混合单位区间图的U-Bubble模型及其应用：MaxCut问题的再审视
U-Bubble Model for Mixed Unit Interval Graphs and its Applications: The MaxCut Problem Revisited
AI总结：
本文提出混合单位区间图的U-Bubble模型，改进了MaxCut问题的求解算法，并揭示了单位区间图MaxCut问题的多项式性证明中的错误。
摘要：
区间图是线段的交集图，在算法研究和特殊结构性质中起着关键作用。单位区间图是其真子类，其中每个区间长度为单位长度，已被广泛研究。我们研究混合单位区间图——单位区间图的推广，其中每个区间仍保持单位长度，但允许不同类型的区间（开区间、闭区间、半开区间）。这种小的修改捕获了更丰富的图类。特别是，混合单位区间图不像单位区间图那样是爪自由的。Heggernes、Meister和Papadopoulos定义了单位区间图的一种表示称为气泡模型，该模型在算法设计中证明很有用。我们扩展了这一模型到混合单位区间图的类，并通过提供一个亚指数时间算法来解决混合单位区间图上的MaxCut问题，以展示这种扩展模型的优势。此外，我们为混合单位区间图的某些子类推导出一个多项式时间算法。我们指出Boyaci、Ekim和Shalom（2017）在单位区间图上MaxCut问题多项式性证明中存在重大错误。因此，该问题在单位区间图上的时间复杂度仍处于开放状态。我们进一步为混合单位区间图的clique-width提供了更好的算法上界。clique-width是最重要的通用图结构参数之一，其中一大组自然问题在给定高效表示时仍能在可处理时间内解决。不幸的是，精确计算clique-width表示的复杂度是NP难的。因此，对clique-width的良好上界非常受欢迎，特别是当这种界具有算法性时。
英文摘要：
Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs---a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyaci, Ekim, and Shalom (2017). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs. Clique-width is one of the most general structural graph parameters, where a large group of natural problems is still solvable in the tractable time when an efficient representation is given. Unfortunately, the exact computation of the clique-width representation is \NP-hard. Therefore, good upper-bounds on clique-width are highly appreciated, in particular, when such a bound is algorithmic.
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