Competitive Capacitated Online Recoloring

TL;DR

This paper introduces new competitive algorithms for online graph recoloring problems, achieving near-optimal bounds for various graph classes and dynamic settings with resource augmentation.

Contribution

It presents the first competitive algorithms for capacitated online recoloring and fully dynamic recoloring, with optimal or near-optimal bounds under resource augmentation.

Findings

Deterministic $O(rac{ ext{log} n}{ ext{resource augmentation}})$-competitive algorithm for bipartite graphs.

$O(n ext{log} n)$-competitive algorithm for fully dynamic recoloring.

Randomized $O(1)$-competitive algorithm for $ ext{max degree} = O( ext{sqrt}(n/ ext{log} n))$.

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 $\sigma$ 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 $\sigma$. 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 $\Omega(\log n)$ 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 $\Omega(n)$ lower bound that holds for an unbounded amount of augmentation. Our second set of results is for $\Delta$-recoloring in an $(1+\varepsilon)$-overprovisioned setting where the maximum degree of $G_t$ is bounded by $(1-\varepsilon)\Delta$ for all $t$, and each color assigned to at most $(1+\varepsilon)\frac{n}{\Delta}$ vertices, for an arbitrary $\varepsilon > 0$. Our main result is an $O(1)$-competitive randomized algorithm for $\Delta = O(\sqrt{n/\log n})$. We also present an $O(\Delta)$-competitive deterministic algorithm for $\Delta \le \varepsilon n/2$. Both results are asymptotically optimal.