Parameterized extension complexity of independent set and related problems

Let G be a graph on n vertices and STABk(G) be the convex hull of characteristic vectors of its independent sets of size at most k. We study extension complexity of STABk(G) with respect to a fixed parameter k (analogously to, e.g., parameterized computational complexity of problems). We show that for graphs G from a class of bounded expansion it holds that xc(STABk(G))⩽O(f(k)⋅n) where the function f 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 no function f such that, for all values of the parameter k and for all graphs on n vertices, the extension complexity of STABk(G) is at most f(k)⋅nO(1). While such results are not surprising since it is known that optimizing over STABk(G) is FPT for graphs of bounded expansion and W[1]-hard in general, they are also not trivial and in both cases stronger than the corresponding computational complexity results.