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

Abstract

We address a version of the Half-Product Problem and its restricted variant with a linear knapsack constraint. For these minimization problems of Boolean programming, we focus on the development of fully polynomial-time approximation schemes with running times that depend quadratically on the number of variables. Applications to various single machine scheduling problems are reported: minimizing the total weighted flow time with controllable processing times, minimizing the makespan with controllable release dates, minimizing the total weighted flow time for two models of scheduling with rejection.

A fast FPTAS for single machine scheduling problem of minimizing total weighted earliness and tardiness about a large common due date

“…In order to find the upper and lower bounds on the value of the objective function, we solve the continuous relaxation of the problem to get a lower bound followed by an appropriate rounding of the obtained fractional solution to get an upper bound. A similar technique has been used in [9,11] by Kellerer and Strusevich, where several scheduling problems have been reduced to a Problem HPAdd with a convex objective function F (x), the lower bound F LB has been found by solving the continuous relaxation and the upper bound F U B has been derived by an appropriate rounding of the fractional solution.…”