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-Erdos bound, and yield FPT algorithms for several graph problems parameterized above lower bounds. © 2014 Elsevier Inc. All rights reserved.
Mnich, M., Philip, G., Saurabh, S., & Suchý, O. (2014). Beyond Max-Cut: λ-extendible properties parameterized above the Poljak-Turzík bound. Journal of Computer and System Sciences, 80(7), 1384–1403. https://doi.org/10.1016/j.jcss.2014.04.011