Computer Science – Data Structures and Algorithms
Scientific paper
2011-07-19
Computer Science
Data Structures and Algorithms
Scientific paper
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 (JCSS 2011), Bodlaender et al. (JCSS 2009) showed that, unless NP \subseteq 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(k^d-eps) for certain problems, assuming NP !\subseteq coNP/poly. Their work implies that even the existence of a co-nondeterministic composition rules out polynomial kernels. In this work we present the first example of how co-nondeterminism can help to make a composition algorithm. We study 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. 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 \in 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 \subseteq coNP/poly, the problem does not admit a kernelization with polynomial size guarantee.
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