How to Pack Your Items When You Have to Buy Your Knapsack

Antonios Antoniadis; Chien-Chung Huang; Sebastian Ott; José Verschae

doi:10.1007/978-3-642-40313-2

How to Pack Your Items When You Have to Buy Your Knapsack

Antonios Antoniadis^*, Chien-Chung Huang, Sebastian Ott, José Verschae

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

8 Citations (Scopus)

Abstract

In this paper we consider a generalization of the classical knapsack problem. While in the standard setting a fixed capacity may not be exceeded by the weight of the chosen items, we replace this hard constraint by a weight-dependent cost function. The objective is to maximize the total profit of the chosen items minus the cost induced by their total weight. We study two natural classes of cost functions, namely convex and concave functions. For the concave case, we show that the problem can be solved in polynomial time; for the convex case we present an FPTAS and a 2-approximation algorithm with the running time of O(n log n) , where n is the number of items. Before, only a 3-approximation algorithm was known.

We note that our problem with a convex cost function is a special case of maximizing a non-monotone, possibly negative submodular function.

Original language	English
Title of host publication	Mathematical Foundations of Computer Science 2013
Subtitle of host publication	38th International Symposium, MFCS 2013, Klosterneuburg, Austria, August 26-30, 2013. Proceedings
Editors	Krishnendu Chatterjee, Jirí Sgall
Place of Publication	Berlin, Heidelberg
Publisher	Springer
Pages	62-73
ISBN (Electronic)	978-3-642-40313-2
ISBN (Print)	978-3-642-40312-5
DOIs	https://doi.org/10.1007/978-3-642-40313-2
Publication status	Published - 2013
Externally published	Yes
Event	38th International Symposium on Mathematical Foundations of Computer Science, MFCS 2013 - Klosterneuburg, Austria Duration: 26 Aug 2013 → 30 Aug 2013 Conference number: 38

Publication series

Name	Lecture Notes in Computer Science
Publisher	Springer
Volume	8087
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	38th International Symposium on Mathematical Foundations of Computer Science, MFCS 2013
Abbreviated title	MFCS
Country/Territory	Austria
City	Klosterneuburg
Period	26/08/13 → 30/08/13

Access to Document

10.1007/978-3-642-40313-2

Cite this

Antoniadis, A., Huang, C.-C., Ott, S., & Verschae, J. (2013). How to Pack Your Items When You Have to Buy Your Knapsack. In K. Chatterjee, & J. Sgall (Eds.), Mathematical Foundations of Computer Science 2013: 38th International Symposium, MFCS 2013, Klosterneuburg, Austria, August 26-30, 2013. Proceedings (pp. 62-73). (Lecture Notes in Computer Science; Vol. 8087). Springer. https://doi.org/10.1007/978-3-642-40313-2

Antoniadis, Antonios ; Huang, Chien-Chung ; Ott, Sebastian et al. / How to Pack Your Items When You Have to Buy Your Knapsack. Mathematical Foundations of Computer Science 2013: 38th International Symposium, MFCS 2013, Klosterneuburg, Austria, August 26-30, 2013. Proceedings. editor / Krishnendu Chatterjee ; Jirí Sgall. Berlin, Heidelberg : Springer, 2013. pp. 62-73 (Lecture Notes in Computer Science).

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title = "How to Pack Your Items When You Have to Buy Your Knapsack",

abstract = "In this paper we consider a generalization of the classical knapsack problem. While in the standard setting a fixed capacity may not be exceeded by the weight of the chosen items, we replace this hard constraint by a weight-dependent cost function. The objective is to maximize the total profit of the chosen items minus the cost induced by their total weight. We study two natural classes of cost functions, namely convex and concave functions. For the concave case, we show that the problem can be solved in polynomial time; for the convex case we present an FPTAS and a 2-approximation algorithm with the running time of O(n log n) , where n is the number of items. Before, only a 3-approximation algorithm was known.We note that our problem with a convex cost function is a special case of maximizing a non-monotone, possibly negative submodular function.",

author = "Antonios Antoniadis and Chien-Chung Huang and Sebastian Ott and Jos{\'e} Verschae",

year = "2013",

doi = "10.1007/978-3-642-40313-2",

language = "English",

isbn = "978-3-642-40312-5",

series = "Lecture Notes in Computer Science",

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Antoniadis, A, Huang, C-C, Ott, S & Verschae, J 2013, How to Pack Your Items When You Have to Buy Your Knapsack. in K Chatterjee & J Sgall (eds), Mathematical Foundations of Computer Science 2013: 38th International Symposium, MFCS 2013, Klosterneuburg, Austria, August 26-30, 2013. Proceedings. Lecture Notes in Computer Science, vol. 8087, Springer, Berlin, Heidelberg, pp. 62-73, 38th International Symposium on Mathematical Foundations of Computer Science, MFCS 2013, Klosterneuburg, Austria, 26/08/13. https://doi.org/10.1007/978-3-642-40313-2

How to Pack Your Items When You Have to Buy Your Knapsack. / Antoniadis, Antonios; Huang, Chien-Chung; Ott, Sebastian et al.
Mathematical Foundations of Computer Science 2013: 38th International Symposium, MFCS 2013, Klosterneuburg, Austria, August 26-30, 2013. Proceedings. ed. / Krishnendu Chatterjee; Jirí Sgall. Berlin, Heidelberg: Springer, 2013. p. 62-73 (Lecture Notes in Computer Science; Vol. 8087).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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AU - Antoniadis, Antonios

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AU - Ott, Sebastian

AU - Verschae, José

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PY - 2013

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N2 - In this paper we consider a generalization of the classical knapsack problem. While in the standard setting a fixed capacity may not be exceeded by the weight of the chosen items, we replace this hard constraint by a weight-dependent cost function. The objective is to maximize the total profit of the chosen items minus the cost induced by their total weight. We study two natural classes of cost functions, namely convex and concave functions. For the concave case, we show that the problem can be solved in polynomial time; for the convex case we present an FPTAS and a 2-approximation algorithm with the running time of O(n log n) , where n is the number of items. Before, only a 3-approximation algorithm was known.We note that our problem with a convex cost function is a special case of maximizing a non-monotone, possibly negative submodular function.

AB - In this paper we consider a generalization of the classical knapsack problem. While in the standard setting a fixed capacity may not be exceeded by the weight of the chosen items, we replace this hard constraint by a weight-dependent cost function. The objective is to maximize the total profit of the chosen items minus the cost induced by their total weight. We study two natural classes of cost functions, namely convex and concave functions. For the concave case, we show that the problem can be solved in polynomial time; for the convex case we present an FPTAS and a 2-approximation algorithm with the running time of O(n log n) , where n is the number of items. Before, only a 3-approximation algorithm was known.We note that our problem with a convex cost function is a special case of maximizing a non-monotone, possibly negative submodular function.

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BT - Mathematical Foundations of Computer Science 2013

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Antoniadis A, Huang CC, Ott S, Verschae J. How to Pack Your Items When You Have to Buy Your Knapsack. In Chatterjee K, Sgall J, editors, Mathematical Foundations of Computer Science 2013: 38th International Symposium, MFCS 2013, Klosterneuburg, Austria, August 26-30, 2013. Proceedings. Berlin, Heidelberg: Springer. 2013. p. 62-73. (Lecture Notes in Computer Science). doi: 10.1007/978-3-642-40313-2

How to Pack Your Items When You Have to Buy Your Knapsack

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