Kernelization Lower Bounds 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.

Sparsification Upper and Lower Bounds for Graph Problems and Not-All-Equal SAT

“…We will prove these sparsification lower bounds using a degree-2 cross-composition, starting from a variation of the Colored Red-Blue Dominating Set problem (Col-RBDS) as described by Dom et alin [9].…”

Parameterized Complexity of Safe Set

“…RDBS becomes trivial when k |R| and thus we assume that k < |R| in what follows. It is known that RBDS parameterized simultaneously by k and |R| does not admit a polynomial kernel unless PH = Σ p 3 [10]. Since RBDS is NP-hard and SS and CSS are in NP, it suffices to present a polynomial-parameter transformation from RBDS parameterized by k + |R| to SS and CSS parameterized by the vertex cover number.…”

Your Rugby Mates Don’t Need to Know Your Colleagues: Triadic Closure with Edge Colors

“…A closer look at the instance (G, c, k, Λ) for VL-Multi-STC constructed from an instance (U, F , t) for Set Cover in the proof of Theorem 5 reveals that c = |F | + 1 and k 1 ≤ |F | − t. It follows that c + k 1 ≤ 2|F | + 1, so the construction is a polynomial-parameter transformation from Set Cover parameterized by |F | to VL-Multi-STC parameterized by (c, k 1 ). By the fact that Set Cover parameterized by |F | does not admit a polynomial kernel unless NP ⊆ coNP/poly [5] we obtain the following.…”