassin (10.1016/j.tcs.2007.03.006_b7) 1992; 17 Sahni (10.1016/j.tcs.2007.03.006_b19) 1976; 23 Lawler (10.1016/j.tcs.2007.03.006_b14) 1990; 26 Lawler (10.1016/j.tcs.2007.03.006_b16) 1969; 16 Håstad (10.1016/j.tcs.2007.03.006_b8) 1999; 182 Lawler (10.1016/j.tcs.2007.03.006_b15) 1993; vol. 4 Ergun (10.1016/j.tcs.2007.03.006_b3) 2002; 83 Lorenz (10.1016/j.tcs.2007.03.006_b17) 2001; 28 |
| References_xml | – year: 1979 ident: b4 article-title: Computers and Intractability – volume: 83 start-page: 287 year: 2002 end-page: 291 ident: b3 article-title: An improved FPTAS for restricted shortest path publication-title: Information Processing Letters – year: 1962 ident: b1 article-title: Applied Dynamic Programming – volume: 16 start-page: 77 year: 1969 end-page: 84 ident: b16 article-title: A functional equation and its application to resource allocation and sequencing problems publication-title: Management Science – volume: 17 start-page: 36 year: 1992 end-page: 42 ident: b7 article-title: Approximation schemes for the restricted shortest path problem publication-title: Mathematics of Operations Research – volume: 26 start-page: 125 year: 1990 end-page: 133 ident: b14 article-title: A dynamic programming algorithm for preemptive scheduling of a single machine to minimize the number of late jobs publication-title: Annals of Operations Research – volume: 23 start-page: 116 year: 1976 end-page: 127 ident: b19 article-title: Algorithms for scheduling independent tasks publication-title: Journal of the ACM – volume: 23 start-page: 317 year: 1976 end-page: 327 ident: b10 article-title: Exact and approximate algorithms for scheduling nonidentical processors publication-title: Journal of the ACM – volume: vol. 4 start-page: 445 year: 1993 end-page: 522 ident: b15 article-title: Sequencing and scheduling: Algorithms and complexity publication-title: Logistics of Production and Inventory – volume: 35 start-page: 70 year: 1987 end-page: 79 ident: b20 article-title: Approximation of pareto optima in multiple-objective shortest path problems publication-title: Operations Research – volume: 3 start-page: 313 year: 1981 end-page: 318 ident: b5 article-title: Fast approximation algorithms for job sequencing with deadlines publication-title: Discrete Applied Mathematics – start-page: 85 year: 1972 end-page: 104 ident: b12 article-title: Reducibility among combinatorial problems publication-title: Complexity of Computer Computations – volume: 5 start-page: 287 year: 1979 end-page: 326 ident: b6 article-title: Optimization and approximation in deterministic sequencing and scheduling: A survey publication-title: Annals of Discrete Mathematics – volume: 12 start-page: 57 year: 2000 end-page: 75 ident: b21 article-title: When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)? publication-title: INFORMS Journal on Computing – volume: 21 start-page: 277 year: 1974 end-page: 292 ident: b9 article-title: Computing partitions with applications to the knapsack problem publication-title: Journal of the ACM – volume: 22 start-page: 463 year: 1975 end-page: 468 ident: b11 article-title: Fast approximation algorithms for the knapsack and sum of subset problems publication-title: Journal of the ACM – volume: 182 start-page: 105 year: 1999 end-page: 142 ident: b8 article-title: Clique is hard to approximate within publication-title: Acta Mathematica – volume: 4 start-page: 339 year: 1979 end-page: 356 ident: b13 article-title: Fast approximation schemes for knapsack problems publication-title: Mathematics of Operations Research – year: 1990 ident: b18 article-title: Knapsack problems: Algorithms and computer implementations – volume: 28 start-page: 213 year: 2001 end-page: 219 ident: b17 article-title: A simple efficient approximation scheme for the restricted shortest path problem publication-title: Operations Research Letters – year: 2001 ident: b2 article-title: Introduction to Algorithms – volume: 83 start-page: 287 year: 2002 ident: 10.1016/j.tcs.2007.03.006_b3 article-title: An improved FPTAS for restricted shortest path publication-title: Information Processing Letters doi: 10.1016/S0020-0190(02)00205-3 – volume: 23 start-page: 317 year: 1976 ident: 10.1016/j.tcs.2007.03.006_b10 article-title: Exact and approximate algorithms for scheduling nonidentical processors publication-title: Journal of the ACM doi: 10.1145/321941.321951 – volume: 35 start-page: 70 year: 1987 ident: 10.1016/j.tcs.2007.03.006_b20 article-title: Approximation of pareto optima in multiple-objective shortest path problems publication-title: Operations Research doi: 10.1287/opre.35.1.70 – year: 1979 ident: 10.1016/j.tcs.2007.03.006_b4 – volume: 182 start-page: 105 year: 1999 ident: 10.1016/j.tcs.2007.03.006_b8 article-title: Clique is hard to approximate within n1−ϵ publication-title: Acta Mathematica doi: 10.1007/BF02392825 – volume: 4 start-page: 339 year: 1979 ident: 10.1016/j.tcs.2007.03.006_b13 article-title: Fast approximation schemes for knapsack problems publication-title: Mathematics of Operations Research doi: 10.1287/moor.4.4.339 – volume: 3 start-page: 313 year: 1981 ident: 10.1016/j.tcs.2007.03.006_b5 article-title: Fast approximation algorithms for job sequencing with deadlines publication-title: Discrete Applied Mathematics doi: 10.1016/0166-218X(81)90008-1 – start-page: 85 year: 1972 ident: 10.1016/j.tcs.2007.03.006_b12 article-title: Reducibility among combinatorial problems – volume: 5 start-page: 287 year: 1979 ident: 10.1016/j.tcs.2007.03.006_b6 article-title: Optimization and approximation in deterministic sequencing and scheduling: A survey publication-title: Annals of Discrete Mathematics doi: 10.1016/S0167-5060(08)70356-X – volume: 17 start-page: 36 year: 1992 ident: 10.1016/j.tcs.2007.03.006_b7 article-title: Approximation schemes for the restricted shortest path problem publication-title: Mathematics of Operations Research doi: 10.1287/moor.17.1.36 – volume: 23 start-page: 116 year: 1976 ident: 10.1016/j.tcs.2007.03.006_b19 article-title: Algorithms for scheduling independent tasks publication-title: Journal of the ACM doi: 10.1145/321921.321934 – volume: 26 start-page: 125 year: 1990 ident: 10.1016/j.tcs.2007.03.006_b14 article-title: A dynamic programming algorithm for preemptive scheduling of a single machine to minimize the number of late jobs publication-title: Annals of Operations Research doi: 10.1007/BF02248588 – year: 1962 ident: 10.1016/j.tcs.2007.03.006_b1 – year: 2001 ident: 10.1016/j.tcs.2007.03.006_b2 – volume: 16 start-page: 77 year: 1969 ident: 10.1016/j.tcs.2007.03.006_b16 article-title: A functional equation and its application to resource allocation and sequencing problems publication-title: Management Science doi: 10.1287/mnsc.16.1.77 – volume: 28 start-page: 213 year: 2001 ident: 10.1016/j.tcs.2007.03.006_b17 article-title: A simple efficient approximation scheme for the restricted shortest path problem publication-title: Operations Research Letters doi: 10.1016/S0167-6377(01)00069-4 – volume: 22 start-page: 463 year: 1975 ident: 10.1016/j.tcs.2007.03.006_b11 article-title: Fast approximation algorithms for the knapsack and sum of subset problems publication-title: Journal of the ACM doi: 10.1145/321906.321909 – volume: 12 start-page: 57 year: 2000 ident: 10.1016/j.tcs.2007.03.006_b21 article-title: When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)? 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| SubjectTerms | Applied sciences Approximation algorithm Approximation scheme Calculus of variations and optimal control Combinatorial optimization Computer science; control theory; systems Computer systems performance. Reliability Exact sciences and technology FPTAS Mathematical analysis Mathematics Miscellaneous Numerical analysis Numerical analysis. Scientific computation Numerical methods in mathematical programming, optimization and calculus of variations Pseudo-polynomial algorithm Scheduling theory Sciences and techniques of general use Software Theoretical computing Worst case analysis |
| Title | Approximation schemes for a class of subset selection problems |
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