There is no EPTAS for two-dimensional knapsack

• Published in: Information processing letters Vol. 110; no. 16; pp. 707 - 710
• Main Authors: Kulik, Ariel, Shachnai, Hadas
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 31.07.2010; Elsevier; Elsevier Sequoia S.A
• Subjects: 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
• DOI: 10.1016/j.ipl.2010.05.031

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.