Balanced allocation on hypergraphs

We consider a variation of balls-into-bins which randomly allocates m balls into n bins. Following Godfrey's model (SODA, 2008), we assume that each ball t, $1⩽t⩽m$, comes with a hypergraph $H^{\left(t\right)}=\{B_1,B_2,\dots ,B_{s_t}\}$, and each edge $B\in H^{\left(t\right)}$ contains at least a logarithmic number of bins. Given $d⩾2$, our d-choice algorithm chooses an edge $B\in H^{\left(t\right)}$, uniformly at random, and then chooses a set D of d random bins from the selected edge B. The ball is allocated to a least-loaded bin from D. We prove that if the hypergraphs $H^{\left(1\right)},\dots ,H^{\left(m\right)}$ satisfy a balancedness condition and have low pair visibility, then after allocating $m=\Theta \left(n\right)$ balls, the maximum load of any bin is at most ${\log }_d\log n+O\left(1\right)$, with high probability. Moreover, we establish a lower bound for the maximum load attained by the balanced allocation for a sequence of hypergraphs in terms of pair visibility.