Parametric Packing of Selfish Items and the Subset Sum Algorithm
The subset sum algorithm is a natural heuristic for the classical Bin Packing problem: In each iteration, the algorithm finds among the unpacked items, a maximum size set of items that fits into a new bin. More than 35 years after its first mention in the literature, establishing the worst-case perf...
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| Vydáno v: | Algorithmica Ročník 74; číslo 1; s. 177 - 207 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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01.01.2016
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| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | The subset sum algorithm is a natural heuristic for the classical Bin Packing problem: In each iteration, the algorithm finds among the unpacked items, a maximum size set of items that fits into a new bin. More than 35 years after its first mention in the literature, establishing the worst-case performance of this heuristic remains, surprisingly, an open problem. Due to their simplicity and intuitive appeal, greedy algorithms are the heuristics of choice of many practitioners. Therefore, better understanding simple greedy heuristics is, in general, an interesting topic in its own right. Very recently, Epstein and Kleiman (Proc. ESA 2008, pp. 368–380) provided another incentive to study the subset sum algorithm by showing that the Strong Price of Anarchy of the game theoretic version of the Bin Packing problem is
precisely
the approximation ratio of this heuristic. In this paper we establish the exact approximation ratio of the subset sum algorithm, thus settling a long standing open problem. We generalize this result to the parametric variant of the Bin Packing problem where item sizes lie on the interval
(
0
,
α
]
for some
α
≤
1
, yielding tight bounds for the Strong Price of Anarchy for all
α
≤
1
. Finally, we study the pure Price of Anarchy of the parametric Bin Packing game for which we show nearly tight upper and lower bounds for all
α
≤
1
. |
|---|---|
| AbstractList | The subset sum algorithm is a natural heuristic for the classical Bin Packing problem: In each iteration, the algorithm finds among the unpacked items, a maximum size set of items that fits into a new bin. More than 35 years after its first mention in the literature, establishing the worst-case performance of this heuristic remains, surprisingly, an open problem. Due to their simplicity and intuitive appeal, greedy algorithms are the heuristics of choice of many practitioners. Therefore, better understanding simple greedy heuristics is, in general, an interesting topic in its own right. Very recently, Epstein and Kleiman (Proc. ESA 2008, pp. 368–380) provided another incentive to study the subset sum algorithm by showing that the Strong Price of Anarchy of the game theoretic version of the Bin Packing problem is
precisely
the approximation ratio of this heuristic. In this paper we establish the exact approximation ratio of the subset sum algorithm, thus settling a long standing open problem. We generalize this result to the parametric variant of the Bin Packing problem where item sizes lie on the interval
(
0
,
α
]
for some
α
≤
1
, yielding tight bounds for the Strong Price of Anarchy for all
α
≤
1
. Finally, we study the pure Price of Anarchy of the parametric Bin Packing game for which we show nearly tight upper and lower bounds for all
α
≤
1
. |
| Author | Kleiman, Elena Epstein, Leah Mestre, Julián |
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| Cites_doi | 10.1016/j.geb.2008.03.005 10.1016/S0167-6377(03)00092-0 10.1145/950620.950621 10.1145/3828.3833 10.1016/j.ipl.2005.04.005 10.1137/0203025 10.1145/506147.506153 10.1080/095372899232894 10.1007/s00453-009-9348-6 10.1007/978-3-540-24777-7 10.1007/s00453-006-0056-1 10.1007/3-540-49116-3_38 10.1007/978-1-4684-2001-2_9 10.1007/978-3-540-73420-8_51 10.1145/1478873.1478901 10.1007/978-3-540-92185-1_50 10.1007/978-3-540-31856-9_53 10.1145/1219944.1219949 10.1109/IPDPS.2006.1639283 |
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| References | Gupta, Ho (CR10) 1999; 10 Lee, Lee (CR18) 1985; 32 Csirik, Leung, Gonzalez (CR6) 2007 CR2 Caprara, Pferschy (CR4) 2005; 96 Roughgarden, Tardos (CR21) 2002; 49 Coffman, Csirik, Gonzalez (CR5) 2007 Andelman, Feldman, Mansour (CR1) 2009; 65 CR7 Kellerer, Pferschy, Pisinger (CR16) 2004 CR17 CR9 CR15 Mavronicolas, Spirakis (CR19) 2007; 48 CR12 CR23 CR11 CR22 Caprara, Pferschy (CR3) 2004; 32 Epstein, Kleiman (CR8) 2011; 60 Roughgarden (CR20) 2005 Johnson, Demers, Ullman, Garey, Graham (CR14) 1974; 3 Jain, Mahdian, Markakis, Saberi, Vazirani (CR13) 2003; 50 9942_CR11 9942_CR22 9942_CR12 DS Johnson (9942_CR14) 1974; 3 9942_CR23 9942_CR15 M Mavronicolas (9942_CR19) 2007; 48 H Kellerer (9942_CR16) 2004 9942_CR17 9942_CR9 9942_CR7 J Csirik (9942_CR6) 2007 9942_CR2 E Coffman Jr (9942_CR5) 2007 A Caprara (9942_CR4) 2005; 96 L Epstein (9942_CR8) 2011; 60 K Jain (9942_CR13) 2003; 50 JND Gupta (9942_CR10) 1999; 10 T Roughgarden (9942_CR20) 2005 T Roughgarden (9942_CR21) 2002; 49 CC Lee (9942_CR18) 1985; 32 A Caprara (9942_CR3) 2004; 32 N Andelman (9942_CR1) 2009; 65 |
| References_xml | – ident: CR22 – year: 2007 ident: CR5 article-title: Performance guarantees for one-dimensional bin packing publication-title: Handbook of approximation algorithms and metaheuristics, chapter 32 – volume: 65 start-page: 289 issue: 2 year: 2009 end-page: 317 ident: CR1 article-title: Strong price of anarchy publication-title: Games Econ. Behav. doi: 10.1016/j.geb.2008.03.005 – volume: 32 start-page: 159 issue: 2 year: 2004 end-page: 166 ident: CR3 article-title: Worst-case analysis of the subset sum algorithm for bin packing publication-title: Oper. Res. Lett. doi: 10.1016/S0167-6377(03)00092-0 – volume: 50 start-page: 795 issue: 6 year: 2003 end-page: 824 ident: CR13 article-title: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP publication-title: J. ACM doi: 10.1145/950620.950621 – year: 2007 ident: CR6 article-title: Variants of classical one-dimensional bin packing publication-title: Handbook of Approximation Algorithms and Metaheuristics, chapter 33 – volume: 32 start-page: 562 year: 1985 end-page: 572 ident: CR18 article-title: A simple online bin packing algorithm publication-title: J. ACM doi: 10.1145/3828.3833 – volume: 96 start-page: 18 issue: 1 year: 2005 end-page: 23 ident: CR4 article-title: Modified subset sum heuristics for bin packing publication-title: Inf. Process. Lett. doi: 10.1016/j.ipl.2005.04.005 – ident: CR15 – ident: CR2 – volume: 3 start-page: 299 issue: 4 year: 1974 end-page: 325 ident: CR14 article-title: Worst-case performance bounds for simple one-dimensional packing algorithms publication-title: SIAM J. Comput. doi: 10.1137/0203025 – ident: CR12 – ident: CR17 – volume: 49 start-page: 236 issue: 2 year: 2002 end-page: 259 ident: CR21 article-title: How bad is selfish routing? publication-title: J. ACM doi: 10.1145/506147.506153 – ident: CR11 – ident: CR9 – volume: 10 start-page: 598 year: 1999 end-page: 603 ident: CR10 article-title: A new heuristic algorithm for the one-dimensional bin-packing problem publication-title: Prod. Plan. Control doi: 10.1080/095372899232894 – volume: 60 start-page: 368 issue: 2 year: 2011 end-page: 394 ident: CR8 article-title: Selfish bin packing publication-title: Algorithmica doi: 10.1007/s00453-009-9348-6 – year: 2005 ident: CR20 publication-title: Selfish routing and the price of anarchy – ident: CR7 – year: 2004 ident: CR16 publication-title: Knapsack problems doi: 10.1007/978-3-540-24777-7 – ident: CR23 – volume: 48 start-page: 91 issue: 1 year: 2007 end-page: 126 ident: CR19 article-title: The price of selfish routing publication-title: Algorithmica doi: 10.1007/s00453-006-0056-1 – ident: 9942_CR17 doi: 10.1007/3-540-49116-3_38 – ident: 9942_CR15 doi: 10.1007/978-1-4684-2001-2_9 – volume: 96 start-page: 18 issue: 1 year: 2005 ident: 9942_CR4 publication-title: Inf. Process. Lett. doi: 10.1016/j.ipl.2005.04.005 – volume: 3 start-page: 299 issue: 4 year: 1974 ident: 9942_CR14 publication-title: SIAM J. Comput. doi: 10.1137/0203025 – volume: 32 start-page: 562 year: 1985 ident: 9942_CR18 publication-title: J. ACM doi: 10.1145/3828.3833 – volume: 60 start-page: 368 issue: 2 year: 2011 ident: 9942_CR8 publication-title: Algorithmica doi: 10.1007/s00453-009-9348-6 – volume-title: Selfish routing and the price of anarchy year: 2005 ident: 9942_CR20 – volume: 48 start-page: 91 issue: 1 year: 2007 ident: 9942_CR19 publication-title: Algorithmica doi: 10.1007/s00453-006-0056-1 – volume: 65 start-page: 289 issue: 2 year: 2009 ident: 9942_CR1 publication-title: Games Econ. Behav. doi: 10.1016/j.geb.2008.03.005 – ident: 9942_CR9 doi: 10.1007/978-3-540-73420-8_51 – ident: 9942_CR11 doi: 10.1145/1478873.1478901 – ident: 9942_CR23 doi: 10.1007/978-3-540-92185-1_50 – ident: 9942_CR12 doi: 10.1007/978-3-540-31856-9_53 – volume: 32 start-page: 159 issue: 2 year: 2004 ident: 9942_CR3 publication-title: Oper. Res. Lett. doi: 10.1016/S0167-6377(03)00092-0 – ident: 9942_CR7 doi: 10.1145/1219944.1219949 – ident: 9942_CR2 doi: 10.1109/IPDPS.2006.1639283 – volume-title: Handbook of Approximation Algorithms and Metaheuristics, chapter 33 year: 2007 ident: 9942_CR6 – volume: 49 start-page: 236 issue: 2 year: 2002 ident: 9942_CR21 publication-title: J. ACM doi: 10.1145/506147.506153 – ident: 9942_CR22 – volume: 50 start-page: 795 issue: 6 year: 2003 ident: 9942_CR13 publication-title: J. ACM doi: 10.1145/950620.950621 – volume-title: Knapsack problems year: 2004 ident: 9942_CR16 doi: 10.1007/978-3-540-24777-7 – volume: 10 start-page: 598 year: 1999 ident: 9942_CR10 publication-title: Prod. Plan. Control doi: 10.1080/095372899232894 – volume-title: Handbook of approximation algorithms and metaheuristics, chapter 32 year: 2007 ident: 9942_CR5 |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Theory of Computation |
| Title | Parametric Packing of Selfish Items and the Subset Sum Algorithm |
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