Introvert Clustering for Distributed Graph Algorithms

We introduce a graph decomposition primitive called introvert clustering, which strengthens standard low-diameter clustering by guaranteeing that every clustered vertex keeps at least a (1/2-ε)-fraction of its relevant neighbors in its own cluster. Repeatedly applying this primitive yields a layered introvert network decomposition with O(log n) layers and weak diameter O(log n). We give two applications in the 𝖫𝖮𝖢𝖠𝖫 model. For every constant ε>0, we obtain a O(log^2 n)-round deterministic algorithm for list (3/2+ε)Δ-edge coloring on graphs of maximum degree Δ_0(ε); for bipartite graphs, the result holds for all Δ. For every constant 0<ε<1/4, we also obtain a O(log^2 n)-round deterministic algorithm for a (1/4-ε)-locally balanced cut, where every vertex has at least a (1/4-ε)-fraction of its neighbors on the opposite side. The resulting algorithms are remarkably simple: edge coloring processes the layers in reverse order and colors each cluster, while locally balanced cut processes them forward and computes a locally maximum cut within each cluster. The introvert guarantee enables these procedures beyond the usual greedy regime of network decomposition. We construct the decomposition in O(log^2 n) randomized rounds using Miller–Peng–Xu low-diameter clustering and a simple trimming procedure, and deterministically in O(log^2 n) rounds via a white-box adaptation of the recursive network decomposition algorithm of Ghaffari and Grunau [FOCS 2024].