A new fully polynomial time approximation scheme for the interval subset sum problem

The interval subset sum problem (ISSP) is a generalization of the well-known subset sum problem. Given a set of intervals [ a i , 1 , a i , 2 ] i = 1 n and a target integer T , the ISSP is to find a set of integers, at most one from each interval, such that their sum best approximates the target T b...

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Veröffentlicht in:	Journal of global optimization Jg. 68; H. 4; S. 749 - 775
Hauptverfasser:	Diao, Rui, Liu, Ya-Feng, Dai, Yu-Hong
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.08.2017 Springer Springer Nature B.V
Schlagworte:	Approximation Comparative analysis Complexity Computer Science Computer simulation Integers Knapsack problem Mathematical analysis Mathematics Mathematics and Statistics Numerical analysis Operations Research/Decision Theory Optimization Production scheduling Real Functions 68Q25 Solution structure Fully polynomial time approximation scheme 90C59 Interval subset sum problem Computational complexity Worst-case performance
ISSN:	0925-5001, 1573-2916
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Abstract	The interval subset sum problem (ISSP) is a generalization of the well-known subset sum problem. Given a set of intervals [ a i , 1 , a i , 2 ] i = 1 n and a target integer T , the ISSP is to find a set of integers, at most one from each interval, such that their sum best approximates the target T but cannot exceed it. In this paper, we first study the computational complexity of the ISSP. We show that the ISSP is relatively easy to solve compared to the 0–1 knapsack problem. We also identify several subclasses of the ISSP which are polynomial time solvable (with high probability), albeit the problem is generally NP-hard. Then, we propose a new fully polynomial time approximation scheme for solving the general ISSP problem. The time and space complexities of the proposed scheme are O n max 1 / ϵ , log n and O n + 1 / ϵ , respectively, where ϵ is the relative approximation error. To the best of our knowledge, the proposed scheme has almost the same time complexity but a significantly lower space complexity compared to the best known scheme. Both the correctness and efficiency of the proposed scheme are validated by numerical simulations. In particular, the proposed scheme successfully solves ISSP instances with n = 100 , 000 and ϵ = 0.1 % within 1 s.
AbstractList	(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) The interval subset sum problem (ISSP) is a generalization of the well-known subset sum problem. Given a set of intervals ... and a target integer T, the ISSP is to find a set of integers, at most one from each interval, such that their sum best approximates the target T but cannot exceed it. In this paper, we first study the computational complexity of the ISSP. We show that the ISSP is relatively easy to solve compared to the 0-1 knapsack problem. We also identify several subclasses of the ISSP which are polynomial time solvable (with high probability), albeit the problem is generally NP-hard. Then, we propose a new fully polynomial time approximation scheme for solving the general ISSP problem. The time and space complexities of the proposed scheme are ... and ... respectively, where ... is the relative approximation error. To the best of our knowledge, the proposed scheme has almost the same time complexity but a significantly lower space complexity compared to the best known scheme. Both the correctness and efficiency of the proposed scheme are validated by numerical simulations. In particular, the proposed scheme successfully solves ISSP instances with ... and ... within 1 s. The interval subset sum problem (ISSP) is a generalization of the well-known subset sum problem. Given a set of intervals [Formula omitted] and a target integer T, the ISSP is to find a set of integers, at most one from each interval, such that their sum best approximates the target T but cannot exceed it. In this paper, we first study the computational complexity of the ISSP. We show that the ISSP is relatively easy to solve compared to the 0-1 knapsack problem. We also identify several subclasses of the ISSP which are polynomial time solvable (with high probability), albeit the problem is generally NP-hard. Then, we propose a new fully polynomial time approximation scheme for solving the general ISSP problem. The time and space complexities of the proposed scheme are [Formula omitted] and [Formula omitted] respectively, where [Formula omitted] is the relative approximation error. To the best of our knowledge, the proposed scheme has almost the same time complexity but a significantly lower space complexity compared to the best known scheme. Both the correctness and efficiency of the proposed scheme are validated by numerical simulations. In particular, the proposed scheme successfully solves ISSP instances with [Formula omitted] and [Formula omitted] within 1 s. The interval subset sum problem (ISSP) is a generalization of the well-known subset sum problem. Given a set of intervals [ a i , 1 , a i , 2 ] i = 1 n and a target integer T , the ISSP is to find a set of integers, at most one from each interval, such that their sum best approximates the target T but cannot exceed it. In this paper, we first study the computational complexity of the ISSP. We show that the ISSP is relatively easy to solve compared to the 0–1 knapsack problem. We also identify several subclasses of the ISSP which are polynomial time solvable (with high probability), albeit the problem is generally NP-hard. Then, we propose a new fully polynomial time approximation scheme for solving the general ISSP problem. The time and space complexities of the proposed scheme are O n max 1 / ϵ , log n and O n + 1 / ϵ , respectively, where ϵ is the relative approximation error. To the best of our knowledge, the proposed scheme has almost the same time complexity but a significantly lower space complexity compared to the best known scheme. Both the correctness and efficiency of the proposed scheme are validated by numerical simulations. In particular, the proposed scheme successfully solves ISSP instances with n = 100 , 000 and ϵ = 0.1 % within 1 s.
Audience	Academic
Author	Diao, Rui Dai, Yu-Hong Liu, Ya-Feng
Author_xml	– sequence: 1 givenname: Rui surname: Diao fullname: Diao, Rui organization: State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences – sequence: 2 givenname: Ya-Feng surname: Liu fullname: Liu, Ya-Feng email: yafliu@lsec.cc.ac.cn organization: State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences – sequence: 3 givenname: Yu-Hong surname: Dai fullname: Dai, Yu-Hong organization: State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
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Cites_doi	10.1109/TPWRS.2006.876672 10.1145/321906.321909 10.1016/S0022-0000(03)00006-0 10.1007/11533719_62 10.1007/s40305-013-0004-0 10.1023/A:1009813105532 10.1145/2455.2461 10.1006/jagm.1999.1034 10.1007/978-1-4684-4730-9_2 10.1007/BF01201999 10.1287/moor.4.4.339 10.1016/0377-2217(81)90175-2 10.1287/opre.28.6.1402
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References_xml	– reference: Brickell, E.F.: Solving low density knapsacks. In: Chaum, D. (ed.) Advances in Cryptology, pp. 25–37. Springer (1984) – reference: MichaelRGJohnsonDSComputers and Intractability: A Guide to the Theory of NP-Completeness1979San FranciscoW. H. Freeman & Co.0411.68039 – reference: ChvátalVHard knapsack problemsOper. Res.19802861402141160996710.1287/opre.28.6.14020447.90063 – reference: KellererHPferschyUA new fully polynomial time approximation scheme for the knapsack problemJ. Comb. Optim.1999315971170246110.1023/A:10098131055320957.90112 – reference: LagariasJCOdlyzkoAMSolving low-density subset sum problemsJ. ACM198532122924683234110.1145/2455.24610632.94007 – reference: LawlerELFast approximation algorithms for knapsack problemsMath. Oper. Res.19794433935654912210.1287/moor.4.4.3390425.90064 – reference: Kothari, A., Suri, S., Zhou, Y.H.: Interval subset sum and uniform-price auction clearing. In: Wang, L. (ed.) Computing and Combinatorics, pp. 608–620. Springer (2005) – reference: SunXLZhengXJLiDRecent advances in mathematical programming with semi-continuous variables and cardinality constraintJ. Oper. Res. Soc. China201311557710.1007/s40305-013-0004-01277.90001 – reference: CosterMJJouxALaMacchiaBAOdlyzkoAMSchnorrCPSternJImproved low-density subset sum algorithmsComput. Complex.199222111128119082510.1007/BF012019990768.11049 – reference: CarriónMArroyoJMA computationally efficient mixed-integer linear formulation for the thermal unit commitment problemIEEE Trans. Power Syst.20062131371137810.1109/TPWRS.2006.876672 – reference: KellererHMansiniRPferschyUSperanzaMGAn efficient fully polynomial approximation scheme for the subset-sum problemJ. Comput. Syst. Sci.2003662349370197601910.1016/S0022-0000(03)00006-01045.68157 – reference: MagazineMJOguzOA fully polynomial approximation algorithm for the 0–1 knapsack problemEur. J. Oper. Res.19818327027363739910.1016/0377-2217(81)90175-20473.90056 – reference: PisingerDLinear time algorithms for knapsack problems with bounded weightsJ. Algorithms1999331114171269010.1006/jagm.1999.10340951.90047 – reference: IbarraOHKimCEFast approximation algorithms for the knapsack and sum of subset problemsJ. ACM197522446346837846310.1145/321906.3219090345.90049 – volume: 21 start-page: 1371 issue: 3 year: 2006 ident: 514_CR2 publication-title: IEEE Trans. Power Syst. doi: 10.1109/TPWRS.2006.876672 – volume: 22 start-page: 463 issue: 4 year: 1975 ident: 514_CR5 publication-title: J. ACM doi: 10.1145/321906.321909 – volume: 66 start-page: 349 issue: 2 year: 2003 ident: 514_CR6 publication-title: J. Comput. Syst. Sci. doi: 10.1016/S0022-0000(03)00006-0 – ident: 514_CR8 doi: 10.1007/11533719_62 – volume: 1 start-page: 55 issue: 1 year: 2013 ident: 514_CR14 publication-title: J. Oper. Res. Soc. China doi: 10.1007/s40305-013-0004-0 – volume: 3 start-page: 59 issue: 1 year: 1999 ident: 514_CR7 publication-title: J. Comb. Optim. doi: 10.1023/A:1009813105532 – volume: 32 start-page: 229 issue: 1 year: 1985 ident: 514_CR9 publication-title: J. ACM doi: 10.1145/2455.2461 – volume: 33 start-page: 1 issue: 1 year: 1999 ident: 514_CR13 publication-title: J. Algorithms doi: 10.1006/jagm.1999.1034 – ident: 514_CR1 doi: 10.1007/978-1-4684-4730-9_2 – volume: 2 start-page: 111 issue: 2 year: 1992 ident: 514_CR4 publication-title: Comput. Complex. doi: 10.1007/BF01201999 – volume: 4 start-page: 339 issue: 4 year: 1979 ident: 514_CR10 publication-title: Math. Oper. Res. doi: 10.1287/moor.4.4.339 – volume: 8 start-page: 270 issue: 3 year: 1981 ident: 514_CR11 publication-title: Eur. J. Oper. Res. doi: 10.1016/0377-2217(81)90175-2 – volume: 28 start-page: 1402 issue: 6 year: 1980 ident: 514_CR3 publication-title: Oper. Res. doi: 10.1287/opre.28.6.1402 – volume-title: Computers and Intractability: A Guide to the Theory of NP-Completeness year: 1979 ident: 514_CR12
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Snippet	The interval subset sum problem (ISSP) is a generalization of the well-known subset sum problem. Given a set of intervals [ a i , 1 , a i , 2 ] i = 1 n and a... The interval subset sum problem (ISSP) is a generalization of the well-known subset sum problem. Given a set of intervals [Formula omitted] and a target... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) The interval subset sum problem (ISSP) is a generalization of the well-known subset...
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SubjectTerms	Approximation Comparative analysis Complexity Computer Science Computer simulation Integers Knapsack problem Mathematical analysis Mathematics Mathematics and Statistics Numerical analysis Operations Research/Decision Theory Optimization Production scheduling Real Functions
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Title	A new fully polynomial time approximation scheme for the interval subset sum problem
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Volume	68
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