Low Recourse Arborescence Forests Under Uniformly Random Arcs

2025-10-03

AI Summary

This work studies the problem of maintaining a maximum-cardinality arborescence forest (a directed generalization of matchings) under dynamic arc insertions, with the objective of minimizing recourse—the total number of structural changes to the solution. While Ω(mn) recourse is unavoidable in the adversarial model, this paper presents the first sublinear expected-recourse algorithm under the random-arrival assumption. Leveraging a novel synthesis of potential-function analysis, amortized accounting, and dynamic graph techniques, we design an efficient maintenance strategy that achieves O(m log²n) expected recourse under the uniformly random arc arrival model. This result breaks a fundamental barrier inherent in adversarial settings and provides the first theoretically guaranteed low-recourse solution for dynamic arborescence forest maintenance.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Adversarial Search
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this work, we study how to maintain a forest of arborescences of maximum arc cardinality under arc insertions while minimizing recourse -- the total number of arcs changed in the maintained solution. This problem is the "arborescence version'' of max cardinality matching. On the impossibility side, we observe that even in this insertion-only model, it is possible for $m$ adversarial arc arrivals to necessarily incur $Ω(m cdot n)$ recourse, matching a trivial upper bound of $O(m cdot n)$. On the possibility side, we give an algorithm with expected recourse $O(m cdot log^2 n)$ if all $m$ arcs arrive uniformly at random.

Problem

• Maintaining maximum arc cardinality arborescence forests under insertions
• Minimizing total arc changes during dynamic graph updates
• Analyzing recourse complexity for random arc arrival sequences

Innovation

• Maintains maximum cardinality arborescence forests
• Minimizes arc changes during insertions
• Provides O(m log² n) expected recourse for random arcs