We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus G has a cut graph of length at most a given value. We prove a time lower bound for this problem of n Ω( g log g ) conditionally to the ETH. In other words, the first n O(g) -time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year-old question of these authors. A multiway cut of an undirected graph G with t distinguished vertices, called terminals , is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n Ω( gt + g 2 + t log ( g + t )) , conditionally to the ETH, for any choice of the genus g ≥ 0 of the graph and the number of terminals t ≥ 4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case). Reductions to planar problems usually involve a gridlike structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value G of the genus.

中文翻译：

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嵌入式图中硬切割问题的几乎严格的下界

对于表面嵌入图上的两个基本但看似非常不同的切割问题，我们在指数时间假设的条件下证明了本质上紧密的下界：最短割图问题和多路切割问题。图的割图G嵌入表面 S 是G从 S 中移除它会留下一个磁盘。我们考虑决定一个未加权的图是否嵌入到一个属的表面上的问题G有一个长度最多为给定值的割图。我们证明了这个问题的时间下界n Ω(G日志G)有条件地到 ETH。换句话说，第一个n （克） Erickson 和 Har-Peled 的时间算法 [SoCG 2002，Discr。计算。几何。2004] 基本上是最优的。我们还证明了当按属参数化时问题是 W[1]-hard，回答了这些作者 17 年前提出的一个问题。无向图的多路割G和吨可区分的顶点，称为终端, 是一组边，其移除会断开所有端子对。我们考虑决定一个未加权图的问题G至多在给定值下具有多路权重切割。我们证明了这个问题的时间下界n Ω(gt+G 2+吨日志 （G+吨)), 对 ETH 有条件, 对于任何种类的选择G≥0的图形和端子数吨≥4。也就是说，第二作者的算法[Algorithmica 2017]（针对更一般的多切问题）本质上是最优的；这扩展了第三作者 [ICALP 2012] 的下限（对于平面案例）。平面问题的简化通常涉及网格状结构。我们的结果的主要新颖想法是了解如果我们想优化利用某个值，需要哪些结构而不是网格G属的。