Directed Feedback Vertex Set is Fixed-Parameter Tractable

We resolve positively a long standing open question regarding the fixed-parameter tractability of the parameterized Directed Feedback Vertex Set problem. In particular, we propose an algorithm which solves this problem in $O(8^kk!*poly(n))$.Comment: 14 page

FPT algoritms providing constant ratio approximation of hypertree width parameters for hypergraphs of bounded rank

We propose an algorithm whose input are parameters $k$ and $r$ and a hypergraph $H$ of rank at most $r$. The algorithm either returns a tree decomposition of $H$ of generalized hypertree width at most $4k$ or 'NO'. In the latter case, it is guaranteed that the hypertree width of $H$ is greater than $k$. Most importantly, the runtime of the algorithm is \emph{FPT} in $k$ and $r$. The approach extends to fractional hypertree width with a slightly worse approximation ($4k+1$ instead of $4k$). We hope that the results of this paper will give rise to a new research direction whose aim is design of FPT algorithms for computation and approximation of hypertree width parameters for restricted classes of hypergraphs

On OBDDs for CNFs of bounded treewidth

In this paper we show that a CNF cannot be compiled into an Ordered Binary Decision Diagram (OBDD) of fixed-parameter size parameterized by the primal graph treewidth of the input CNF. Thus we provide a parameterized separation between OBDDs and Sentential Decision Diagrams for which such fixed-parameter compilation is possible. We also show that the best existing parameterized upper bound for OBDDs in fact holds for the incidence graph treewidth parameterization

Regular resolution for CNF of bounded incidence treewidth with few long clauses

We demonstrate that Regular Resolution is FPT for two restricted families of CNFs of bounded incidence treewidth. The first includes CNFs having at most $p$ clauses whose removal results in a CNF of primal treewidth at most $k$. The parameters we use in this case are $p$ and $k$. The second class includes CNFs of bounded one-sided (incidence) treewdth, a new parameter generalizing both primal treewidth and incidence pathwidth. The parameter we use in this case is the one-sided treewidth