Kernelization Lower Bounds Through Colors and IDs

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 nonexistence of polynomial kernels for specific problems has been developed by Bodlaender et al. [2009] and Fortnow and Santhanam [2008]. With few exceptions, all known kernelization lower bounds results have been obtained by directly applying this framework. In this article, we show how to combine these results with combinatorial reductions that use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems. To follow 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 with at most 2k vertices.

—Alon and Gutner [2008] 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. 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 [2007] obtain a “compression algorithm” for the Sparse Subset Sum problem. 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 previously was an open problem explicitly stated in the literature [Alon and Gutner 2008; Betzler 2006; Guo and Niedermeier 2007; Guo et al. 2007; Moser et al. 2007]. Many of our results also rule out the existence of compression algorithms, a notion similar to kernelization defined by Harnik and Naor [2007], for the problems in question.