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What if anything is known about the parameterized complexity of computing the intersection number of a graph (the smallest number of cliques needed to cover all its edges)?

It has long been known to be NP-complete, and it's obviously FPT because it has a kernel: if you can cover a graph with $k$ cliques then there are at most $2^k$ different closed neighborhoods of vertices (two vertices have the same neighborhoods if they belong to the same set of cliques), and you might as well keep only one vertex per neighborhood. Is this observation in the literature somewhere? What kind of dependence on $k$ is known?

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The problem has been studied under the name Edge Clique Cover, and the observations you make regarding data reduction have been used to get a kernel with 2^k vertices. It is a long standing open problem whether a polynomial kernel exists. I don't know about good bounds on the running time, see http://theinf1.informatik.uni-jena.de/publications/clique-cover-jea07.pdf

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    $\begingroup$ Evidently a polynomial kernel is infeasible, according to some fairly recent developments: arxiv.org/abs/1111.0570 $\endgroup$
    – Neeldhara
    Nov 16 '11 at 12:24
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Answering my own question, there's now a preprint on arXiv showing that double-exponential is the correct dependence, assuming the exponential time hypothesis. See "Known algorithms for EDGE CLIQUE COVER are probably optimal", Marek Cygan, Marcin Pilipczuk, and Michał Pilipczuk, arXiv:1203.1754 and SODA 2013

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