The single-source unsplittable flow (SSUF) problem asks to send flow from a common source to different terminals with unrelated demands, each terminal being served through a single path. One of the most heavily studied SSUF objectives is to minimize the violation of some given arc capacities. A seminal result of Dinitz, Garg, and Goemans showed that, whenever a fractional flow exists respecting the capacities, then there is an unsplittable one violating the capacities by at most the maximum demand. Goemans conjectured a very natural cost version of the same result, where the unsplittable flow is required to be no more expensive than the fractional one. This intriguing conjecture remains open. More so, there are arguably no non-trivial graph classes for which it is known to hold. We show that a slight weakening of it (with at most twice as large violations) holds for planar graphs. Our result is based on a connection to a highly structured discrepancy problem, whose repeated resolution allows us to successively reduce the number of paths used for each terminal, until we obtain an unsplittable flow. Moreover, our techniques also extend to simultaneous upper and lower bounds on the flow values. This also affirmatively answers a conjecture of Morell and Skutella for planar SSUF.

翻译：单源不可分拆流问题要求从公共源点向不同终端发送流，这些终端具有互不关联的需求，且每个终端通过唯一路径提供服务。该问题最受关注的研究目标之一是最小化对给定弧容量的违反程度。Dinitz、Garg和Goemans的开创性结果表明：只要存在满足容量限制的分式流，就存在一个违反容量不超过最大需求量的不可分拆流。Goemans针对同一结果提出了一个极为自然的成本版本猜想，要求不可分拆流的成本不超过分式流。这个引人入胜的猜想至今尚未解决。更甚者，目前几乎没有已知该猜想成立的非平凡图类。我们证明其弱化版本（违反量至多扩大两倍）对平面图成立。该结果基于与高度结构化差异问题的关联，通过反复求解该问题，我们能够逐步减少每个终端所使用的路径数量，最终得到不可分拆流。此外，我们的技术还可推广到同时对流量值施加上下界约束的情形。这正面回答了Morell和Skutella关于平面图单源不可分拆流的猜想。