Maintaining Bipartite Colourings on Temporal Graphs on a Budget

2025-11-25

AI Summary

This paper investigates the budget-constrained dynamic bipartite coloring problem on temporal graphs, aiming to maintain a valid bipartition using at most $k$ color changes over time—thereby preventing resource access conflicts and breaking symmetry in dynamic networks. We are the first to extend static bipartite coloring to the temporal setting; we prove that even when every snapshot is bipartite, the problem remains NP-hard and admits no constant-factor approximation. We propose two algorithms: (i) an exact algorithm based on dynamic programming and connected-component tracking, with time complexity $O(T|E|2^k + nT2^{2k})$, and (ii) an efficient approximation algorithm achieving an $O(sqrt{log(nT)})$ approximation ratio. Extensive experiments on real-world temporal networks demonstrate both effectiveness and scalability of our approaches.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

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

Abstract

Graph colouring is a fundamental problem for networks, serving as a tool for avoiding conflicts via symmetry breaking, for example, avoiding multiple computer processes simultaneously updating the same resource. This paper considers a generalisation of this problem to emph{temporal graphs}, i.e., to graphs whose structure changes according to an ordered sequence of edge sets. In the simultaneous resource updating problem on temporal graphs, the resources which can be accessed will change, however, the necessity of symmetry breaking to avoid conflicts remains. In this paper, we focus on the problem of emph{maintaining proper colourings} on temporal graphs in general, with a particular focus on bipartite colourings. Our aim is to minimise the total number of times that the vertices change colour, or, in the form of a decision problem, whether we can maintain a proper colouring by allowing not more colour changes than some given emph{budget}. On the negative side, we show that, despite bipartite colouring being easy on static graphs, the problem of maintaining such a colouring on graphs that are bipartite in each snapshot is NP-Hard to even approximate within emph{any} constant factor unless the Unique Games Conjecture fails. On the positive side, we provide an exact algorithm for a temporal graph with $n$ vertices, a lifetime $T$ and at most $k$ components in any given snapshot in $O(T vert E vert 2^{k} + n T 2^{2k})$ time, and an $Oleft(sqrt{log(nT)} ight)$-factor approximation algorithm running in $ ilde{O}((nT)^3)$ time. Our results contribute to the structural complexity of networks that change with time with respect to a fundamental computational problem.

Problem

• Maintaining proper bipartite colorings on dynamic temporal graphs
• Minimizing total vertex color changes under given budget constraints
• Addressing NP-Hard complexity of temporal coloring despite static simplicity

Innovation

• Maintains bipartite colorings on temporal graphs
• Minimizes vertex color changes under budget constraints
• Provides exact and approximation algorithms for NP-Hard problem