Faster Existential FO Model Checking on Posets

Abstract

We prove that the model checking problem for the existential fragment of first order (FO) logic on partially ordered sets is fixed-parameter tractable (FPT) with respect to the formula and the width of a poset (the maximum size of an antichain). While there is a long line of research into FO model checking on graphs, the study of this problem on posets has been initiated just recently by Bova, Ganian and Szeider (LICS 2014), who proved that the existential fragment of FO has an FPT algorithm for a poset of fixed width. We improve upon their result in two ways: (1) the runtime of our algorithm is $$O(f(|\phi |,w)\cdot n^2)$$ on $$n$$ -element posets of width $$w$$ , compared to $$O(g(|\phi |)\cdot n^{h(w)})$$ of Bova et al., and (2) our proofs are simpler and easier to follow. We complement this result by showing that, under a certain complexity-theoretical assumption, the existential FO model checking problem does not have a polynomial kernel.