Abstract
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.
Highlights
In the last decade decision diagrams (DDs) have shown to be a powerful tool in combinatorial optimization (Andersen et al, 2007; Bergman et al, 2014; Cire and van Hoeve, 2013)
For the type of problem with both selection and sequencing decisions we consider, it is natural to build upon multi-valued DDs (MDDs) similar to those from Cire and van Hoeve (2013), as solutions can be represented by permutations of the chosen elements
We show for the PC-JSOCMSR that the relaxed MDDs obtained by the AÃ-based method yield substantially stronger bounds than relaxed MDDs of comparable size constructed by two standard techniques
Summary
In the last decade decision diagrams (DDs) have shown to be a powerful tool in combinatorial optimization (Andersen et al, 2007; Bergman et al, 2014; Cire and van Hoeve, 2013). For the type of problem with both selection and sequencing decisions we consider, it is natural to build upon MDDs similar to those from Cire and van Hoeve (2013), as solutions can be represented by permutations of the chosen elements. A particularity of our case is that feasible solutions may have arbitrary size in terms of the number of selected elements This leads us to a novel technique for constructing relaxed MDDs. The method is inspired by AÃ search, a commonly used algorithm in path planning and problem solving (Hart et al, 1968).
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