Maintaining Bipartite Colourings on Temporal Graphs on a Budget
Duncan Adamson, George B. Mertzios, Paul G. Spirakis
TL;DR
This work studies maintaining a proper bipartite colouring on temporal graphs while minimising recolourings across time, formalised as Temporal Recolouring (TRec). It establishes strong hardness results, showing NP-hardness to approximate within any constant factor under standard assumptions, even for bipartite snapshots, and provides constructive algorithms: a fixed-parameter tractable algorithm parameterised by the maximum number of components per snapshot with running time $O(T|E|2^{k} + nT2^{2k})$, and a $O\left(\sqrt{\log(nT)}\right)$-approximation achievable in $\tilde{O}((nT)^3)$ by reducing to a MinUnCut instance via an auxiliary graph $\alpha(\mathcal{G})$. The paper also yields a linear-time test for zero-budget feasibility in bipartite underlying graphs and presents a hardness result for the non-bipartite case. Together, these results illuminate the structural complexity of dynamic networks with respect to a fundamental colouring problem and provide both exact and approximation tools for temporal recolouring tasks.
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 $O\left(\sqrt{\log(nT)}\right)$-factor approximation algorithm running in $\tilde{O}((nT)^3)$ time. Our results contribute to the structural complexity of networks that change with time with respect to a fundamental computational problem.