To analyze the worst-case running time of branching algorithms, the majority of work in exponential time algorithms focuses on designing complicated branching rules over developing better analysis methods for simple algorithms. In the mid-$2000$s, Fomin et al. [2005] introduced measure & conquer, an advanced general analysis method, sparking widespread adoption for obtaining tighter worst-case running time upper bounds for many fundamental NP-complete problems. Yet, much potential in this direction remains untapped, as most subsequent work applied it without further advancement. Motivated by this, we present piecewise analysis, a new general method that analyzes the running time of branching algorithms. Our approach is to define a similarity ratio that divides instances into groups and then analyze the running time within each group separately. The similarity ratio is a scale between two parameters of an instance I. Instead of relying on a single measure and a single analysis for the whole instance space, our method allows to take advantage of different intrinsic properties of instances with different similarity ratios. To showcase its potential, we reanalyze two $17$-year-old algorithms from Fomin et al. [2007] that solve $4$-Coloring and #$3$-Coloring respectively. The original analysis in their paper gave running times of $O(1.7272^n)$ and $O(1.6262^n)$ respectively for these algorithms, our analysis improves these running times to $O(1.7207^n)$ and $O(1.6225^n)$.

翻译：在分析分支算法的最坏情况运行时间时，指数时间算法领域的大部分工作集中于设计复杂的分支规则，而非为简单算法开发更好的分析方法。21世纪初，Fomin等人[2005]提出了测度征服法，这是一种先进通用分析方法，推动了为许多基本NP完全问题获取更紧致最坏情况运行时间上界的广泛应用。然而，该方向仍有巨大潜力未被发掘，因为后续研究大多直接应用而未作进一步推进。受此启发，我们提出分段分析法——一种分析分支算法运行时间的新通用方法。我们的核心思路是定义一个相似比，将实例划分为不同组别，随后对各组分别进行运行时间分析。相似比是实例I的两个参数之间的比例标度。与依赖单一测度并对整个实例空间进行单一分析的传统方法不同，我们的方法能够利用具有不同相似比的实例所具备的不同内在特性。为展示其潜力，我们重新分析了Fomin等人[2007]提出的两个已有17年历史的算法，分别用于解决4-着色问题和#3-着色问题。原论文给出的运行时间分析结果分别为$O(1.7272^n)$和$O(1.6262^n)$，而我们的分析将运行时间改进为$O(1.7207^n)$和$O(1.6225^n)$。