Parameterized Extension Complexity of Independent Set and Related Problems
Jakub Gajarsk\'y, Petr Hlin\v{e}n\'y, Hans Raj Tiwary

TL;DR
This paper investigates the extension complexity of the independent set polytope constrained by size in graphs, showing polynomial bounds for bounded expansion classes and proving super-polynomial lower bounds for general graphs.
Contribution
It establishes bounds on extension complexity for bounded expansion graphs and proves super-polynomial lower bounds for general graphs, advancing understanding of polyhedral complexity in parameterized graph problems.
Findings
Polynomial extension complexity for bounded expansion graphs
Super-polynomial lower bounds for general graphs
Stronger results than known computational complexity classifications
Abstract
Let be a graph on vertices and be the convex hull of characteristic vectors of its independent sets of size at most . We study extension complexity of with respect to a fixed parameter (analogously to, e.g., parameterized computational complexity of problems). We show that for graphs from a class of bounded expansion it holds that where the function depends only on the class. This result can be extended in a simple way to a wide range of similarly defined graph polytopes. In case of general graphs we show that there is {\em no function } such that, for all values of the parameter and for all graphs on vertices, the extension complexity of is at most While such results are not surprising since…
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