Powerful results from the theory of integer programming have recently led to substantial advances in parameterized complexity. However, our perception is that, except for Lenstra’s algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community of the strengths and limitations of the available tools. This is understandable: it is often difficult to infer exact runtimes or even the distinction between $\mathsf{FPT}$ and $\mathsf{XP}$ algorithms, and some knowledge is simply unwritten folklore in a different community. We wish to make a step in remedying this situation. To that end, we first provide an easy to navigate quick reference guide of integer programming algorithms from the perspective of parameterized complexity. Then, we show their applications in three case studies, obtaining $\mathsf{FPT}$ algorithms with runtime $f\left(k\right)$poly$\left(n\right)$. We focus on:

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Modeling: since the algorithmic results follow by applying existing algorithms to new models, we shift the focus from the complexity result to the modeling result, highlighting common patterns and tricks which are used.

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Optimality program: after giving an $\mathsf{FPT}$ algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups.

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Minding the poly$\left(n\right)$: reducing $f\left(k\right)$ often has the unintended consequence of increasing poly$\left(n\right)$; so we highlight the common trade-offs and show how to get the best of both worlds.

整数编程理论的强大结果最近导致参数化复杂性的实质性进步。但是，我们的理解是，除了Lenstra的算法用于求解固定维数的整数线性规划外，在参数化复杂度社区中仍然对可用工具的优势和局限性了解甚少。这是可以理解的：通常很难推断出确切的运行时间，甚至很难区分$\mathsf{FPT}$ 和 $\mathsf{经验值}$算法和一些知识，只是在不同社区中不成文的民俗知识。我们希望采取步骤纠正这种情况。为此，我们首先从参数化复杂性的角度提供易于浏览的整数编程算法的快速参考指南。然后，我们在三个案例研究中展示了它们的应用，$\mathsf{FPT}$ 运行时的算法 $F（ķ）$聚$（ñ）$。我们专注于：

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建模：由于算法结果是将现有算法应用于新模型，因此我们将重点从复杂度结果转移到建模结果，重点介绍常用的模式和技巧。

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最优方案：给人一种后$\mathsf{FPT}$算法，我们有兴趣减少对参数的依赖；我们展示了哪些算法和技巧通常对加速有用。

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介意聚$（ñ）$：减少 $F（ķ）$ 通常会增加聚变的意想不到的结果$（ñ）$; 因此，我们重点介绍了常见的取舍，并展示了如何兼顾两者。