通过树深度和顶点完整性计算双宽度
Computing Twin-Width via Treedepth and Vertex Integrity
AI总结：
本文证明，当参数化为树深度时，近似双宽度是固定参数可解的；当参数化为顶点完整性时，精确计算双宽度是固定参数可解的，首次为非平凡参数化算法提供最优收缩序列。
摘要：
双宽度是一个图参数，已成为解释一阶模型检验在许多图类上固定参数可解性的核心。尽管其算法重要性，计算双宽度仍然知之甚少：甚至识别双宽度至多为4的图是NP难的，并且没有已知的以双宽度本身为参数的固定参数近似。最近突破这一障碍的方法侧重于首先开发以不同于双宽度的参数化来计算或近似双宽度的固定参数算法。我们的第一个结果表明，当以树深度为参数时，近似双宽度是固定参数可解的，从而打破了所有先前可处理的参数化都基于删除距离的长期障碍。证明通过有向双宽度进行，首次提供了该变体可能在算法上更易处理的构造性证据。作为第二个主要结果，我们表明，以顶点完整性为参数时，精确计算双宽度是固定参数可解的。这构成了计算最优收缩序列的第一个非平凡参数化算法。
英文摘要：
Twin-width is a graph parameter that has become central to explaining the fixed-parameter tractability of first-order model checking across many graph classes. Despite its algorithmic importance, computing twin-width remains poorly understood: even recognizing graphs of twin-width at most four is NP-hard, and no fixed-parameter approximations parameterized by twin-width itself are known. A recent approach towards breaking this barrier focuses on first developing fixed-parameter algorithms for computing or approximating twin-width under parameterizations distinct from twin-width. Our first result establishes that approximating twin-width is fixed-parameter tractable when parameterized by treedepth, thereby breaking the long-standing barrier that all previous tractable parameterizations were based on deletion distance. The proof proceeds via oriented twin-width, yielding the first constructive evidence that this variant may be easier to handle algorithmically. As our second main result, we show that computing twin-width exactly is fixed-parameter tractable with respect to vertex integrity. This constitutes the first non-trivial parameterized algorithm for computing optimal contraction sequences.
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