Locally computing edge orientations

We consider the question of orienting the edges in a graph $G$ such that every vertex has bounded out-degree. For graphs of arboricity $α$, there is an orientation in which every vertex has out-degree at most $α$ and, moreover, the best possible maximum out-degree of an orientation is at least $α- 1$. We are thus interested in algorithms that can achieve a maximum out-degree of close to $α$. A widely studied approach for this problem in the distributed algorithms setting is a ``peeling algorithm'' that provides an orientation with maximum out-degree $α(2+ε)$ in a logarithmic number of iterations. We consider this problem in the local computation algorithm (LCA) model, which quickly answers queries of the form ``What is the orientation of edge $(u,v)$?'' by probing the input graph. When the peeling algorithm is executed in the LCA setting by applying standard techniques, e.g., the Parnas-Ron paradigm, it requires $Ω(n)$ probes per query on an $n$-vertex graph. In the case where $G$ has unbounded degree, we show that any LCA that orients its edges to yield maximum out-degree $r$ must use $Ω(\sqrt n/r)$ probes to $G$ per query in the worst case, even if $G$ is known to be a forest (that is, $α=1$). We also show several algorithms with sublinear probe complexity when $G$ has unbounded degree. When $G$ is a tree such that the maximum degree $Δ$ of $G$ is bounded, we demonstrate an algorithm that uses $Δn^{1-\log_Δr + o(1)}$ probes to $G$ per query. To obtain this result, we develop an edge-coloring approach that ultimately yields a graph-shattering-like result. We also use this shattering-like approach to demonstrate an LCA which $4$-colors any tree using sublinear probes per query.