Integer Programming in Parameterized Complexity: Three Miniatures

TL;DR

This paper explores the application of integer programming techniques to parameterized complexity, providing a guide and three case studies that develop fixed-parameter tractable algorithms for specific graph problems.

Contribution

It offers a comprehensive reference of integer programming algorithms for parameterized complexity and demonstrates their use in three novel case studies with FPT algorithms.

Findings

Developed FPT algorithms for Capacitated Dominating Set, Sum Coloring, and Max-q-Cut.

Highlighted modeling patterns and tricks for applying integer programming in parameterized problems.

Discussed trade-offs between runtime dependence on parameters and polynomial factors.

Abstract

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 FPT and 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 FPT algorithms with runtime $f(k)poly(n)$. We focus on: * 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. * Optimality program: after giving an FPT algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups. * Minding the poly(n): reducing $f(k)$ often has the unintended consequence of increasing poly(n); 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 FPT algorithms for Capacitated Dominating Set, Sum Coloring, and Max-q-Cut by modeling them as convex programs in fixed dimension, n-fold integer programs, bounded dual treewidth programs, and indefinite quadratic programs in fixed dimension.