Faster Minimization of Tardy Processing Time on a Single Machine

Abstract

This paper is concerned with the 1|| pjUj problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also a very important problem from a theoretical point of view as it generalizes the Subset Sum problem and is closely related to the 0/1-Knapsack problem. The problem is well-known to be NP-hard, but only in a weak sense, meaning it admits pseudo-polynomial time algorithms. The fastest known pseudo-polynomial time algorithm for the problem is the famous Lawler and Moore algorithm which runs in O(P • n) time, where P is the total processing time of all n jobs in the input. This algorithm has been developed in the late 60s, and has yet to be improved to date.In this paper we develop two new algorithms for 1|| pjUj, each improving on Lawler and Moore's algorithm in a different scenario:Our first algorithm runs in Õ(P 7/4 ) time 1 , and outperforms Lawler and Moore's algorithm in instances where n = ω(P 3/4 ).Our second algorithm runs in Õ(min{P • D # , P + D}) time, where D # is the number of different due dates in the instance, and D is the sum of all different due dates. This algorithm improves on Lawler and Moore's algorithm when n = ω(D # ) or n = ω(D/P ). Further, it extends the known Õ(P ) algorithm for the single due date special case of 1|| pjUj in a natural way.Both algorithms rely on basic primitive operations between sets of integers and vectors of integers for the speedup in their running times. The second algorithm relies on fast polynomial multiplication as its main engine, while for the first algorithm we define a new "skewed" version of (max, min)convolution which is interesting in its own right. ACM Subject Classification Theory of computation → Design and analysis of algorithmsKeywords and phrases Weighted number of tardy jobs, sumsets, convolutions1 Throughout the paper we use Õ(•) to suppress logarithmic factors.

Quick Minimization of Tardy Processing Time on a Single Machine

“…Our notations follow the notations in [2]. The input to the MTPT problem is a set of n jobs J = {J 1 , J 2 , .…”

“…Next, we recall the definition of convolutions and describe the techniques developed in [2] and used by our algorithm. Given two vectors A and B of dimension n + 1 and two binary operations • and •, the (•, •)-convolution applied on A and B results in a 2n + 1 dimensional vector C, defined as:…”

“…1 The notation Õ(•) hides all logarithmic factors. Bringmann et al [2] apply an equivalent form of (max, min)-skewed-convolution defined as…”

“…Bringmann et al [2] introduced a new convolution variant called a (max, min)-skewed-convolution. They gave an algorithm for MTPT that uses (max, min)-skewed-convolutions, and proved that up to logarithmic factors, the running time of this algorithm is equivalent to the time it takes to compute a (max, min)-skewed-convolution of integers that sum to O(P).…”

Single Machine Weighted Number of Tardy Jobs Minimization With Small Weights

“…The problem is known to be polynomial-time solvable when either the number of different processing times p # , or the number of different weights w # , is bounded by a constant, and fixed parameter tractable when parameterized by either p # + w # , p # + d # , or w # + d # [11]. Finally, the special case where w j = p j for all jobs j, the 1 || p j U j problem, was recently shown to be solvable in O(p 7/4 ), O(pd # ), and O(p + d) time [5].…”