By a tight tour in a $k$-uniform hypergraph $H$ we mean any sequence of its vertices $(w_0,w_1,\ldots,w_{s-1})$ such that for all $i=0,\ldots,s-1$ the set $e_i=\{w_i,w_{i+1}\ldots,w_{i+k-1}\}$ is an edge of $H$ (where operations on indices are computed modulo $s$) and the sets $e_i$ for $i=0,\ldots,s-1$ are pairwise different. A tight tour in $H$ is a tight Euler tour if it contains all edges of $H$. We prove that the problem of deciding if a given $3$-uniform hypergraph has a tight Euler tour is NP-complete, and that it cannot be solved in time $2^{o(m)}$ (where $m$ is the number of edges in the input hypergraph), unless the ETH fails. We also present an exact exponential algorithm for the problem, whose time complexity matches this lower bound, and the space complexity is polynomial. In fact, this algorithm solves a more general problem of computing the number of tight Euler tours in a given uniform hypergraph.
Discrete Mathematics & Theoretical Computer Science