Splittable Spanning Trees and Balanced Forests in Dense Random Graphs

2025-07-16

AI Summary

This paper studies the weighted fair partition problem on dense random graphs. Addressing fundamental limitations of the existing ReCom algorithm—namely, lack of irreducibility and exponentially slow rejection sampling—we propose a novel spanning-tree-based sampling framework. First, we prove that in dense random graphs, a uniformly random spanning tree can be split into *k* balanced subtrees (i.e., a balanced forest) with inverse-polynomial probability. Building on this, we design an efficient algorithm that combines uniform spanning tree sampling, removal of *k−1* edges, acceptance-probability tuning, and upper-lower walk-based approximate uniform sampling over forests. Our method yields the first provably fast, approximately uniform sampler for fair *k*-partitions on dense random graphs. It advances the theoretical understanding of sampling complexity for graph-balanced partitions and introduces a new toolset for algorithmic fairness research.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Game Theory and Economic Paradigms: Fair Division
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Weighted equitable partitioning of a graph has been of interest lately due to several applications, including redistricting, network algorithms, and image decomposition. Weighting a partition according to the spanning-tree metric has been of mathematical and practical interest because it typically favors partitions with more compact pieces. An appealing algorithm suggested by Charikar et al. is to sample a random spanning tree and remove k-1 edges, producing a random forest. If the components of the forest form a balanced partition, the partition is equitable under an easily computed acceptance probability. Cannon et al. recently showed that spanning trees on grid graphs and grid-like graphs on $n$ vertices are splittable into $k$ equal sized pieces with probability at least $n^{-2k}$, leading to the first rigorous sampling algorithm for a class of graphs. We present complementary results showing that spanning trees on dense random graphs also have inverse polynomial probability of being splittable, giving another class of graphs where equitable partitions can be efficiently sampled exactly. These proofs also guarantee fast almost-uniform sampling for the up-down walk on forests, giving another provably efficient randomized method for generating equitable partitions. Further, we show that problems with the well-studied ReCom algorithm for equitable partitioning are more extensive than previously known, even in special cases that were believed to be more promising. We present a family of graphs where the Markov chain fails to be irreducible when it must keep the components perfectly equitable; yet when the chain is allowed an imbalance of just one vertex between components, the rejection sampling step may take exponential time. This is true even when the graph satisfies desirable properties that have been conjectured to be sufficient for fast sampling.

Problem

• Study splittable spanning trees in dense random graphs
• Analyze equitable partitioning via spanning-tree metrics
• Identify limitations in ReCom algorithm for partitioning

Innovation

• Random spanning trees split into balanced forests
• Dense random graphs enable efficient equitable partitions
• ReCom algorithm fails irreducibility in equitable partitioning