Colorful Vertex Recoloring of Bipartite Graphs

TL;DR

This paper studies online vertex recoloring in bipartite graphs, proposing algorithms that leverage additional colors to improve performance and establishing bounds on competitiveness, especially for graphs with bounded bonds.

Contribution

It introduces algorithms for bipartite graphs with additional colors, analyzing their competitiveness and providing lower bounds for online recoloring performance.

Findings

Algorithms achieve better competitiveness with more colors.

Lower bounds show limits of online algorithms with extra colors.

Performance depends on graph properties like bond size and maximum degree.

Abstract

In vertex recoloring, we are given $n$ vertices with their initial coloring, and edges arrive in an online fashion. The algorithm must maintain a valid coloring by recoloring vertices, at a cost. The problem abstracts a scenario of job placement in machines (possibly in the cloud), where vertices represent jobs, colors represent machines, and edges represent ``anti affinity'' (disengagement) constraints. Online recoloring is a hard problem. One family of instances which is fairly well-understood is bipartite graphs, in which two colors are sufficient to satisfy all constraints. In this case it is known that the competitive ratio of vertex recoloring is $\Theta(\log n)$. We propose a generalization of the problem, which allows using additional colors (possibly at a higher cost), to improve overall performance. We analyze the simple case of bipartite graphs of bounded largest \emph{bond} (a bond of a connected graph is an edge-cut that partitions the graph into two connected components). First, we propose two algorithms. One exhibits a trade-off for the uniform-cost case: given $\Omega(\log\beta)\le c\le O(\log n)$ colors, the algorithm guarantees that its cost is at most $O(\frac{\log n}{c})$ times the optimal offline cost for two colors, where $n$ is the number of vertices and $\beta$ is the size of the largest bond. The other algorithm is for the case where the additional colors come at a higher cost, $D>1$: given $\Delta$ additional colors, where $\Delta$ is the maximum degree in the graph, the algorithm guarantees $O(\log D)$ competitiveness. As to lower bounds, we show that if the cost of the extra colors is $D>1$, no (randomized) algorithm can achieve a competitive ratio of $o(\log D)$. We also show that for bipartite graphs of unbounded bond size, any deterministic online algorithm has competitive ratio $\Omega(\min(D,\log n))$.