Incompressibility through Colors and IDs

Abstract

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.

A Multivariate Complexity Analysis of Lobbying in Multiple Referenda

“…To prove our conditional non-existence result for Lobbying parameterized by (m, k), we employ an "incompressibility result" for Set Cover due to the work of Dom, Lokshtanov, and Saurabh (2009).…”

“…Dom et al (2009) showed that, unless NP ⊆ coNP/poly, Set Cover (SC) does not admit a polynomial kernel with respect to the combined parameter (|U|, h). To show that this result transfers to Lobbying with respect to (m, k) we describe a polynomial time and parameter transformation from SC with respect to (|U|, h).…”

“…Since Lobbying is contained in NP and since SC is NP-hard, a polynomial kernel for Lobbying with respect to (m, k) would imply a polynomial kernel for SC with respect to (|U|, h) which is not possible unless NP ⊆ coNP/poly (Dom et al, 2009). …”

“…Dom et al (2009) showed that, unless NP ⊆ coNP/poly, HS does not admit a polynomial kernel with respect to |U H |.…”

On Problem Kernels for Possible Winner Determination under the k-Approval Protocol

“…Dom et al [9] provide a framework to build composition algorithms by using "identifiers". One of the necessary conditions is the existence of an algorithm running in 2 p γ · poly time for the considered parameter p and a fixed constant γ.…”