Short hypercycles require brute force under hyperclique hypotheses, longer cycles admit matrix-multiplication speedups, and counting subhypergraphs or database query answers on random data is as hard as the corresponding worst-case counting.
Tight cycles in hypergraphs
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Worst-Case and Average-Case Hardness of Hypercycle and Database Problems
Short hypercycles require brute force under hyperclique hypotheses, longer cycles admit matrix-multiplication speedups, and counting subhypergraphs or database query answers on random data is as hard as the corresponding worst-case counting.