Beyond Max-Cut: lambda-Extendible Properties Parameterized Above the Poljak-Turzik Bound

Poljak and Turzík (Discrete Math. 1986) introduced the notion of λ-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0 < λ < 1 and λ-extendible property Π, any connected graph G on n vertices and m edges contains a spanning subgraph H ∈ Π with at least λm+ 1−λ 2 (n−1) edges. The property of being bipartite is λ-extendible for λ = 1/2, and thus the Poljak-Turzík bound generalizes the well-known EdwardsErdős bound for Max-Cut. We define a variant, namely strong λ-extendibility, to which the Poljak-Turzík bound applies. For a strongly λ-extendible graph property Π, we define the parameterized Above PoljakTurzík (Π) problem as follows: 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−λ 2 (n− 1) + k edges? The parameter is k, the surplus over the number of edges guaranteed by the Poljak-Turzík bound. We consider properties Π for which the Above Poljak-Turzík (Π) problem is fixedparameter tractable (FPT) on graphs which are O(k) vertices away from being a graph in which each block is a clique. We show that for all such properties, Above Poljak-Turzík (Π) is FPT for all 0 < λ < 1. Our results hold for properties of oriented graphs and graphs with edge labels. Our results 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. For instance, we get that the above-guarantee Max q-Colorable Subgraph problem is FPT. Our results also imply that the parameterized above-guarantee Oriented Max Acyclic Digraph problem is FPT, thus solving an open question of Raman and Saurabh (Theor. Comput. Sci. 2006).