Online Multistage Subset Maximization Problems
Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems : One is given a ground set N = { 1 , ⋯ , n } , a collection F ⊆ 2 N of subsets thereof such that ∅ ∈ F , and an objective (profit) function p : F → R + . The task...
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| Published in: | Algorithmica Vol. 83; no. 8; pp. 2374 - 2399 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
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01.08.2021
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| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as
subset maximization problems
: One is given a ground set
N
=
{
1
,
⋯
,
n
}
, a collection
F
⊆
2
N
of subsets thereof such that
∅
∈
F
, and an objective (profit) function
p
:
F
→
R
+
. The task is to choose a set
S
∈
F
that maximizes
p
(
S
). We consider the
multistage
version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function
p
t
(and possibly the set of feasible solutions
F
t
) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of
n
and the Hamming distance between the two characteristic vectors. We study multistage subset maximization problems in the
online
setting, that is,
p
t
(along with possibly
F
t
) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future. We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one. |
|---|---|
| AbstractList | Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems: One is given a ground set N = {1, . . . , n}, a collection F ⊆ 2N of subsets thereof such that ∅ ∈ F, and an objective (profit) function p : F → R+. The task is to choose a set S ∈ F that maximizes p(S). We consider the multistage version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function pt (and possibly the set of feasible solutions Ft) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of n and the Hamming distance between the two characteristic vectors. We study multistage subset maximization problems in the online setting, that is, pt (along with possibly Ft) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future. We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one. Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems: One is given a ground set N={1,⋯,n}, a collection F⊆2N of subsets thereof such that ∅∈F, and an objective (profit) function p:F→R+. The task is to choose a set S∈F that maximizes p(S). We consider the multistage version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function pt (and possibly the set of feasible solutions Ft) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of n and the Hamming distance between the two characteristic vectors. We study multistage subset maximization problems in the online setting, that is, pt (along with possibly Ft) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future. We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one. Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems : One is given a ground set N = { 1 , ⋯ , n } , a collection F ⊆ 2 N of subsets thereof such that ∅ ∈ F , and an objective (profit) function p : F → R + . The task is to choose a set S ∈ F that maximizes p ( S ). We consider the multistage version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function p t (and possibly the set of feasible solutions F t ) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of n and the Hamming distance between the two characteristic vectors. We study multistage subset maximization problems in the online setting, that is, p t (along with possibly F t ) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future. We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one. |
| Author | Escoffier, Bruno Bampis, Evripidis Teiller, Alexandre Schewior, Kevin |
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| ContentType | Journal Article |
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| Issue | 8 |
| Keywords | Online algorithms Multistage Optimization Multistage optimization On-line algorithms |
| Language | English |
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| References | Bampis, E., Escoffier, B., Lampis, M., Paschos, V.T.: Multistage matchings. In: Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pp. 1–13 (2018) PruhsKWoegingerGJApproximation schemes for a class of subset selection problemsTheor. Comput. Sci.20073822151156235211010.1016/j.tcs.2007.03.006 Bampis, E., Escoffier, B., Mladenovic, S.: Fair resource allocation over time. In: International Conference on Autonomous Agents and MultiAgent Systems (AAMAS), pp. 766–773 (2018) Buchbinder, N., Chen, S., Naor, J.: Competitive analysis via regularization. In: ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 436–444 (2014) GuAGuptaAKumarAThe power of deferral: maintaining a constant-competitive steiner tree onlineSIAM J. Comput.2016451128343977810.1137/140955276 Albers, S., Quedenfeld, J.: Optimal algorithms for right-sizing data centers. In: ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 363–372 (2018) Gupta, A., Talwar, K., Wieder, U.: Changing bases: Multistage optimization for matroids and matchings. In: International Colloquium on Automata, Languages, and Programming (ICALP), pp. 563–575 (2014) CohenECormodeGDuffieldNGLundCOn the tradeoff between stability and fitACM Trans. Algorithms2016131124359810610.1145/2963103 Bampis, E., Escoffier, B., Schewior, K., Teiller, A.: Online multistage subset maximization problems. In: M.A. Bender, O. Svensson, G. Herman (eds.) 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany., LIPIcs, vol. 144, pp. 11:1–11:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019). https://doi.org/10.4230/LIPIcs.ESA.2019.11 Antoniadis, A., Schewior, K.: A tight lower bound for online convex optimization with switching costs. In: Workshop on Approximation and Online Algorithms (WAOA), pp. 164–175 (2017) BuchbinderNChenSNaorJShamirOUnified algorithms for online learning and competitive analysisMath. Oper. Res.2016412612625348681010.1287/moor.2015.0742 Anthony, B.M., Gupta, A.: Infrastructure leasing problems. In: Conference on Integer Programming and Combinatorial Optimization (IPCO), pp. 424–438 (2007) AnHNorouzi-FardASvenssonODynamic facility location via exponential clocksACM Trans. Algorithms2017132120365924310.1145/2928272 MegowNSkutellaMVerschaeJWieseAThe power of recourse for online MST and TSPSIAM J. Comput.2016453859880351687210.1137/130917703 LinMWiermanAAndrewLLHThereskaEDynamic right-sizing for power-proportional data centersIEEE/ACM Trans. Netw.20132151378139110.1109/TNET.2012.2226216 NagarajanCWilliamsonDPOffline and online facility leasingDiscret. Optim.2013104361370312862310.1016/j.disopt.2013.10.001 SleatorDDTarjanREAmortized efficiency of list update and paging rulesCommun. ACM198528220220877738510.1145/2786.2793 Eisenstat, D., Mathieu, C., Schabanel, N.: Facility location in evolving metrics. In: International Colloquium on Automata, Languages, and Programming (ICALP), pp. 459–470 (2014) Joseph, V., de Veciana, G.: Jointly optimizing multi-user rate adaptation for video transport over wireless systems: Mean-fairness-variability tradeoffs. In: IEEE International Conference on Computer Communications (INFOCOM), pp. 567–575 (2012) Rottner, C.: Combinatorial aspects of the unit commitment problem. Ph.D. thesis, Sorbonne Université (2018) Bansal, N., Gupta, A., Krishnaswamy, R., Pruhs, K., Schewior, K., Stein, C.: A 2-competitive algorithm for online convex optimization with switching costs. In: Workshop on Approximation, Randomization, and Combinatorial Optimization Algorithms and Techniques (APPROX/RANDOM), pp. 96–109 (2015) Blanchard, N.K., Schabanel, N.: Dynamic sum-radii clustering. In: International Conference and Workshops on Algorithms and Computation (WALCOM), pp. 30–41 (2017) Liu, Z., Liu, I., Low, S.H., Wierman, A.: Pricing data center demand response. In: ACM International Conference on Measurement and Modeling of Computer Systems (SIGMETRICS), pp. 111–123 (2014) Olver, N., Pruhs, K., Sitters, R., Schewior, K., Stougie, L.: The itinerant list-update problem. In: Workshop on Approximation and Online Algorithms (WAOA), pp. 310–326 (2018) 834_CR23 834_CR13 834_CR10 834_CR21 N Megow (834_CR19) 2016; 45 834_CR9 834_CR18 834_CR6 N Buchbinder (834_CR11) 2016; 41 A Gu (834_CR14) 2016; 45 834_CR16 834_CR5 834_CR8 834_CR7 834_CR15 M Lin (834_CR17) 2013; 21 834_CR1 834_CR4 834_CR3 K Pruhs (834_CR22) 2007; 382 DD Sleator (834_CR24) 1985; 28 C Nagarajan (834_CR20) 2013; 10 H An (834_CR2) 2017; 13 E Cohen (834_CR12) 2016; 13 |
| References_xml | – reference: AnHNorouzi-FardASvenssonODynamic facility location via exponential clocksACM Trans. Algorithms2017132120365924310.1145/2928272 – reference: CohenECormodeGDuffieldNGLundCOn the tradeoff between stability and fitACM Trans. Algorithms2016131124359810610.1145/2963103 – reference: Bampis, E., Escoffier, B., Lampis, M., Paschos, V.T.: Multistage matchings. In: Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pp. 1–13 (2018) – reference: Blanchard, N.K., Schabanel, N.: Dynamic sum-radii clustering. In: International Conference and Workshops on Algorithms and Computation (WALCOM), pp. 30–41 (2017) – reference: Eisenstat, D., Mathieu, C., Schabanel, N.: Facility location in evolving metrics. In: International Colloquium on Automata, Languages, and Programming (ICALP), pp. 459–470 (2014) – reference: LinMWiermanAAndrewLLHThereskaEDynamic right-sizing for power-proportional data centersIEEE/ACM Trans. Netw.20132151378139110.1109/TNET.2012.2226216 – reference: MegowNSkutellaMVerschaeJWieseAThe power of recourse for online MST and TSPSIAM J. Comput.2016453859880351687210.1137/130917703 – reference: PruhsKWoegingerGJApproximation schemes for a class of subset selection problemsTheor. Comput. Sci.20073822151156235211010.1016/j.tcs.2007.03.006 – reference: Liu, Z., Liu, I., Low, S.H., Wierman, A.: Pricing data center demand response. In: ACM International Conference on Measurement and Modeling of Computer Systems (SIGMETRICS), pp. 111–123 (2014) – reference: Bansal, N., Gupta, A., Krishnaswamy, R., Pruhs, K., Schewior, K., Stein, C.: A 2-competitive algorithm for online convex optimization with switching costs. In: Workshop on Approximation, Randomization, and Combinatorial Optimization Algorithms and Techniques (APPROX/RANDOM), pp. 96–109 (2015) – reference: Gupta, A., Talwar, K., Wieder, U.: Changing bases: Multistage optimization for matroids and matchings. In: International Colloquium on Automata, Languages, and Programming (ICALP), pp. 563–575 (2014) – reference: Buchbinder, N., Chen, S., Naor, J.: Competitive analysis via regularization. In: ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 436–444 (2014) – reference: Olver, N., Pruhs, K., Sitters, R., Schewior, K., Stougie, L.: The itinerant list-update problem. In: Workshop on Approximation and Online Algorithms (WAOA), pp. 310–326 (2018) – reference: Bampis, E., Escoffier, B., Schewior, K., Teiller, A.: Online multistage subset maximization problems. In: M.A. Bender, O. Svensson, G. Herman (eds.) 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany., LIPIcs, vol. 144, pp. 11:1–11:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019). https://doi.org/10.4230/LIPIcs.ESA.2019.11 – reference: Albers, S., Quedenfeld, J.: Optimal algorithms for right-sizing data centers. In: ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 363–372 (2018) – reference: NagarajanCWilliamsonDPOffline and online facility leasingDiscret. Optim.2013104361370312862310.1016/j.disopt.2013.10.001 – reference: BuchbinderNChenSNaorJShamirOUnified algorithms for online learning and competitive analysisMath. Oper. Res.2016412612625348681010.1287/moor.2015.0742 – reference: GuAGuptaAKumarAThe power of deferral: maintaining a constant-competitive steiner tree onlineSIAM J. Comput.2016451128343977810.1137/140955276 – reference: Rottner, C.: Combinatorial aspects of the unit commitment problem. Ph.D. thesis, Sorbonne Université (2018) – reference: SleatorDDTarjanREAmortized efficiency of list update and paging rulesCommun. ACM198528220220877738510.1145/2786.2793 – reference: Antoniadis, A., Schewior, K.: A tight lower bound for online convex optimization with switching costs. In: Workshop on Approximation and Online Algorithms (WAOA), pp. 164–175 (2017) – reference: Joseph, V., de Veciana, G.: Jointly optimizing multi-user rate adaptation for video transport over wireless systems: Mean-fairness-variability tradeoffs. In: IEEE International Conference on Computer Communications (INFOCOM), pp. 567–575 (2012) – reference: Anthony, B.M., Gupta, A.: Infrastructure leasing problems. In: Conference on Integer Programming and Combinatorial Optimization (IPCO), pp. 424–438 (2007) – reference: Bampis, E., Escoffier, B., Mladenovic, S.: Fair resource allocation over time. In: International Conference on Autonomous Agents and MultiAgent Systems (AAMAS), pp. 766–773 (2018) – ident: 834_CR6 – ident: 834_CR5 – ident: 834_CR21 doi: 10.1007/978-3-030-04693-4_19 – volume: 382 start-page: 151 issue: 2 year: 2007 ident: 834_CR22 publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2007.03.006 – ident: 834_CR8 – ident: 834_CR10 doi: 10.1137/1.9781611973402.32 – volume: 45 start-page: 1 issue: 1 year: 2016 ident: 834_CR14 publication-title: SIAM J. Comput. doi: 10.1137/140955276 – ident: 834_CR3 doi: 10.1007/978-3-540-72792-7_32 – ident: 834_CR23 – volume: 21 start-page: 1378 issue: 5 year: 2013 ident: 834_CR17 publication-title: IEEE/ACM Trans. Netw. doi: 10.1109/TNET.2012.2226216 – ident: 834_CR4 doi: 10.1007/978-3-319-89441-6_13 – ident: 834_CR15 doi: 10.1007/978-3-662-43948-7_47 – ident: 834_CR9 doi: 10.1007/978-3-319-53925-6_3 – volume: 13 start-page: 1 issue: 1 year: 2016 ident: 834_CR12 publication-title: ACM Trans. Algorithms doi: 10.1145/2963103 – volume: 45 start-page: 859 issue: 3 year: 2016 ident: 834_CR19 publication-title: SIAM J. Comput. doi: 10.1137/130917703 – volume: 10 start-page: 361 issue: 4 year: 2013 ident: 834_CR20 publication-title: Discret. 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| Snippet | Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as
subset maximization problems
: One is given a ground... Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems: One is given a ground... |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Combinatorial analysis Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Matching Mathematics of Computing Maximization Optimization Similarity Similarity measures Theory of Computation Upper bounds |
| Title | Online Multistage Subset Maximization Problems |
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