Beyond Max-Cut: λ-extendible properties parameterized above the Poljak-Turzík bound.

We define strong λ-extendibility as a variant of the notion of λ-extendible properties of graphs (Poljak and Turzík, Discrete Mathematics, 1986). We show that the parameterized APT(Π) problem — given a connected graph G on n vertices and m edges and an integer parameter k, does there exist a spanning subgraph H of G such that H∈Π and H has at least λm+1−λ2(n−1)+k edges — is fixed-parameter tractable (FPT) for all 0<λ<1, for all strongly λ-extendible graph properties Π for which the APT(Π) problem is FPT on graphs which are O(k) vertices away from being a graph in which each block is a clique. Our results hold for properties of oriented graphs and graphs with edge labels, generalize the recent result of Crowston et al. (ICALP 2012) on Max-Cut parameterized above the Edwards–Erdős bound, and yield FPT algorithms for several graph problems parameterized above lower bounds.