In this paper, we bound the integrality gap and the approximation ratio for maximum plane multiflow problems and deduce bounds on the flow-multicut-gap. We consider instances where the union of the supply and demand graphs is planar and prove that there exists a multiflow of value at least half the capacity of a minimum multicut. We then show how to convert any multiflow into a half-integer flow of value at least half the original multiflow. Finally, we round any half-integer multiflow into an integer multiflow, losing at most half the value thus providing a 1/4-approximation algorithm and integrality gap for maximum integer multiflows in the plane.

中文翻译：

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整数平面多流最大化：四分之一近似和间隙

在本文中，我们限制了最大平面多流问题的完整性间隙和近似比，并推导出了流多切割间隙的界限。我们考虑供需图的联合是平面的情况，并证明存在至少是最小多重切割容量的一半的多重价值流。然后，我们将展示如何将任何多流转换为至少是原始多流一半的半整数值流。最后，我们将任何半整数多流舍入为整数多流，最多损失一半的值，从而为平面中的最大整数多流提供 1/4 近似算法和完整性差距。