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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Shrnutí: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.
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ISSN:0020-0190
1872-6119
DOI:10.1016/j.ipl.2010.05.031