Competitive Capacitated Online Recoloring
Rajmohan Rajaraman, Omer Wasim
TL;DR
This work addresses capacitated online recoloring and its fully dynamic variant, framing the problem with weighted vertices, color capacities, and online edge requests. It introduces phase-based, component-centric algorithms and a Rebalance subroutine (an FPTAS) to manage capacity while controlling recolorings, alongside Follow-Greedy strategies for online settings. The results include the first competitive algorithms for capacitated online 2-recoloring with $(1+\varepsilon)$-resource augmentation achieving $O(\log n)$-competitiveness, and an $O(n\log n)$ fully dynamic algorithm; for capacitated $\Delta$-recoloring, it delivers an $O(1)$-competitive randomized algorithm in the overprovisioned regime and $O(\Delta)$-competitive deterministic bounds for small $\Delta$, with matching lower bounds. The findings advance practical VM scheduling under anti-affinity constraints and offer tight, theory-grounded performance guarantees with techniques that blend bipartite-component reasoning, vertex-cover based lower bounds, and martingale/ Chernoff-type analyses for concentration results.
Abstract
In this paper, we revisit the online recoloring problem introduced recently by Azar et al. In online recoloring, there is a fixed set $V$ of $n$ vertices and an initial coloring $c_0: V\rightarrow [k]$ for some $k\in \mathbb{Z}^{>0}$. Under an online sequence $σ$ of requests where each request is an edge $(u_t,v_t)$, a proper vertex coloring $c$ of the graph $G_t$ induced by requests until time $t$ needs to be maintained for all $t$; i.e., for any $(u,v)\in G_t$, $c(u)\neq c(v)$. The objective is to minimize the total weight of vertices recolored for the sequence $σ$. We obtain the first competitive algorithms for capacitated online recoloring and fully dynamic recoloring. Our first set of results is for $2$-recoloring using algorithms that are $(1+\varepsilon)$-resource augmented where $\varepsilon\in (0,1)$ is an arbitrarily small constant. Our main result is an $O(\log n)$-competitive deterministic algorithm for weighted bipartite graphs, which is asymptotically optimal in light of an $Ω(\log n)$ lower bound that holds for an unbounded amount of augmentation. We also present an $O(n\log n)$-competitive deterministic algorithm for fully dynamic recoloring, which is optimal within an $O(\log n)$ factor in light of a $Ω(n)$ lower bound that holds for an unbounded amount of augmentation. Our second set of results is for $Δ$-recoloring in an $(1+\varepsilon)$-overprovisioned setting where the maximum degree of $G_t$ is bounded by $(1-\varepsilon)Δ$ for all $t$, and each color assigned to at most $(1+\varepsilon)\frac{n}Δ$ vertices, for an arbitrary $\varepsilon > 0$. Our main result is an $O(1)$-competitive randomized algorithm for $Δ= O(\sqrt{n/\log n})$. We also present an $O(Δ)$-competitive deterministic algorithm for $Δ\le \varepsilon n/2$. Both results are asymptotically optimal.