In parameterized complexity each problem instance comes with a parameter k, and a parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the non-existence of polynomial kernels for specific problems has been developed by Bodlaender et al. [6] and Fortnow and Santhanam [17]. With few exceptions, all known kernelization lower bounds result have been obtained by directly applying this framework. In this paper we show how to combine these results with combinatorial reductions which use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems. Below we give a summary of our main results. All results are under the assumption that the polynomial hierarchy does not collapse to the third level.• We show that the STEINER TREE problem parameterized by the number of terminals and solution size k, and the CONNECTED VERTEX COVER and CAPACITATED VERTEX COVER problems do not admit a polynomial kernel. The two latter results are surprising because the closely related VERTEX COVER problem admits a kernel of with at most 2k vertices.• Alon and Gutner obtain a k poly(h) kernel for DOMINATING SET IN H -MINOR FREE GRAPHS parameterized by h = |H| and solution size k, and ask whether kernels of smaller size exist [3]. We partially resolve this question by showing that DOMINATING SET IN H -MINOR FREE GRAPHS does not admit a kernel with size polynomial in k + h.• Harnik and Naor obtain a "compression algorithm" for the SPARSE SUBSET SUM problem [21]. We show that their algorithm is essentially optimal by showing that the instances cannot be compressed further.• The HITTING SET and SET COVER problems are among the most studied problems in algorithmics. Both problems admit a kernel of size k O(d) when parameterized by solution size k and maximum set size d. We show that neither of them, along with the UNIQUE COVERAGE and BOUNDED RANK DISJOINT SETS problems, admits a polynomial kernel.The existence of polynomial kernels for several of the problems mentioned above were open problems explicitly stated in the literature [3,4,19,20,26]. Many of our results also rule out the existence of compression algorithms, a notion similar to kernelization defined by Harnik and Naor [21], for the problems in question.

In parameterized complexity each problem instance comes with a parameter k, and a parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the non-existence of polynomial kernels for specific problems has been developed by Bodlaender et al. [6] and Fortnow and Santhanam [17]. With few exceptions , all known kernelization lower bounds result have been obtained by directly applying this framework. In this paper we show how to combine these results with combinatorial reductions which use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems. Below we give a summary of our main results. All results are under the assumption that the polynomial hierarchy does not collapse to the third level. • We show that the STEINER TREE problem parameterized by the number of terminals and solution size k, and the CONNECTED VERTEX COVER and CAPACITATED VERTEX COVER problems do not admit a polynomial kernel. The two latter results are surprising because the closely related VERTEX COVER problem admits a kernel of with at most 2k vertices. • Alon and Gutner obtain a k poly(h) kernel for DOMINATING SET IN H-MINOR FREE GRAPHS parameterized by h = |H| and solution size k, and ask whether kernels of smaller size exist [3]. We partially resolve this question by showing that DOMINATING SET IN H-MINOR FREE GRAPHS does not admit a kernel with size polynomial in k + h. • Harnik and Naor obtain a "compression algorithm" for the SPARSE SUBSET SUM problem [21]. We show that their algorithm is essentially optimal by showing that the instances cannot be compressed further. • The HITTING SET and SET COVER problems are among the most studied problems in algorithmics. Both problems admit a kernel of size k O(d) when parameterized by solution size k and maximum set size d. We show that neither of them, along with the UNIQUE COVERAGE and BOUNDED RANK DISJOINT SETS problems, admits a polynomial kernel. The existence of polynomial kernels for several of the problems mentioned above were open problems explicitly stated in the literature [3, 4, 19, 20, 26]. Many of our results also rule out the existence of compression algorithms, a notion similar to kernelization defined by Harnik and Naor [21], for the problems in question.

Capacitated versions of Dominating Set and Vertex Cover have been studied intensively in terms of polynomial time approximation algorithms. Although the problems Dominating Set and Vertex Cover have been subjected to considerable scrutiny in the parameterized complexity world, this is not true for the capacitated versions. Here we make an attempt to understand the behavior of the problems Capacitated Dominating Set and Capacitated Vertex Cover from the perspective of parameterized complexity.The original versions of these problems, Vertex Cover and Dominating Set, are known to be fixed parameter tractable when parameterized by a structure of the graph called the treewidth (tw). In this paper we show that the capacitated versions of these problems behave differently. Our results are:• Capacitated Dominating Set is W[1]-hard when parameterized by treewidth. In fact, Capacitated Dominating Set is W[1]-hard when parameterized by both treewidth and solution size k of the capacitated dominating set.• Capacitated Vertex Cover is W[1]-hard when parameterized by treewidth.• Capacitated Vertex Cover can be solved in time 2 O(tw log k) n O(1) where tw is the treewidth of the input graph and k is the solution size. As a corollary, we show that the weighted version of Capacitated Vertex Cover in general graphs can be solved in time 2 O(k log k) n O(1) . This improves the earlier algorithm of Guo et al.[15] running in time O(1.2We would also like to point out that our W[1]-hardness result for Capacitated Vertex Cover, when parameterized by treewidth, makes it (to the best of our knowledge) the first known "subset problem" which has turned out to be fixed parameter tractable when parameterized by solution size but W[1]-hard when parameterized by treewidth.

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Complementing recent progress on classical complexity and polynomial-time approximability of feedback set problems in (bipartite) tournaments, we extend and improve fixedparameter tractability results for these problems. We show that Feedback Vertex Set in tournaments (FVST) is amenable to the novel iterative compression technique, and we provide a depth-bounded search tree for Feedback Arc Set in bipartite tournaments based on a new forbidden subgraph characterization. Moreover, we apply the iterative compression technique to d-Hitting Set, which generalizes Feedback Vertex Set in tournaments, and obtain improved upper bounds for the time needed to solve 4-Hitting Set and 5-Hitting Set. Using our parameterized algorithm for Feedback Vertex Set in tournaments, we also give an exact (not parameterized) algorithm for it running in O (1.709 n ) time, where n is the number of input graph vertices, answering a question of Woeginger [G.J. Woeginger, Open problems around exact algorithms, Discrete Appl. Math. 156 (3) (2008) 397-405].