{Polynomial Kernels for λ\lambda-extendible Properties Parameterized Above the {Poljak--Turz{\'{i}}k} Bound}

Poljak and Turzik (Discrete Mathematics 1986) introduced the notion of {\lambda}-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0 < {\lambda} < 1 and {\lambda}-extendible property {\Pi}, any connected graph G on n vertices and m edges contains a spanning subgraph H in {\Pi} with at least {\lambda}m + (1-{\lambda})(n-1)/2 edges. The property of being bipartite is {\lambda}-extendible for {\lambda} = 1/2, and so the Poljak-Turzik bound generalizes the well-known Edwards-Erdos bound for Max-Cut. Other examples of {\lambda}-extendible properties include: being an acyclic oriented graph, a balanced signed graph, or a q-colorable graph for some integer q. Mnich et. al. (FSTTCS 2012) defined the closely related notion of strong {\lambda}-extendibility. They showed that the problem of finding a subgraph satisfying a given strongly {\lambda}-extendible property {\Pi} is fixed-parameter tractable (FPT) when parameterized above the Poljak-Turzik bound - does there exist a spanning subgraph H of a connected graph G such that H in {\Pi} and H has at least {\lambda}m + (1-{\lambda})(n-1)/2 + k edges? - subject to the condition that the problem is FPT on a certain simple class of graphs called almost-forests of cliques. In this paper we settle the kernelization complexity of nearly all problems parameterized above Poljak-Turzik bounds, in the affirmative. We show that these problems admit quadratic kernels (cubic when {\lambda} = 1/2), without using the assumption that the problem is FPT on almost-forests of cliques. Thus our results not only remove the technical condition of being FPT on almost-forests of cliques from previous results, but also unify and extend previously known kernelization results in this direction. Our results add to the select list of generic kernelization results known in the literature

Polynomial Kernels for {\lambda}-extendible Properties Parameterized Above the Poljak-Turz\'ik Bound

Poljak and Turzik (Discrete Mathematics 1986) introduced the notion of {\lambda}-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0 < {\lambda} < 1 and {\lambda}-extendible property {\Pi}, any connected graph G on n vertices and m edges contains a spanning subgraph H in {\Pi} with at least {\lambda}m + (1-{\lambda})(n-1)/2 edges. The property of being bipartite is {\lambda}-extendible for {\lambda} = 1/2, and so the Poljak-Turzik bound generalizes the well-known Edwards-Erdos bound for Max-Cut. Other examples of {\lambda}-extendible properties include: being an acyclic oriented graph, a balanced signed graph, or a q-colorable graph for some integer q. Mnich et. al. (FSTTCS 2012) defined the closely related notion of strong {\lambda}-extendibility. They showed that the problem of finding a subgraph satisfying a given strongly {\lambda}-extendible property {\Pi} is fixed-parameter tractable (FPT) when parameterized above the Poljak-Turzik bound - does there exist a spanning subgraph H of a connected graph G such that H in {\Pi} and H has at least {\lambda}m + (1-{\lambda})(n-1)/2 + k edges? - subject to the condition that the problem is FPT on a certain simple class of graphs called almost-forests of cliques. In this paper we settle the kernelization complexity of nearly all problems parameterized above Poljak-Turzik bounds, in the affirmative. We show that these problems admit quadratic kernels (cubic when {\lambda} = 1/2), without using the assumption that the problem is FPT on almost-forests of cliques. Thus our results not only remove the technical condition of being FPT on almost-forests of cliques from previous results, but also unify and extend previously known kernelization results in this direction. Our results add to the select list of generic kernelization results known in the literature

Directed Acyclic Subgraph Problem Parameterized above the Poljak-Turzik Bound

An oriented graph is a directed graph without directed 2-cycles. Poljak and Turzik (1986) proved that every connected oriented graph G on n vertices and m arcs contains an acyclic subgraph with at least m/2+(n-1)/4 arcs. Raman and Saurabh (2006) gave another proof of this result and left it as an open question to establish the parameterized complexity of the following problem: does G have an acyclic subgraph with least m/2 + (n-1)/4 + k arcs, where k is the parameter? We answer this question by showing that the problem can be solved by an algorithm of runtime (12k)!n^{O(1)}. Thus, the problem is fixed-parameter tractable. We also prove that there is a polynomial time algorithm that either establishes that the input instance of the problem is a Yes-instance or reduces the input instance to an equivalent one of size O(k^2)

Polynomial Kernels for λ-extendible Properties Parameterized Above the Poljak-Turzík Bound

Poljak and Turzík (Discrete Mathematics 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 &lt; λ &lt; 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 so the Poljak-Turzík bound generalizes the well-known Edwards-Erdős bound for Max-Cut. Other examples of λ-extendible properties include: being an acyclic oriented graph, a balanced signed graph, or a q-colorable graph for some q ∈ N. Mnich et al. (FSTTCS 2012) defined the closely related notion of strong λ-extendibility. They showed that the problem of finding a subgraph satisfying a given strongly λ-extendible property Π is fixed-parameter tractable (FPT) when parameterized above the Poljak-Turzík bound—does there exist a spanning subgraph H of a connected graph G such that H ∈ Π and H has at least λm+ 1−