A∗-based construction of decision diagrams for a prize-collecting scheduling problem

• Published in: Computers & operations research Vol. 126; pp. 105125 - 21
• Main Authors: Horn, Matthias, Maschler, Johannes, Raidl, Günther R., Rönnberg, Elina
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01.02.2021; Pergamon Press Inc
• Subjects: [formula omitted] search; A search; Combinatorial analysis; Decision diagrams; Heuristic methods; Integer programming; Linear programming; Lower bounds; Mixed integer; Nodes; Operations research; Optimization; Scheduling; Sequencing; Substructures; A search; Decision diagrams; Scheduling; Sequencing
• ISSN: 0305-0548; 1873-765X; 1873-765X; 0305-0548
• DOI: 10.1016/j.cor.2020.105125

•A novel A*-based compilation scheme for relaxed multi-valued decision diagrams.•Considering a prize-collecting scheduling problem.•Obtained relaxed decision diagrams provide stronger bounds than decision diagrams compiled by standard methods.•Exploiting structural information of relaxed decision diagrams to construct restricted decision diagrams.•New state of the art results obtained for considered scheduling problem. Decision diagrams (DDs) have proven to be useful tools in combinatorial optimization. Relaxed DDs represent discrete relaxations of problems, can encode essential structural information in a compact form, and may yield strong dual bounds. We propose a novel construction scheme for relaxed multi-valued DDs for a scheduling problem in which a subset of elements has to be selected from a ground set and the selected elements need to be sequenced. The proposed construction scheme builds upon A* search guided by a fast-to-calculate problem-specific dual bound heuristic. In contrast to traditional DD compilation methods, the new approach does not rely on a correspondence of DD layers to decision variables. For the considered kind of problem, this implies that multiple nodes representing the same state at different layers can be avoided, and consequently also many redundant isomorphic substructures. For keeping the relaxed DD compact, a new mechanism for merging nodes in a layer-independent way is suggested. For our prize-collecting job sequencing problem, experimental results show that the DDs from our A*-based approach provide substantially better bounds while frequently being an order-of-magnitude smaller than DDs obtained from traditional compilation methods, given about the same time. To obtain a heuristic solution and a corresponding lower bound, we further propose to construct a restricted DD based on the relaxed one, thereby substantially exploiting already gained information. This approach outperforms a standalone restricted DD construction, basic constraint programming and mixed integer linear programming approaches, and a variable neighborhood search in terms of solution quality on most of our benchmark instances.