Maintaining Random Assignments under Adversarial Dynamics

TL;DR

This paper develops advanced resampling schemes to maintain near-random assignments in dynamic graphs under adversarial changes, addressing biases caused by adaptive adversaries and enabling efficient graph colorings and random walk maintenance.

Contribution

It introduces a new 'temporal aggregation' principle and two resampling schemes to counteract biases from adaptive adversaries, improving dynamic graph algorithms.

Findings

Maintains $O(\Delta)$ coloring in general graphs with sublinear work per update.

Achieves $O(\frac{\Delta}{\ln \Delta})$ coloring in triangle-free graphs with efficient updates.

Provides methods for maintaining random walks in dynamic graphs.

Abstract

We study and further develop powerful general-purpose schemes to maintain random assignments under adversarial dynamic changes. The goal is to maintain assignments that are (approximately) distributed similarly as a completely fresh resampling of all assignments after each change, while doing only a few resamples per change. This becomes particularly interesting and challenging when dynamics are controlled by an adaptive adversary. Our work builds on and further develops the proactive resampling technique [Bhattacharya, Saranurak, and Sukprasert ESA'22]. We identify a new ``temporal selection'' attack that adaptive adversaries can use to cause biases, even against proactive resampling. We propose a new ''temporal aggregation'' principle that algorithms should follow to counteract these biases, and present two powerful new resampling schemes based on this principle. We give various applications of our new methods. The main one in maintaining proper coloring of the graph under adaptive adversarial modifications: we maintain $O(\Delta)$ coloring for general graphs with maximum degree $\Delta$ and $O(\frac{\Delta}{\ln \Delta})$ coloring for triangle free graphs, both with sublinear in the number of vertices average work per modification. Other applications include efficiently maintaining random walks in dynamically changing graphs.