Discrete Applied Mathematics - Combinatorics and complexity
Fixed-parameter tractability of graph modification problems for hereditary properties
Information Processing Letters
Parameterized complexity of finding subgraphs with hereditary properties
Theoretical Computer Science
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Incompressibility through Colors and IDs
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
On problems without polynomial kernels
Journal of Computer and System Sciences
Chordal Deletion is Fixed-Parameter Tractable
Algorithmica
Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses
Proceedings of the forty-second ACM symposium on Theory of computing
Infeasibility of instance compression and succinct PCPs for NP
Journal of Computer and System Sciences
On the Compressibility of $\mathcal{NP}$ Instances and Cryptographic Applications
SIAM Journal on Computing
Measuring indifference: unit interval vertex deletion
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Kernel bounds for disjoint cycles and disjoint paths
Theoretical Computer Science
Kernelization of packing problems
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Weak compositions and their applications to polynomial lower bounds for kernelization
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Co-nondeterminism in compositions: a kernelization lower bound for a Ramsey-type problem
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Obtaining a Planar Graph by Vertex Deletion
Algorithmica
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This work further explores the applications of co-nondeterminism for showing kernelization lower bounds. The only known example excludes polynomial kernelizations for the Ramsey problem of finding an independent set or a clique of at least k vertices in a given graph (Kratsch 2012, SODA). 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 non-trivial choices of Π by a classic result of Lewis and Yannakakis (JCSS 1980). The parameterized complexity of this problem was classified by Khot and Raman (TCS 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 trivial 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: one using a co-nondeterministic variant of 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. E.g., for Π-Induced Subgraph our compositions may assume existing solutions of size k−1. We believe this to be useful for further lower bound proofs, 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.