几乎相等最大流问题的算法和复杂性
Algorithms and Complexity for the Almost Equal Maximum Flow Problem
摘要：
在等最大流问题（EMFP）中，我们的目标是一个最大流，我们需要在一些给定的子集的边缘上的所有边缘相同的流量值。本文研究了与之密切相关的几乎相等最大流问题（AEMFP），其中一个同源边集的边上的流值最多相差一个所谓的偏差函数$Δ$的赋值.我们证明了整数几乎相等最大流问题（整数AEMFP）是一般$\mathcal{NP}$ -完全的，并表明，即使是在凸偏差函数的情况下，找到一个分数最大流的问题也是$\mathcal{NP}$ -完全的。这与EMFP相反，EMFP在分数情况下是多项式时间可解的。我们提供不可逼近的结果积分AEMFP。对于整数AEMFP，我们国家的多项式算法的恒定偏差和凹的情况下，一个固定数量的同源集。
英文摘要：
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~$Δ$. We prove that the integer almost equal maximum flow problem (integer AEMFP) is in general $\mathcal{NP}$-complete, and show that even the problem of finding a fractional maximum flow in the case of convex deviation functions is also $\mathcal{NP}$-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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