There is no EPTAS for two-dimensional knapsack

In the d-dimensional ( vector) knapsack problem given is a set of items, each having a d-dimensional size vector and a profit, and a d-dimensional bin. The goal is to select a subset of the items of maximum total profit such that the sum of all vectors is bounded by the bin capacity in each dimensio...

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Vydáno v:	Information processing letters Ročník 110; číslo 16; s. 707 - 710
Hlavní autoři:	Kulik, Ariel, Shachnai, Hadas
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier B.V 31.07.2010 Elsevier Elsevier Sequoia S.A
Témata:	Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Approximation Approximations and expansions Computer science; control theory; systems Efficient polynomial-time approximation schemes Exact sciences and technology Knapsack problem Mail order Mathematical analysis Mathematics Miscellaneous Parameterized complexity Probabilistic analysis Problem solving Running Sciences and techniques of general use Studies Theoretical computing Theory of computation Two dimensional Two-dimensional knapsack Vectors (mathematics) Theory of computation Efficient polynomial-time approximation schemes Two-dimensional knapsack Parameterized complexity Computer theory Polynomial approximation Approximation algorithm Complexity Knapsack problem Polynomial time Maximum Information processing Two-dimensional calculations Vector Algorithm analysis
ISSN:	0020-0190, 1872-6119
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Abstract	In the d-dimensional ( vector) knapsack problem given is a set of items, each having a d-dimensional size vector and a profit, and a d-dimensional bin. The goal is to select a subset of the items of maximum total profit such that the sum of all vectors is bounded by the bin capacity in each dimension. It is well known that, unless P = NP , there is no fully polynomial-time approximation scheme for d-dimensional knapsack, already for d = 2 . The best known result is a polynomial-time approximation scheme ( PTAS) due to Frieze and Clarke [A.M. Frieze, M. Clarke, Approximation algorithms for the m-dimensional 0–1 knapsack problem: worst-case and probabilistic analyses, European J. Operat. Res. 15 (1) (1984) 100–109] for the case where d ⩾ 2 is some fixed constant. A fundamental open question is whether the problem admits an efficient PTAS ( EPTAS). In this paper we resolve this question by showing that there is no EPTAS for d-dimensional knapsack, already for d = 2 , unless W [ 1 ] = FPT . Furthermore, we show that unless all problems in SNP are solvable in sub-exponential time, there is no approximation scheme for two-dimensional knapsack whose running time is f ( 1 / ε ) \| I \| o ( 1 / ε ) , for any function f. Together, the two results suggest that a significant improvement over the running time of the scheme of Frieze and Clarke is unlikely to exist.
AbstractList	In the d-dimensional ( vector) knapsack problem given is a set of items, each having a d-dimensional size vector and a profit, and a d-dimensional bin. The goal is to select a subset of the items of maximum total profit such that the sum of all vectors is bounded by the bin capacity in each dimension. It is well known that, unless P = NP , there is no fully polynomial-time approximation scheme for d-dimensional knapsack, already for d = 2 . The best known result is a polynomial-time approximation scheme ( PTAS) due to Frieze and Clarke [A.M. Frieze, M. Clarke, Approximation algorithms for the m-dimensional 0–1 knapsack problem: worst-case and probabilistic analyses, European J. Operat. Res. 15 (1) (1984) 100–109] for the case where d ⩾ 2 is some fixed constant. A fundamental open question is whether the problem admits an efficient PTAS ( EPTAS). In this paper we resolve this question by showing that there is no EPTAS for d-dimensional knapsack, already for d = 2 , unless W [ 1 ] = FPT . Furthermore, we show that unless all problems in SNP are solvable in sub-exponential time, there is no approximation scheme for two-dimensional knapsack whose running time is f ( 1 / ε ) \| I \| o ( 1 / ε ) , for any function f. Together, the two results suggest that a significant improvement over the running time of the scheme of Frieze and Clarke is unlikely to exist. In the d-dimensional (vector) knapsack problem given is a set of items, each having a d-dimensional size vector and a profit, and a d-dimensional bin. The goal is to select a subset of the items of maximum total profit such that the sum of all vectors is bounded by the bin capacity in each dimension. It is well known that, unless P=NP, there is no fully polynomial-time approximation scheme for d-dimensional knapsack, already for d=2. The best known result is a polynomial-time approximation scheme (PTAS) due to Frieze and Clarke [A.M. Frieze, M. Clarke, Approximation algorithms for the m-dimensional 0-1 knapsack problem: worst-case and probabilistic analyses, European J. Operat. Res. 15 (1) (1984) 100-109] for the case where d≥2 is some fixed constant. A fundamental open question is whether the problem admits an efficient PTAS (EPTAS). In this paper the authors resolve this question by showing that there is no EPTAS for d-dimensional knapsack, already for d=2, unless W[1]=FPT. Furthermore, they show that unless all problems in SNP are solvable in sub-exponential time, there is no approximation scheme for two-dimensional knapsack whose running time is ..., for any function f. Together, the two results suggest that a significant improvement over the running time of the scheme of Frieze and Clarke is unlikely to exist. (ProQuest: ... denotes formulae/symbols omitted.) In the d-dimensional (vector) knapsack problem given is a set of items, each having a d-dimensional size vector and a profit, and a d-dimensional bin. The goal is to select a subset of the items of maximum total profit such that the sum of all vectors is bounded by the bin capacity in each dimension. It is well known that, unless P=NP, there is no fully polynomial-time approximation scheme for d-dimensional knapsack, already for d=2. The best known result is a polynomial-time approximation scheme (PTAS) due to Frieze and Clarke [A.M. Frieze, M. Clarke, Approximation algorithms for the m-dimensional 0-1 knapsack problem: worst-case and probabilistic analyses, European J. Operat. Res. 15 (1) (1984) 100-109] for the case where d[greater-or-equal, slanted]2 is some fixed constant. A fundamental open question is whether the problem admits an efficient PTAS (EPTAS). In this paper we resolve this question by showing that there is no EPTAS for d-dimensional knapsack, already for d=2, unless W[1]=FPT. Furthermore, we show that unless all problems in SNP are solvable in sub-exponential time, there is no approximation scheme for two-dimensional knapsack whose running time is [MathML equation], for any function f. Together, the two results suggest that a significant improvement over the running time of the scheme of Frieze and Clarke is unlikely to exist.
Author	Kulik, Ariel Shachnai, Hadas
Author_xml	– sequence: 1 givenname: Ariel surname: Kulik fullname: Kulik, Ariel email: kulik@cs.technion.ac.il – sequence: 2 givenname: Hadas surname: Shachnai fullname: Shachnai, Hadas email: hadas@cs.technion.ac.il
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Cites_doi	10.1287/moor.9.2.244 10.1023/A:1009813105532 10.1016/S0377-2217(99)00261-1 10.1016/0304-3975(94)00097-3 10.1007/978-1-4612-0515-9 10.1145/1008861.1008867 10.1016/0377-2217(84)90053-5 10.1016/0022-0000(91)90023-X 10.1023/B:JOCO.0000021934.29833.6b
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Snippet	In the d-dimensional ( vector) knapsack problem given is a set of items, each having a d-dimensional size vector and a profit, and a d-dimensional bin. The... In the d-dimensional (vector) knapsack problem given is a set of items, each having a d-dimensional size vector and a profit, and a d-dimensional bin. The goal...
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SubjectTerms	Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Approximation Approximations and expansions Computer science; control theory; systems Efficient polynomial-time approximation schemes Exact sciences and technology Knapsack problem Mail order Mathematical analysis Mathematics Miscellaneous Parameterized complexity Probabilistic analysis Problem solving Running Sciences and techniques of general use Studies Theoretical computing Theory of computation Two dimensional Two-dimensional knapsack Vectors (mathematics)
Title	There is no EPTAS for two-dimensional knapsack
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