Co-nondeterminism in compositions: a kernelization lower bound for a Ramsey-type problem

Abstract

Until recently, techniques for obtaining lower bounds for kernelization were one of the most sought after tools in the field of parameterized complexity. Now, after a strong influx of techniques, we are in the fortunate situation of having tools available that are even stronger than what has been required in their applications so far. Based on a result of Fortnow and Santhanam (STOC 2008, JCSS 2011), Bodlaender et al. (ICALP 2008, JCSS 2009) showed that, unless NP ⊆ coNP/poly, the existence of a deterministic polynomial-time composition algorithm, i.e., an algorithm which outputs an instance of bounded parameter value which is yes if and only if one of t input instances is yes, rules out the existence of polynomial kernels for a problem. Dell and van Melkebeek (STOC 2010) continued this line of research and, amongst others, were able to rule out kernels of size O(kd−e) for certain problems, assuming NP n coNP/poly. It is an immediate consequence of their work that even the existence of a co-nondeterministic composition rules out polynomial kernels. 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: 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, and 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 e O(log 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 build around the search for such graphs. Thus we show that, unless NP ⊆ coNP/poly (and the polynomial hierarchy collapses), the problem does not admit a kernelization with polynomial size guarantee.