Subset feedback vertex set is fixed parameter tractable

The classical Feedback Vertex Set problem asks, for a given undirected graph G and an integer k, to find a set of at most k vertices that hits all the cycles in the graph G. Feedback Vertex Set has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fixed-parameter algorithms have been a rich source of ideas in the field. In this paper we consider a more general and difficult version of the problem, named Subset Feedback Vertex Set (SUBSET-FVS in short) where an instance comes additionally with a set S ? V of vertices, and we ask for a set of at most k vertices that hits all simple cycles passing through S. Because of its applications in circuit testing and genetic linkage analysis SUBSET-FVS was studied from the approximation algorithms perspective by Even et al. [SICOMP'00, SIDMA'00]. The question whether the SUBSET-FVS problem is fixed-parameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixed-parameter tractable when parametrized by |S|. Next we present an algorithm which reduces the given instance to 2^k n^O(1) instances with the size of S bounded by O(k^3), using kernelization techniques such as the 2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two facts allow us to give a 2^O(k log k) n^O(1) time algorithm solving the Subset Feedback Vertex Set problem, proving that it is indeed fixed-parameter tractable.Comment: full version of a paper presented at ICALP'1

Even Faster Algorithm for Set Splitting!

In the p-Set Splitting problem we are given a universe U, a family F of subsets of U and a positive integer k and the objective is to find a partition of U into W and B such that there are at least k sets in F that have non-empty intersection with both B and W. In this paper we study p-Set Splitting from the view point of kernelization and parameterized algorithms. Given an instance (U, F, k) of p-Set Splitting, our kernelization algorithm obtains an equivalent instance with at most 2k sets and k elements in polynomial time. Finally, we give a fixed parameter tractable algorithm for p-Set Splitting running in time O(1.9630k + N), where N is the size of the instance. Both our kernel and our algorithm improve over the best previously known results. Our kernelization algorithm utilizes a classical duality theorem for a connectivity notion in hypergraphs. We believe that the duality theorem we make use of could become an important tool in obtaining kernelization algorithms.

Significant-presence range queries in categorial data

In traditional colored range-searching problems, one wants to store a set of n objects with m distinct colors for the following queries: report all colors such that there is at least one object of that color intersecting the query range. Such an object, however, could be an ‘outlier’ in its color class. Therefore we consider a variant of this problem where one has to report only those colors such that at least a fraction t of the objects of that color intersects the query range, for some parameter t. Our main results are on an approximate version of this problem, where we are also allowed to report those colors for which a fraction (1 - e)t intersects the query range, for some fixed e> 0. We present efficient data structures for such queries with orthogonal query ranges in sets of colored points, and for point stabbing queries in sets of colored rectangles

The zigzag path of a pseudo-triangulation

We define the zigzag path of a pseudo-triangulation, a concept generalizing the path of a triangulation of a point set. The pseudo-triangulation zigzag path allows us to use divide-and-conquer type of approaches for suitable (i.e., decomposable) problems on pseudo-triangulations. For this we provide an algorithm that enumerates all pseudo-triangulation zigzag paths (of all pseudo-triangulations of a given point set with respect to a given line) in O(n 2) time per path and O(n 2) space, where n is the number of points. We illustrate applications of our scheme which include a novel algorithm to count the number of pseudo-triangulations of a point set

A complexity dichotomy for finding disjoint solutions of vertex deletion problems

We investigate the computational complexity of a general "compression task" centrally occurring in the recently developed technique of iterative compression for exactly solving NP-hard minimization problems. The core issue (particularly but not only motivated by iterative compression) is to determine the computational complexity of, given an already inclusion-minimal solution for an underlying (typically NP-hard) vertex deletion problem in graphs, to find a better disjoint solution. The complexity of this task is so far lacking a systematic study. We consider a large class of vertex deletion problems on undirected graphs and show that, except for few cases which are polynomial-time solvable, the others are NP-complete. This class includes problems such as vertex cover (here the corresponding compression task is decidable in polynomial time) or undirected feedback vertex set (here the corresponding compression task is NP-complete)

A Single-Exponential FPT Algorithm for the K 4-Minor Cover Problem

Given an input graph G on $n$ vertices and an integer k, the parameterized \textscK_4-minor cover} problem asks whether there is a set S of at most k vertices whose deletion results in a K_4-minor free graph or, equivalently, in a graph of treewidth at most 2. The problem can thus also be called \textsc{Treewidth-2 Vertex Deletion}. This problem is inspired by two well-studied parameterized vertex deletion problems, \textsc{Vertex Cover} and \textsc{Feedback Vertex Set}, which can be expressed as \textsc{Treewidth-t Vertex Deletion} problems: t=0 for {\sc Vertex Cover} and t=1 for {\sc Feedback Vertex Set}. While a single-exponential FPT algorithm has been known for a long time for \textsc{Vertex Cover}, such an algorithm for \textsc{Feedback Vertex Set} was devised comparatively recently. While it is known to be unlikely that \textsc{Treewidth-t Vertex Deletion} can be solved in time c^{o(k)}⋅ n^{O(1)}, it was open whether the \textsc{K_4-minor cover} could be solved in single-exponential FPT time, i.e. in c^k⋅ n^{O(1) time. This paper answers this question in the affirmative

Alternative Parameterizations for Cluster Editing

Given an undirected graph G and a nonnegative integer k, the NP-hard Cluster Editing problem asks whether G can be transformed into a disjoint union of cliques by applying at most k edge modifications. In the field of parameterized algorithmics, Cluster Editing has almost exclusively been studied parameterized by the solution size k. Contrastingly, in many real-world instances it can be observed that the parameter k is not really small. This observation motivates our investigation of parameterizations of Cluster Editing different from the solution size k. Our results are as follows. Cluster Editing is fixedparameter tractable with respect to the parameter “size of a minimum cluster vertex deletion set of G”, a typically much smaller parameter than k. Cluster Editing remains NP-hard on graphs with maximum degree six. A restricted but practically relevant version of Cluster Editing is fixed-parameter tractable with respect to the combined parameter “number of clusters in the target graph” and “maximum number of modified edges incident to any vertex in G”. Many of our results also transfer to the NP-hard Cluster Deletion problem, where only edge deletions are allowed