论颜色编码的描述复杂性
On the Descriptive Complexity of Color Coding
摘要：
颜色编码是参数化复杂性理论中使用的一种算法技术，用于检测图中的“小”结构。这个想法是去随机化算法，首先随机着色图，然后搜索一个容易检测的小颜色模式。我们转移颜色编码的描述复杂性理论的世界的特征-纯粹的语法结构描述公式-当强大的二阶量词表示随机着色可以被替换为等价的，简单的一阶公式。在此基础上，我们确定的一阶量词，可以从描述参数化问题的公式中消除的语法特性。这个结果适用于许多包装和嵌入问题，但也适用于长路径问题。再加上一个新的结果参数化的复杂性公式家庭只涉及一个固定数量的变量，我们得到，许多问题在于FPT只是因为它们通常使用逻辑公式描述的方式。
英文摘要：
Color coding is an algorithmic technique used in parameterized complexity theory to detect "small" structures inside graphs. The idea is to derandomize algorithms that first randomly color a graph and then search for an easily-detectable, small color pattern. We transfer color coding to the world of descriptive complexity theory by characterizing -- purely in terms of the syntactic structure of describing formulas -- when the powerful second-order quantifiers representing a random coloring can be replaced by equivalent, simple first-order formulas. Building on this result, we identify syntactic properties of first-order quantifiers that can be eliminated from formulas describing parameterized problems. The result applies to many packing and embedding problems, but also to the long path problem. Together with a new result on the parameterized complexity of formula families involving only a fixed number of variables, we get that many problems lie in FPT just because of the way they are commonly described using logical formulas.
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