The symmetric quadratic knapsack problem: approximation and scheduling applications

Abstract

This paper reviews two problems of Boolean non-linear programming: the Symmetric Quadratic Knapsack Problem and the Half-Product Problem. The problems are related since they have a similar quadratic non-separable objective function. For these problems, we focus on the development of fully polynomial-time approximation schemes, especially of those with strongly polynomial time, and on their applications to various scheduling problems.Keywords Quadratic knapsack · Half-product · Single machine scheduling · FPTAS MSC classification (2000) 90-02 · 90C09 · 90C20 · 90C59 · 90B35

Approximation schemes for scheduling on a single machine subject to cumulative deterioration and maintenance

“…In particular, for the problem with a constant MP, we show that the variable part of the objective function is related to the Subset-sum problem, a version of the linear knapsack problem; see Kellerer et al (2004). On the other hand, if the duration of the MP depends linearly on its start time, we establish its link to a problem of quadratic Boolean programming, known as the Half-product problem, see Badics and Boros (1998) and Kellerer and Strusevich (2011). Although each of the mentioned Boolean programming problems admits an FPTAS, a challenge remains to adapt such an FPTAS to handling the original objective function.…”

“…It is known that an FPTAS for minimizing the function H(x) does not necessarily behave as an FPTAS for minimizing the function F (x). This is due to the fact the optimal value of H(x) is negative; see Erel and Ghosh (2008) and Kellerer and Strusevich (2011) for discussion and examples. Suppose that a lower bound LB and an upper bound U B on the optimal value of the function F (x) are available, i.e., LB ≤ F (x * ) ≤ U B.…”

Fast approximation schemes for Boolean programming and scheduling problems related to positive convex Half-Product

“…, n, as proved by Badics and Boros (1998). It has numerous applications, mainly to machine scheduling; see Erel and Ghosh (2008) and Kellerer and Strusevich (2012) for reviews. Notice that in those applications a scheduling objective function usually is written in the form…”

“…While for many problems of this range purpose-built approximation schemes have been developed, a general framework has been identified based on reformulation of the original scheduling problems in terms of minimization problems of quadratic Boolean programming. The Half-Product Problem and the closely related Symmetric Quadratic Knapsack Problem appear to be among the most suitable models, see recent reviews by Kacem et al (2011) and Kellerer and Strusevich (2012). This paper studies a version of the Half-Product Problem and its modification with the knapsack constraint, establishes conditions under which the problems admit fast fully polynomial-time approximation schemes, describes the relevant algorithms and discusses their scheduling applications.…”

“…We use the term "symmetric" because both the quadratic and the linear parts of the objective function are separated into two terms, one depending on the variables x j , and the other depending on the variables (1 − x j ). A comprehensive review of the results on Problem SQK and its scheduling applications is given by Kellerer and Strusevich (2012). Table 1 summarizes the notation introduced above for all Boolean programming problems under consideration.…”