Abstract
The subset sum problem is one of the simplest and most fundamental NP-hard problems in combinatorial optimization. We consider two extensions of this problem: The subset sum problem with digraph constraint (SSG) and subset sum problem with weak digraph constraint (SSGW). In both problems there is given a digraph with sizes assigned to the vertices. Within SSG we want to find a subset of vertices whose total size does not exceed a given capacity and which contains a vertex if at least one of its predecessors is part of the solution. Within SSGW we want to find a subset of vertices whose total size does not exceed a given capacity and which contains a vertex if all its predecessors are part of the solution. SSG and SSGW have been introduced by Gourvès et al. who studied their complexity for directed acyclic graphs and oriented trees. We show that both problems are NP-hard even on oriented co-graphs and minimal series-parallel digraphs. Further, we provide pseudo-polynomial solutions for SSG and SSGW with digraph constraints given by directed co-graphs and series-parallel digraphs.
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Notes
- 1.
The proofs of the results marked with a \(\bigstar \) are omitted due to space restrictions.
- 2.
The value s = 0 is for choosing an empty solution in digraph(X 1 ⊘ X 2).
- 3.
The value s = 0 is for choosing an empty solution in digraph(X 1 ⊘ X 2).
- 4.
The value s = 0 is for choosing an empty solution in digraph(X 1 ⊗ X 2).
- 5.
The value s = 0 is for choosing an empty solution in digraph(X 1 × X 2).
- 6.
The value s = s′ = 0 is for choosing an empty solution in digraph(X 1 × X 2). The values s > s′ = 0 are for choosing a solution without sinks in digraph(X 1 × X 2)
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Gurski, F., Komander, D., Rehs, C. (2020). Subset Sum Problems with Special Digraph Constraints. In: Neufeld, J.S., Buscher, U., Lasch, R., Möst, D., Schönberger, J. (eds) Operations Research Proceedings 2019. Operations Research Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-030-48439-2_41
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DOI: https://doi.org/10.1007/978-3-030-48439-2_41
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