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

Content Provider	ACM Digital Library
Author	Rai, Ashutosh Pilipczuk, Marcin Raman, Venkatesh Kratsch, Stefan
Copyright Year	2015
Abstract	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 $\textit{k}$ vertices in a given graph [Kratsch 2012]. We study the more general problem of finding induced subgraphs on $\textit{k}$ 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 $\textit{k}.$ 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 $\textit{co-nondeterministic}$ 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.
Starting Page	1
Ending Page	18
Page Count	18
File Format	PDF
ISSN	19423454
e-ISSN	19423462
DOI	10.1145/2691321
Volume Number	7
Issue Number	1
Journal	ACM Transactions on Computation Theory (TOCT)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2015-01-13
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Kernelization lower bounds Co-nondeterminism Hereditary graph properties
Content Type	Text
Resource Type	Article
Subject	Computational Theory and Mathematics Theoretical Computer Science

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