Showing papers by "Yngve Villanger published in 2009"

Kernel(s) for Problems with No Kernel: On Out-Trees with Many Leaves

Abstract: The {\sc $k$-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 {\sc $k$-Leaf-Out-Branching} problem. We give the first polynomial kernel for {\sc Rooted $k$-Leaf-Out-Branching}, a variant of {\sc $k$-Leaf-Out-Branching} where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics. For the {\sc $k$-Leaf-Out-Branching} problem, we show that no polynomial kernel is possible unless the polynomial hierarchy collapses to third level by applying a recent breakthrough result by Bodlaender et al. (ICALP 2008) in a non-trivial fashion. However, our positive results for {\sc Rooted $k$-Leaf-Out-Branching} immediately imply that the seemingly intractable {\sc $k$-Leaf-Out-Branching} problem admits a data reduction to $n$ independent $O(k^3)$ kernels. These two results, tractability and intractability side by side, are the first ones separating {\it many-to-one kernelization} from {\it Turing kernelization}. This answers affirmatively an open problem regarding ``cheat kernelization'' raised by Mike Fellows and Jiong Guo independently.

Local search: is brute-force avoidable?

Abstract: Many local search algorithms are based on searching in the k-exchange neighborhood. This is the set of solutions that can be obtained from the current solution by exchanging at most k elements. As a rule of thumb, the larger k is, the better are the chances of finding an improved solution. However, for inputs of size n, a naive brute-force search of the k-exchange neighborhood requires nO(k) time, which is not practical even for very small values of k. We show that for several classes of sparse graphs, like planar graphs, graphs of bounded vertex degree and graphs excluding some fixed graph as a minor, an improved solution in the k-exchange neighborhood for many problems can be found much more efficiently. Our algorithms run in time O(τ (k) ċ nc), where τ is a function depending on k only and c is a constant independent of k. We demonstrate the applicability of this approach on different problems like r-CENTER, VERTEX COVER, ODD CYCLE TRANSVERSAL, MAX-CUT, and MIN-BISECTION. In particular, on planar graphs, all our algorithms searching for a k- local improvement run in time O(2O(k) ċ n2), which is polynomial for k = O(log n). We also complement the algorithms with complexity results indicating that--brute force search is unavoidable--in more general classes of sparse graphs.

Computing Pathwidth Faster Than 2n

Abstract: Computing the Pathwidth of a graph is the problem of finding a tree decomposition of minimum width, where the decomposition tree is a path. It can be easily computed in $\mathcal{O}^*(2^n)$ time by using dynamic programming over all vertex subsets. For some time now there has been an open problem if there exists an algorithm computing Pathwidth with running time $\mathcal{O}^*(c^n)$ for c < 2. In this paper we show that such an algorithm with c = 1.9657 exists, and that there also exists an approximation algorithm and a constant ? such that an opt + ? approximation can be obtained in $\mathcal{O}^*(1.89^ n)$ time.

Kernel(s) for problems with no kernel: on o ut-trees with many leaves (extended abstract)

Abstract: 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 has cubic size and is obtained using extremal combinatorics. For the k-Leaf-Out-Branching problem, we show that no polynomial kernel is pos- sible unless the polynomial hierarchy collapses to third level by applying a recent break- through result by Bodlaender et al. (ICALP 2008) in a non-trivial fashion. 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 in- dependent O(k 3 ) kernels. These two results, tractability and intractability side by side, are the first ones separating many-to-one 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

Abstract: Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulations 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 Pi_G, and an integer t, it is possible in time |Pi_G| * n^(O(t)) 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 O(1.734601^n), this yields that both problems are solvable in time 1.734601^n * n^(O(t)).