TY - JOUR

T1 - Competitive Vertex Recoloring

T2 - (Online Disengagement)

AU - Azar, Yossi

AU - Machluf, Chay

AU - Patt-Shamir, Boaz

AU - Touitou, Noam

N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2023/7

Y1 - 2023/7

N2 - Motivated by placement of jobs in physical machines, we introduce and analyze the problem of online recoloring, or online disengagement. In this problem, we are given a set of n weighted vertices and a k-coloring of the vertices (vertices represent jobs, and colors represent physical machines). Edges, representing conflicts between jobs, are inserted in an online fashion. After every edge insertion, the algorithm must output a proper k-coloring of the vertices. The cost of recoloring a vertex is the vertex’s weight. Our aim is to minimize the competitive ratio of the algorithm, i.e., the ratio between the cost paid by the online algorithm and the cost paid by an optimal, offline algorithm. We consider a couple of polynomially-solvable coloring variants. Specifically, for 2-coloring bipartite graphs we present an O(log n) -competitive deterministic algorithm and an Ω (log n) lower bound on the competitive ratio of randomized algorithms. For (Δ + 1) -coloring, where Δ is the maximal node degree, we present tight bounds of Θ (Δ) and Θ (log Δ) on the competitive ratios of deterministic and randomized algorithms, respectively (where Δ denotes the maximum degree). We also consider the fully dynamic case which allows edge deletions as well as insertions. All our algorithms are applicable to the case where vertices are arbitrarily weighted, and all our lower bounds hold even in the uniform weights (unweighted) case.

AB - Motivated by placement of jobs in physical machines, we introduce and analyze the problem of online recoloring, or online disengagement. In this problem, we are given a set of n weighted vertices and a k-coloring of the vertices (vertices represent jobs, and colors represent physical machines). Edges, representing conflicts between jobs, are inserted in an online fashion. After every edge insertion, the algorithm must output a proper k-coloring of the vertices. The cost of recoloring a vertex is the vertex’s weight. Our aim is to minimize the competitive ratio of the algorithm, i.e., the ratio between the cost paid by the online algorithm and the cost paid by an optimal, offline algorithm. We consider a couple of polynomially-solvable coloring variants. Specifically, for 2-coloring bipartite graphs we present an O(log n) -competitive deterministic algorithm and an Ω (log n) lower bound on the competitive ratio of randomized algorithms. For (Δ + 1) -coloring, where Δ is the maximal node degree, we present tight bounds of Θ (Δ) and Θ (log Δ) on the competitive ratios of deterministic and randomized algorithms, respectively (where Δ denotes the maximum degree). We also consider the fully dynamic case which allows edge deletions as well as insertions. All our algorithms are applicable to the case where vertices are arbitrarily weighted, and all our lower bounds hold even in the uniform weights (unweighted) case.

KW - Anti-affinity constraints

KW - Coloring with recourse

KW - Online migration

UR - http://www.scopus.com/inward/record.url?scp=85145509055&partnerID=8YFLogxK

U2 - 10.1007/s00453-022-01076-x

DO - 10.1007/s00453-022-01076-x

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AN - SCOPUS:85145509055

SN - 0178-4617

VL - 85

SP - 2001

EP - 2027

JO - Algorithmica

JF - Algorithmica

IS - 7

ER -