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Suppose we have a multigraph (later, a multihypergraph). An edge-clique is a set of edges which all pairwise intersect (have at least one common vertex). Then any edge-clique $C$ in a multigraph always falls into one of two categories:

  • A star: there is a vertex such that every edge of $C$ contains it
  • A triangle: there are three vertices such that every edge of $C$ goes between two of them

This leads to an easy $O(n^3)$-time algorithm to compute the largest edge-clique.

I am pretty sure one can show more generally, that for every $r$, in multihypergraphs with maximum edge size $r$, you can prove a certain structure theorem for hyperedge-cliques, and get a polynomial time algorithm to find the maximum clique.

Is there anything related to this result known? Also, the algorithm I have in mind is extremely high degree polynomial; it would be nice to get something with running time $n^{\mathrm{poly}(r)}$ or better.

I found this interesting since the maximum edge-clique is a lower bound on the edge-chromatic number (a.k.a chromatic index).

Edit: In the cross-post, the reference about kernels leads to a $2^{2^{\mathrm{exp}(r)}}n^{\mathrm{exp}(r)}$-time algorithm: guess the kernel and guess the restriction of the clique to the kernel.

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You are almost at extremal set theory, e.g. intersecting systems. Check out Cameron's "Combinatorics" and references therein. I'm not sure about the algorithmics, though. –  RJK Oct 1 '10 at 12:34
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Cross-post asking for a structure theorem: mathoverflow.net/questions/41123/cliques-of-hyperedges –  daveagp Oct 5 '10 at 10:28
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1 Answer

Every graph G is a line graph of some hypergraph, for instance the one that has the edges of G as its vertices and the sets of edges neighboring each vertex of G as its hyperedges. Therefore, finding cliques in line graphs of hypergraphs can be no easier than finding cliques in arbitrary graphs.

Of course with your limitation of edge size $r$, one can only do this starting from a graph $G$ with degree $r$, and max clique in bounded degree graphs is not so hard, so this doesn't rule out the $n^{poly(r)}$ algorithm you're hoping for.

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