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 $O(1.7215^n)$ and $O(1.6232^n)$. Among the two improvements, our new running time $O(1.7215^n)$ is the first improvement in the best known running time for the 4-Coloring problem since 2007.

翻译：为分析分支算法的最坏情况运行时间，指数时间算法领域的多数工作专注于设计复杂的分支规则，而较少关注改进简单算法的分析方法。在2005年前后，Fomin等人[2005]提出了“测度与征服”这一先进的通用分析方法，该方法被广泛采用，为许多基本NP完全问题获得了更紧的最坏情况运行时间上界。然而，这一方向仍存在大量未开发的潜力，因为后续大多数工作仅应用该方法而未进一步推进其发展。受此启发，我们提出了分段分析——一种分析分支算法运行时间的通用新方法。我们的方法是通过定义相似度比率将实例划分为不同组别，然后分别分析每组内的运行时间。相似度比率反映了实例I的两个参数之间的尺度关系。与依赖单一测度对整个实例空间进行单一分析不同，我们的方法能够利用具有不同相似度比率的实例各自的内在属性。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，它们分别解决4-着色问题和#3-着色问题。原始论文中针对这些算法的分析给出的运行时间分别为O(1.7272^n)和O(1.6262^n)，而我们的分析将这些运行时间改进至O(1.7215^n)和O(1.6232^n)。在这两项改进中，新的运行时间O(1.7215^n)是自2007年以来在4-着色问题已知最佳运行时间上的首次改进。