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In the Equal Maximum Flow Problem (EMFP), we aim for a maximum flow where we require the same flow value on all edges in some given subsets of the edge set. In this paper, we study the closely related Almost Equal Maximum Flow Problems (AEMFP) where the flow values on edges of one homologous edge set differ at most by the valuation of a so called deviation function ∆. We prove that the integer almost equal maximum flow problem (integer AEMFP) is in general N P-complete, and show that even the problem of finding a fractional maximum flow in the case of convex deviation functions is also N P-complete. This is in contrast to the EMFP, which is polynomial time solvable in the fractional case. We provide inapproximability results for the integral AEMFP. For the integer AEMFP we state a polynomial algorithm for the constant deviation and concave case for a fixed number of homologous sets.

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The Set Cover Problem (SCP) and the Hitting Set Problem (HSP) are well-studied optimization problems. In this paper we introduce the Reward-Penalty-Selection Problem which can be understood as a combination of the SCP and the HSP where the objectives of both problems are contrary to each other. Applications of the RPSP can be found in the context of combinatorial exchanges in order to solve the corresponding winner determination problem. We give complexity results for the minimization and the maximization problem as well as for several variants with additional restrictions. Further, we provide an algorithm that runs in polynomial time for the special case of laminar sets and a dynamic programming approach for the case where the instance can be represented by a tree or a graph with bounded tree-width. We further present a graph theoretical generalization of this problem and results regarding its complexity.

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