The Lazy Bureaucrat Problem with Common Arrivals and Deadlines: Approximation and Mechanism Design

Abstract

Abstract. We study the Lazy Bureaucrat scheduling problem (Arkin, Bender, Mitchell and Skiena [1]) in the case of common arrivals and deadlines. In this case the goal is to select a subset of given jobs in such a way that the total processing time is minimized and no other job can fit into the schedule. Our contribution comprises a linear time 4/3-approximation algorithm and an FPTAS, which respectively improve on a linear time 2-approximation algorithm and a PTAS given for the more general case of common deadlines [2,3]. We then consider a selfish perspective, in which jobs are submitted by players who may falsely report larger processing times, and show a tight upper bound of 2 on the approximation ratio of strategyproof mechanisms, even randomized ones. We conclude by introducing a maximization version of the problem and a dedicated greedy algorithm.

The Lazy Matroid Problem

“…On free matroids, Lazy Greedy coincides with the shortest job first scheduling policy introduced in [8] for the common deadline case of the Lazy Bureaucrat Problem and provides, in the worst case, a 2-approximation [8]. As proved in [2], a slight modification of this greedy algorithm gives a 4/3approximation for Lazy Bureaucrat Problem in linear time.…”

“…She has a certain budget that she can spend on these projects and she wants to select projects in such a way that as much money as possible are saved (remain unused), yet not enough for any left-out project. This is in fact a 'reincarnation' of the Lazy Bureaucrat Problem [1][2][3][4][5] in which a lazy worker wants to select a set of tasks of minimum total duration in such a way that his remaining working time does not suffice to add any task. Assume further that the Minister has to deal with additional constraints, e.g.…”

“…To the best of our knowledge, this is the first study on the matroidal version of the Lazy Bureaucrat scheduling problem; the latter was defined by Arkin, Bender, Mitchell and Skiena [6,1] under various optimization objectives. In fact, Lazy Matroid is a generalization of Lazy Bureaucrat with common arrivals and deadlines, which was shown to admit an FPTAS in [2]; note that in the common arrivals case the two most studied objectives, namely makespan and time-spent coincide. The (weak) NP-hardness of this case was shown by Gai and Zhang [7,3].…”

Subset sum problems with digraph constraints

“…Concerning the Maximal version of SUBSET SUM, this problem is called the LAZY BUREAU-CRAT PROBLEM with common arrivals and deadlines in the literature [15,18] where the problem has been proved NP-hard and approximable with a FPTAS. This latter problem has also several generalizations known as the LAZY BUREAUCRAT SCHEDULING PROBLEM [2,14,15] and the LAZY MATROID PROBLEM [17].…”

“…To the best of our knowledge, these problems are new, except SSG which is a special case of the PARTIALLY-ORDERED KNAPSACK problem (also known as the PRECEDENCE-CONSTRAINED KNAPSACK PROBLEM and it will define later) [22,23,19]. MAXIMAL SSG and MAXIMAL SSGW generalize the LAZY BUREAUCRAT PROBLEM with common deadlines and arrivals [2,14,15,18] representing the maximal version of SS. We aim at extending this problem with (weak) digraph constraints on digraphs.…”