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

The input to the NP-hard Point Line Cover problem (PLC) consists of a set P of n points on the plane and a positive integer k, and the question is whether there exists a set of at most k lines which pass through all points in P. By straightforward reduction rules one can efficiently reduce any input to one with at most k 2 points. We show that this easy reduction is already essentially tight under standard assumptions. More precisely, unless the polynomial hierarchy collapses to its third level, for any ε > 0, there is no polynomial-time algorithm that reduces every instance (P, k) of PLC to an equivalent instance with O(k 2−ε ) points. This answers, in the negative, an open problem posed by Lokshtanov (PhD Thesis, 2009).Our proof uses the notion of a kernel from parameterized complexity, and the machinery for deriving lower bounds on the size of kernels developed by Dell and van Melkebeek (STOC 2010). It has two main ingredients: We first show, by reduction from Vertex Cover, that-unless the polynomial hierarchy collapses-PLC has no kernel of total size O(k 2−ε ) bits. This does not directly imply the claimed lower bound on the number of points, since the best known polynomial-time encoding of a PLC instance with n points requires ω(n 2 ) bits. To get around this hurdle we build on work of Goodman, Pollack and Sturmfels (STOC 1989) and devise an oracle communication protocol of cost O(n log n) for PLC; its main building blocks are a bound of O(n O(n) ) for the order types of n points that are not necessarily in general position and an explicit (albeit slow) algorithm that enumerates a superset of size n O(n) of all possible order types of n points. This protocol, together with the lower bound on the total size (which also holds for such protocols), yields the stated lower bound on the number of points.While a number of essentially tight polynomial lower bounds on total sizes of kernels are known, our result is-to the best of our knowledge-the first to show a nontrivial lower bound for structural/secondary parameters.

We show that for any fixed j ≥ i ≥ 1, the k-DOMINATING SET problem restricted to graphs that do not have K i, j (the complete bipartite graph on (i + j) vertices, where the two parts have i and j vertices, respectively) as a subgraph is fixed parameter tractable (FPT) and has a polynomial kernel. We describe a polynomial-time algorithm that, given a K i, j -free graph G and a nonnegative integer k, constructs a graph H (the "kernel") and an integer k ′ such that (1) G has a dominating set of size at most k if and only if H has a dominating set of size at most k ′ , (2) H has The most general class of graphs for which a polynomial kernel was previously known for k-DOMINATING SET is the class of K h -topologicalminor-free graphs [21]. Graphs of bounded degeneracy are the most general class of graphs for which an FPT algorithm was previously known for this problem. K h -topological-minor-free graphs are K h,h -free (but not vice versa), and so our results show that k-DOMINATING SET has both FPT algorithms and polynomial kernels in more general classes of graphs.Using the same techniques, we also obtain an O jk i vertex-kernel for the k-INDEPENDENT DOMINATING SET problem on K i, j -free graphs.

The standard parameterization of the Vertex Cover problem (Given an undirected graph G and k ∈ N as input, does G have a vertex cover of size at most k?) has the solution size k as the parameter. The following more challenging parameterization of Vertex Cover stems from the observation that the size MM of a maximum matching of G lower-bounds the size of any vertex cover of G: Does G have a vertex cover of size at most MM + kμ? The parameter is the excess kμ of the solution size over the lower bound MM.Razgon TALG 2014). The last two bounds were in fact proven for a different parameter: namely, the excess k λ of the solution size over LP , the value of the linear programming relaxation of the standard LP formulation of Vertex Cover. Since LP ≥ MM for any graph, we have that k λ ≤ kμ for Yes instances. This is thus a stricter parameterization-the new parameter is, in general, smaller-and the running times carry over directly to the parameter kμ.We investigate an even stricter parameterization of Vertex Cover, namely the excessk of the solution size over the quantity (2LP − MM). We ask: Given a graph G and k ∈ N as input, does G have a vertex cover of size at most (2LP − MM) +k? The parameter isk. It can be shown that (2LP − MM) is a lower bound on vertex cover size, and since LP ≥ MM we have that (2LP − MM) ≥ LP , and hence thatk ≤ k λ holds for Yes instances. Further, (k λ −k) could be as large as (LP − MM) and-to the best of our knowledge-this difference cannot be expressed as a function of k λ alone. These facts motivate and justify our choice of parameter: this is indeed a stricter parameterization whose tractability does not follow directly from known results.We show that Vertex Cover is fixed-parameter tractable for this stricter parameterk: We derive an algorithm which solves Vertex Cover in time O (3k), thus pushing the envelope further on the parameterized tractabil- * Work done at the

When modeling an application of practical relevance as an instance of a combinatorial problem X, we are often interested not merely in finding one optimal solution for that instance, but in finding a sufficiently diverse collection of good solutions. In this work we initiate a systematic study of diversity from the point of view of fixed-parameter tractability theory. We consider an intuitive notion of diversity of a collection of solutions which suits a large variety of combinatorial problems of practical interest. Our main contribution is an algorithmic framework which --automatically-- converts a tree-decomposition-based dynamic programming algorithm for a given combinatorial problem X into a dynamic programming algorithm for the diverse version of X. Surprisingly, our algorithm has a polynomial dependence on the diversity parameter.