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：为了分析分支算法的最坏情况运行时间，指数时间算法领域的大多数工作聚焦于设计复杂的分支规则，而非发展更优的简单算法分析方法。在2000年代中期，Fomin等人[2005]提出了测度与征服这一先进通用分析方法，该方法被广泛采用，为许多基础NP完全问题获得了更紧的最坏情况运行时间上界。然而，该方向仍有大量潜力未被发掘，因为后续大多数工作仅将其作为工具使用而未进一步发展。受此启发，我们提出分段分析这一新的通用方法，用于分析分支算法的运行时间。我们的方法是定义一个相似比，将实例划分为不同组别，然后分别分析每组内的运行时间。相似比是实例I两个参数之间的一个标度。与对整个实例空间依赖单一测度和单一分析不同，我们的方法能够利用具有不同相似比的实例的不同内在特性。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，它们分别求解4-着色问题和#3-着色问题。原论文中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为O(1.7215^n)和O(1.6232^n)。在这两个改进中，新的运行时间O(1.7215^n)是自2007年以来4-着色问题已知最优运行时间的首次改进。