Constant-Time Approximation Algorithms for the Optimum Branching Problem on Sparse Graphs

We propose constant-time algorithms for approximating the weight of the maximum weight branching in the general graph model. A directed graph is called a branching if it is acyclic and each vertex has at most one incoming edge. An edge-weighted digraph G, in which weights are given in real values in [0, 1], of average degree d is given as an oracle access, and we are allowed to ask degrees and incoming edges for every vertex through the oracle. Then, with high probability, our algorithm estimates the weight of the maximum weight branching in G with an absolute error of at most εn with query complexity O(d/ε³), where n is the number of vertices. We also show a lower bound of Ω(d/ε²). Additionally, our algorithm can be modified to run with query complexity O(1/ε⁴) for unweighted digraphs, i.e., it runs in time independent of the input size even for digraphs with Ω(n²) edges. In contrast, we show that it requires Ω(n) queries to approximate the weight of the minimum (or maximum) spanning arborescence in a weighted digraph.