We investigate the parameterized complexity of Vertex Cover parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an O * ((2.618) k ) algorithm for the problem. Here k is the excess of the vertex cover size over the LP optimum, and we write O * (f (k)) for a time complexity of the form O(f (k)n O(1) ), where f (k) grows exponentially with k. We proceed to show that a more sophisticated branching algorithm achieves a runtime of O * (2.3146 k ).Following this, using known and new reductions, we give O * (2.3146 k ) algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion and Almost 2-SAT, and an O * (1.5214 k ) algorithm for Konig Vertex Deletion, Vertex Cover Param by OCT and Vertex Cover Param by KVD. These algorithms significantly improve the best known bounds for these problems. The most notable improvement is the new bound for Odd Cycle Transversal -this is the first algorithm which beats the dependence on k of the seminal O * (3 k ) algorithm of Reed, Smith and Vetta. Finally, using our algorithm, we obtain a kernel for the standard parameterization of Vertex Cover with at most 2k − O(log k) vertices. Our kernel is simpler than previously known kernels achieving the same size bound. Vertex CoverInstance: An undirected graph G and a positive integer k. Parameter: k.Problem: Does G have a vertex cover of of size at most k?We start with a few basic definitions regarding parameterized complexity. For decision problems with input size n, and a parameter k, the goal in parameterized complexity is to design an algorithm with runtime f (k)n O(1) where f is a function of k alone, as contrasted with a trivial n k+O(1) algorithm. Problems which admit such algorithms are said to be fixed parameter tractable (FPT). The theory of parameterized complexity was developed by Downey and Fellows [6]. For recent developments, see the book by Flum and Grohe [7].Vertex Cover was one of the first problems that was shown to be FPT [6]. After a long race, the current best algorithm for Vertex Cover runs in time O(1.2738 k + kn) [3]. However, when k < m, the size of the maximum matching, the Vertex Cover problem is not interesting, as the answer is trivially NO. Hence, when m is large (for example when the graph has a perfect matching), the running time bound of the standard FPT algorithm is not practical, as k, in this case, is quite large. This led to the following natural "above guarantee" variant of the Vertex Cover problem. Above Guarantee Vertex Cover (agvc)Instance: An undirected graph G, a maximum matching M and a positive integer k. Parameter: k − |M |.Problem: Does G have a vertex cover of of size at most k?In addition to being a natural parameterization of the classical Vertex Cover pr...

In this paper we propose a new framework for analyzing the performance of preprocessing algorithms. Our framework builds on the notion of kernelization from parameterized complexity. However, as opposed to the original notion of kernelization, our definitions combine well with approximation algorithms and heuristics. The key new definition is that of a polynomial size α-approximate kernel. Loosely speaking, a polynomial size α-approximate kernel is a polynomial time pre-processing algorithm that takes as input an instance (I, k) to a parameterized problem, and outputs another instance (I , k ) to the same problem, such that |I | + k ≤ k O(1) . Additionally, for every c ≥ 1, a c-approximate solution s to the pre-processed instance (I , k ) can be turned in polynomial time into a (c · α)-approximate solution s to the original instance (I, k).Our main technical contribution are α-approximate kernels of polynomial size for three problems, namely Connected Vertex Cover, Disjoint Cycle Packing and Disjoint Factors. These problems are known not to admit any polynomial size kernels unless NP ⊆ coNP/Poly. Our approximate kernels simultaneously beat both the lower bounds on the (normal) kernel size, and the hardness of approximation lower bounds for all three problems. On the negative side we prove that Longest Path parameterized by the length of the path and Set Cover parameterized by the universe size do not admit even an α-approximate kernel of polynomial size, for any α ≥ 1, unless NP ⊆ coNP/Poly. In order to prove this lower bound we need to combine in a non-trivial way the techniques used for showing kernelization lower bounds with the methods for showing hardness of approximation.

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The \sc Colorful Motif} problem asks if, given a vertex-colored graph G, there exists a subset S of vertices of G such that the graph induced by G on S is connected and contains every color in the graph exactly once. The problem is motivated by applications in computational biology and is also well-studied from the theoretical point of view. In particular, it is known to be NP-complete even on trees of maximum degree three~[Fellows et al, ICALP 2007]. In their pioneering paper that introduced the color-coding technique, Alon et al.~[STOC 1995] show, {\em inter alia}, that the problem is FPT on general graphs. More recently, Cygan et al.~[WG 2010] showed that {\sc Colorful Motif} is NP-complete on {\em comb graphs}, a special subclass of the set of trees of maximum degree three. They also showed that the problem is not likely to admit polynomial kernels on forests. We continue the study of the kernelization complexity of the {\sc Colorful Motif problem restricted to simple graph classes. Surprisingly, the infeasibility of polynomial kernelization persists even when the input is restricted to comb graphs. We demonstrate this by showing a simple but novel composition algorithm. Further, we show that the problem restricted to comb graphs admits polynomially many polynomial kernels. To our knowledge, there are very few examples of problems with many polynomial kernels known in the literature. We also show hardness of polynomial kernelization for certain variants of the problem on trees; this rules out a general class of approaches for showing many polynomial kernels for the problem restricted to trees. Finally, we show that the problem is unlikely to admit polynomial kernels on another simple graph class, namely the set of all graphs of diameter two. As an application of our results, we settle the classical complexity of \cds{} on graphs of diameter two --- specifically, we show that it is \NPC

A vertex-subset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions Ss and St of size k, whether it is possible to transform Ss into St by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k.We study reconfiguration variants of two classical vertex-subset problems, namely Independent Set and Dominating Set. We denote the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete on graphs of bounded bandwidth and W[1]-hard parameterized by k on general graphs. We show that ISR is fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that ISR is fixed-parameter tractable on planar graphs and graphs of bounded degree. For DSR, we show the problem fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes graphs of bounded degeneracy and nowhere-dense graphs.