Parameterized Extension Complexity of Independent Set and Related Problems

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Year of publication 2018
Type Article in Periodical
Magazine / Source Discrete Applied Mathematics
MU Faculty or unit

Faculty of Informatics

Field Informatics
Keywords parameterized complexity; extension complexity; independent set; FO model checking
Description Let G be a graph on n vertices and STAB_k(G) be the convex hull of characteristic vectors of its independent sets of size at most k. We study extension complexity of STAB_k(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(STAB_k(G))<=O(f(k).n)$ where the function f depends only on the 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 STAB_k(G) is at most f(k).n^{O(1)}. While such results are not surprising since it is known that optimizing over STAB_k(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.
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