The Reward-Penalty-Selection Problem

Heller, Till; Krumke, Sven O.; Küfer, Karl‐Heinz

doi:10.48550/arxiv.2106.14601

Cited by 1 publication

(2 citation statements)

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“…On the other hand, end products can only be produced if all the materials required for them are available -only in this case can a profit be made. This problem can be formalized as the reward-penalty-selection problem (RPSP) which was recently introduced in [6]. It can be viewed as a combination of two classical, well-known combinatorial problems, the set cover problem and the hitting set problem (cf.…”

Section: Introductionmentioning

confidence: 99%

“…In the example given above, this corresponds to selecting a subset of the raw materials such that the net profit defined by the profit obtained by selling the final products minus the cost for the necessary raw material packages is maximized. For an analysis on the complexity of the RPSP and its variants, as well as algorithmic approaches, we refer to [6,7].…”

Section: Introductionmentioning

confidence: 99%

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On Reward-Penalty-Selection Games

Gräf¹,

Heller²,

Krumke³

2022

Preprint

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The Reward-Penalty-Selection Problem (RPSP) can be seen as a combination of the Set Cover Problem (SCP) and the Hitting Set Problem (HSP). Given a set of elements, a set of reward sets, and a set of penalty sets, one tries to find a subset of elements such that as many reward sets as possible are covered, i.e. all elements are contained in the subset, and at the same time as few penalty sets as possible are hit, i.e. the intersection of the subset with the penalty set is non-empty. In this paper we define a cooperative game based on the RPSP where the elements of the RPSP are the players. We prove structural results and show that RPS games are convex, superadditive and totally balanced. Furthermore, the Shapley value can be computed in polynomial time. In addition to that, we provide a characterization of the core elements as a feasible flow in a network graph depending on the instance of the underlying RPSP. By using this characterization, a core element can be computed efficiently.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%