Yngve Villanger

Improved algorithms for feedback vertex set problems

We present improved parameterized algorithms for the feedback vertex set problem on both unweighted and weighted graphs. Both algorithms run in time O(5^kkn^2). The algorithms construct a feedback vertex set of size at most k (in the weighted case this set is of minimum weight among the feedback vertex sets of size at most k) in a given graph G of n vertices, or report that no such feedback vertex set exists in G.

Kernel(s) for problems with no kernel: On out-trees with many leaves

The k-Leaf Out-Branching problem is to find an out-branching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the k-Leaf-Out-Branching problem. We give the first polynomial kernel for Rooted k-Leaf-Out-Branching, a variant of k-Leaf-Out-Branching where the root of the tree searched for is also a part of the input. Our kernel with O(k3) vertices is obtained using extremal combinatorics. For the k-Leaf-Out-Branching problem, we show that no polynomial-sized kernel is possible unless coNP is in NP/poly. However, our positive results for Rooted k-Leaf-Out-Branching immediately imply that the seemingly intractable k-Leaf-Out-Branching problem admits a data reduction to n independent polynomial-sized kernels. These two results, tractability and intractability side by side, are the first ones separating Karp kernelization from Turing kernelization. This answers affirmatively an open problem regarding “cheat kernelization” raised by Mike Fellows and Jiong Guo independently.

Finding Induced Subgraphs via Minimal Triangulations

Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulation problems including Minimum Fill-in and Treewidth. We discover unexpected applications of these notions to the field of moderate exponential algorithms. In particular, we show that given an n-vertex graph G together with its set of potential maximal cliques, and an integer t, it is possible in time the number of potential maximal cliques times

Exact Algorithms for Treewidth and Minimum Fill-In

O(n^{O(t)})

Interval Completion Is Fixed Parameter Tractable

to find a maximum induced subgraph of treewidth t in G and for a given graph F of treewidth t, to decide if G contains an induced subgraph isomorphic to F. Combined with an improved algorithm enumerating all potential maximal cliques in time

Local search: Is brute-force avoidable?

O(1.734601^n)

Subexponential Parameterized Algorithm for Minimum Fill-In

, this yields that both the problems are solvable in time

Treewidth Computation and Extremal Combinatorics

1.734601^n

Interval completion with few edges

*

A vertex incremental approach for maintaining chordality

n^{O(t)}