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 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 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 subset of 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 This was first observed in unpublished work of Chen and Muller, and it is an explicit conclusion of recent results by Dell and van Melkebeek (STOC 2010).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 G and an integer k, the question is whether G contains an independent set or a clique of size at least k. It was asked by Rod Downey whether this problem admits a polynomial kernelization with respect to 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 H. The crux is that the host graph H needs to observe a bound of l is an element of O(logt) 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 t; the co-nondeterministic composition is built around the search for such graphs. Thus, we show that, unless NP c coNP/poly, the problem does not admit a kernelization with polynomial size guarantee.
