Integer programming in parameterized complexity: Five miniatures

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 $\mathrm{FPT}$ and $\mathrm{XP}$ 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 $\mathrm{FPT}$ 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 $\mathrm{FPT}$ 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.

Specifically, we consider graphs of bounded neighborhood diversity which are in a sense the simplest of dense graphs, and we show several $\mathrm{FPT}$ algorithms for Capacitated Dominating Set, Sum Coloring, Max- $q$ -Cut, and certain other coloring problems by modeling them as convex programs in fixed dimension, $n$-fold integer programs, bounded dual treewidth programs, indefinite quadratic programs in fixed dimension, parametric integer programs in fixed dimension, and 2-stage stochastic integer programs.