NDLI: Co-Nondeterminism in Compositions: A Kernelization Lower Bound for a Ramsey-Type Problem

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Co-Nondeterminism in Compositions: A Kernelization Lower Bound for a Ramsey-Type Problem

Content Provider	ACM Digital Library
Author	Kratsch, Stefan
Copyright Year	2014
Abstract	The field of kernelization offers a rigorous way of studying the ubiquitous technique of data reduction and preprocessing for combinatorially hard problems. A widely accepted definition of useful data reduction is that of a polynomial kernelization where the output instance is guaranteed to be of size polynomial in some parameter of the input. The fairly recent development of a framework for kernelization lower bounds has made this notion even more attractive as we can now classify many problems into admitting or not admitting polynomial kernelizations. The central notion of the framework is that of a polynomial-time composition algorithm due to Bodlaender et al. (ICALP 2008, JSCC 2009): given $\textit{t}$ input instances, an or-composition algorithm returns a single-output instance with bounded parameter value that is yes if and only if one of $\textit{t}$ input instances is yes; it encodes the logical OR of the input instances. Based on a result of Fortnow and Santhanam (STOC 2008, JSCC 2011), Bodlaender et al. show that an or-composition for an NP-hard problem rules out polynomial kernelizations for it unless NP ⊆ coNP/poly (which is known to imply a collapse of the polynomial hierarchy). It is implicit in the work of Fortnow and Santhanam that even co-nondeterministic composition algorithms suffice to rule out polynomial kernelizations. This was first observed in unpublished work of Chen and Müller, and it is an explicit conclusion of recent results by Dell and van Melkebeek (STOC 2010). However, in contrast to the numerous applications of deterministic composition, the added power of co-nondeterminism has not yet been harnessed to obtain kernelization lower bounds. In this work, we present the first example of how co-nondeterminism can help to make a composition algorithm. We study the existence of polynomial kernels for a Ramsey-type problem where, given a graph $\textit{G}$ and an integer $\textit{k},$ the question is whether $\textit{G}$ contains an independent set or a clique of size at least $\textit{k}.$ It was asked by Rod Downey whether this problem admits a polynomial kernelization with respect to $\textit{k};$ such a result would greatly speed up the computation of Ramsey numbers. We provide a co-nondeterministic composition based on embedding t instances into a single host graph $\textit{H}.$ The crux is that the host graph $\textit{H}$ needs to observe a bound of ℓ ∈ $\textit{O}(log$ $\textit{t})$ on both its maximum independent set and maximum clique size, while also having a cover of its vertex set by independent sets and cliques all of size ℓ; the co-nondeterministic composition is built around the search for such graphs. Thus, we show that, unless NP ⊆ coNP/poly, the problem does not admit a kernelization with polynomial size guarantee.
Starting Page	1
Ending Page	16
Page Count	16
File Format	PDF
ISSN	15496325
e-ISSN	15496333
DOI	10.1145/2635808
Volume Number	10
Issue Number	4
Journal	ACM Transactions on Algorithms (TALG)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2014-08-11
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Kernelization Ramsey numbers Nondeterminism Parameterized complexity
Content Type	Text
Resource Type	Article
Subject	Mathematics

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