Algorithms for determining the existence of subdesigns in a combinatorial design are examined. When λ=1, the existence of a subdesign of order d in a design of order v can be determined in O(vlogd time. The order of the smallest subdesign can be computed in polynomial time. In addition, determining whether a design has a subdesign of maximal possible order (a “head”) requires polynomial time. When λ>1, the problems are apparently significantly more difficult: we show that deciding whether a block design has any non-trivial subdesign is NP-complete.
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