The Complexity Landscape of Resource-Constrained Scheduling

Abstract

<jats:p>The Resource-Constrained Project Scheduling Problem (RCPSP) and its extension via activity modes (MRCPSP) are well-established scheduling frameworks that have found numerous applications in a broad range of settings related to artificial intelligence. Unsurprisingly, the problem of finding a suitable schedule in these frameworks is known to be NP-complete; however, aside from a few results for special cases, we have lacked an in-depth and comprehensive understanding of the complexity of the problems from the viewpoint of natural restrictions of the considered instances. In the first part of our paper, we develop new algorithms and give hardness-proofs in order to obtain a detailed complexity map of (M)RCPSP that settles the complexity of all 1024 considered variants of the problem defined in terms of explicit restrictions of natural parameters of instances. In the second part, we turn to implicit structural restrictions defined in terms of the complexity of interactions between individual activities. In particular, we show that if the treewidth of a graph which captures such interactions is bounded by a constant, then we can solve MRCPSP in polynomial time.</jats:p>

Equitable Scheduling on a Single Machine

“…. , i (d) 9 and then the jobs of all remaining clients in an arbitrary order. Variable Days: For variable x j we create Day j + 2 with a precedence DAG that consists of two directed paths.…”

“…Parameterized complexity studies of NP-hard scheduling problems gained more and more interest over the recent years[4,9,12,22]; we contribute to this field with several of our results.…”

One-Machine Scheduling with Time-Dependent Capacity via Efficient Memetic Algorithms

“…This way, the schedule computed depends on the sequence of choices made. For example, considering the problem instance in Example 1, the sequence of choices π 1 = (3,12,10,7,1,9,11,4,8,6,2,5) would result in the schedule shown in Figure 1. The sequence π 2 = (3, 10, 12, 7, 1, 11, 9, 4, 8, 6, 2, 5) would lead the schedule builder to compute the same schedule, so this mapping is many-to-one.…”

“…This results in a feasible left-shifted schedule. For example, considering the problem instance in Example 1, the chromosome (3,12,10,7,1,9,11,4,8,6,2,5) would lead to the schedule shown in Figure 1.…”

“…Over the last few decades, scheduling problems have become ubiquitous in a growing number of domains, including manufacturing, transportation or cloud computing, among others [1,2]. These problems often exhibit a high computational complexity [3][4][5][6], what makes them an interesting subject of study to several scientific disciplines, as artificial intelligence, operations research or applied mathematics. As a consequence, numerous solving methods, both exact and approximate, have been proposed in the literature, capable of solving increasingly challenging problems.…”