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In the Directed Feedback Vertex Set (DFVS) problem, we are given a digraph D on n vertices and a positive integer k and the objective is to check whether there exists a set of vertices S of size at most k such that F = D − S is a directed acyclic digraph. In a recent paper, Mnich and van Leeuwen [STACS 2016 ] considered the kernelization complexity of DFVS with an additional restriction on F , namely that F must be an out-forest (Out-Forest Vertex Deletion Set), an out-tree (Out-Tree Vertex Deletion Set), or a (directed) pumpkin (Pumpkin Vertex Deletion Set). Their objective was to shed some light on the kernelization complexity of the DFVS problem, a well known open problem in the area of Parameterized Complexity. In this article, we improve the kernel sizes of Out-Forest Vertex Deletion Set from O(k 3 ) to O(k 2 ) and of Pumpkin Vertex Deletion Set from O(k 18 ) to O(k 3 ). We also prove that the former kernel size is tight under certain complexity theoretic assumptions.

We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a d -degenerate graph G and an integer k , outputs an independent set Y , such that for every independent set X in G of size at most k , the probability that X is a subset of Y is at least (( (d+1)k k ) . k (d+1)) -1 . The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph G , a set T = {s_1, t_1} , {s_2, t_2}, …. , {s_ℓ , t_ℓ} of terminal pairs, and an integer k , returns an induced subgraph G* of G that maintains all the inclusion minimal multicuts of G of size at most k and does not contain any ( k +2)-vertex connected set of size 2 O(k) . In particular, G* excludes a clique of size 2 O(k) as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for S TABLE s-t S EPARATOR , S TABLE O DD C YCLE T RANSVERSAL , and S TABLE M ULTICUT on general graphs, and for S TABLE D IRECTED F EEDBACK V ERTEX S ET on d -degenerate graphs, resolving two problems left open by Marx et al. [ ACM Transactions on Algorithms, 2013{. All of our algorithms can be derandomized at the cost of a small overhead in the running time.

The family of judicious partitioning problems, introduced by Bollobás and Scott to the field of extremal combinatorics, has been extensively studied from a structural point of view for over two decades. This rich realm of problems aims to counterbalance the objectives of classical partitioning problems such as Min Cut, Min Bisection and Max Cut. While these classical problems focus solely on the minimization/maximization of the number of edges crossing the cut, judicious (bi)partitioning problems ask the natural question of the minimization/maximization of the number of edges lying in the (two) sides of the cut. In particular, Judicious Bipartition (JB) seeks a bipartition that is "judicious" in the sense that neither side is burdened by too many edges, and Balanced JB also requires that the sizes of the sides themselves are "balanced" in the sense that neither of them is too large. Both of these problems were defined in the work by Bollobás and Scott, and have received notable scientific attention since then. In this paper, we shed light on the study of judicious partitioning problems from the viewpoint of algorithm design. Specifically, we prove that BJB is FPT (which also proves that JB is FPT).

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