Locally computing edge orientations

TL;DR

This work investigates the problem of orienting graph edges to bound out-degree within the Local Computation Algorithm (LCA) model, focusing on graphs with arboricity $α$ and target out-degree $r$. It establishes a fundamental lower bound of $Ω(n^{1/2}/r)$ probes and delivers sublinear LCAs across regimes: an $O(1)$-probe scheme for very large $r$, a ${\tilde{O}}(α n/r^2)$-probe scheme for medium $r$, and a ${\tilde{O}}(n/r)$-probe scheme for forests; in the bounded-degree forest setting, it achieves $Δ\,n^{1-\log_Δ r+o(1)}$ probes. The paper also develops a graph-shattering-inspired approach to enable sublinear 4-coloring of trees and discusses how these techniques might extend to minor-free graphs. Overall, it advances sublinear-time, locally computable orientations and colorings, illuminating fundamental limits and practical algorithms for low-out-degree orientations in LCAs.

Abstract

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.

Figures (1)

• Figure 1: Graph $G$ used to demonstrate lower bound

Theorems & Definitions (20)

• Theorem 1.1: Rephrasing of \ref{['prop:lower-bound']}
• Theorem 1.2: Rephrasing of \ref{['prop:ub-1', 'prop:ub-2', 'prop:ub-3']}
• Theorem 1.3: Rephrasing of \ref{['thm:ub-bdd']}
• Theorem 1.4
• Theorem 2.1: Yao's minimax principle
• Theorem 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3