Kernel Lower Bounds using Co-Nondeterminism: Finding Induced Hereditary Subgraphs

This work further explores the applications of co-nondeterminism for showing kernelization lower bounds. The only known example prior to this work excludes polynomial kernelizations for the so-called Ramsey problem of finding an independent set or a clique of at least k vertices in a given graph [Kratsch 2012]. We study the more general problem of finding induced subgraphs on k vertices fulfilling some hereditary property Π, called Π-Induced Subgraph. The problem is NP-hard for all nontrivial choices of Π by a classic result of Lewis and Yannakakis [1980]. The parameterized complexity of this problem was classified by Khot and Raman [2002] depending on the choice of Π. The interesting cases for kernelization are for Π containing all independent sets and all cliques, since the problem is trivially polynomial time solvable or W[1]-hard otherwise.

Our results are twofold. Regarding Π-Induced Subgraph, we show that for a large choice of natural graph properties Π, including chordal, perfect, cluster, and cograph, there is no polynomial kernel with respect to k. This is established by two theorems, each one capturing different (but not necessarily exclusive) sets of properties: one using a co-nondeterministic variant of OR-cross-composition and one by a polynomial parameter transformation from Ramsey.

Additionally, we show how to use improvement versions of NP-hard problems as source problems for lower bounds, without requiring their NP-hardness. For example, for Π-Induced Subgraph our compositions may assume existing solutions of size k--1. This follows from the more general fact that source problems for OR-(cross-)compositions need only be NP-hard under co-nondeterministic reductions. We believe this to be useful for further lower-bound proofs, for example, since improvement versions simplify the construction of a disjunction (OR) of instances required in compositions. This adds a second way of using co-nondeterminism for lower bounds.