Monotone Properties of k -Uniform Hypergraphs Are Weakly Evasive

Abstract

A Boolean function in n variables is weakly evasive if its decision-tree complexity is Ω( n ). By k -graphs, we mean k -uniform hypergraphs. A k-graph property on v vertices is a Boolean function on n = v k variables corresponding to the k -subsets of a v -set that is invariant under the v ! permutations of the v -set (isomorphisms of k -graphs). In 1976, Rivest and Vuillemin proved that all nonconstant monotone graph properties ( k = 2) are weakly evasive, confirming a conjecture of Aanderaa and Rosenberg in 1973. Then, in 2013, Kulkarni, Qiao, and Sun (KQS) proved the analogous result for 3-graphs. We extend these results to k -graphs for every fixed k . From this, we show that monotone Boolean functions invariant under the action of a large primitive group are weakly evasive. Although KQS employ the powerful topological approach of Kahn et al. in 1984 combined with heavy number theory, our argument is elementary and self-contained (modulo some basic group theory). Inspired by the outline of the KQS approach, we formalize the general framework of “orbit augmentation sequences” of sets with group actions. We show that a parameter of such sequences, called the spacing , is a lower bound on the decision-tree complexity for any nontrivial monotone property that is Γ-invariant for all groups Γ involved in the orbit augmentation sequence, assuming all those groups are p -groups. We develop operations on such sequences such as composition and direct product that will provide helpful machinery for our applications. We apply this general technique to k -graphs via certain liftings of k -graphs with wreath product action of p -groups.