TY - GEN
T1 - On the max coloring problem
AU - Epstein, Leah
AU - Levin, Asaf
PY - 2008
Y1 - 2008
N2 - We consider max coloring on hereditary graph classes. The problem is defined as follows. Given a graph G∈=∈(V,E) and positive node weights w:V →[1,∈∞∈), the goal is to find a proper node coloring of G whose color classes C 1,C 2, ..., C k minimize . We design a general framework which allows to convert approximation algorithms for standard node coloring into algorithms for max coloring. The approximation ratio increases by a multiplicative factor of at most e for deterministic offline algorithms and for randomized online algorithms, and by a multiplicative factor of at most 4 for deterministic online algorithms. We consider two specific hereditary classes which are interval graphs and perfect graphs. For interval graphs, we study the problem in several online environments. In the List Model, intervals arrive one by one, in some order. In the Time Model, intervals arrive one by one, sorted by their left endpoint. For the List Model we design a deterministic 12-competitive algorithm, a randomized 3e-competitive algorithm, and prove a lower bound of 4 on the (deterministic or randomized) competitive ratio. For the Time Model, we use simplified versions of the algorithm and the lower bound of the List Model, to achieve a deterministic 4-competitive algorithm, a randomized e-competitive algorithm, and lower bounds of φ∈≈∈1.618 on the deterministic competitive ratio and on the randomized competitive ratio. The former lower bounds hold even for unit intervals. For unit intervals in the List Model, we obtain a deterministic 8-competitive algorithm, a randomized 2e-competitive algorithm and lower bounds of 2 on the deterministic competitive ratio and on the randomized competitive ratio. Finally, we employ our framework to obtain an offline e-approximation algorithm for max coloring of perfect graphs, improving and simplifying a recent result of Pemmaraju and Raman.
AB - We consider max coloring on hereditary graph classes. The problem is defined as follows. Given a graph G∈=∈(V,E) and positive node weights w:V →[1,∈∞∈), the goal is to find a proper node coloring of G whose color classes C 1,C 2, ..., C k minimize . We design a general framework which allows to convert approximation algorithms for standard node coloring into algorithms for max coloring. The approximation ratio increases by a multiplicative factor of at most e for deterministic offline algorithms and for randomized online algorithms, and by a multiplicative factor of at most 4 for deterministic online algorithms. We consider two specific hereditary classes which are interval graphs and perfect graphs. For interval graphs, we study the problem in several online environments. In the List Model, intervals arrive one by one, in some order. In the Time Model, intervals arrive one by one, sorted by their left endpoint. For the List Model we design a deterministic 12-competitive algorithm, a randomized 3e-competitive algorithm, and prove a lower bound of 4 on the (deterministic or randomized) competitive ratio. For the Time Model, we use simplified versions of the algorithm and the lower bound of the List Model, to achieve a deterministic 4-competitive algorithm, a randomized e-competitive algorithm, and lower bounds of φ∈≈∈1.618 on the deterministic competitive ratio and on the randomized competitive ratio. The former lower bounds hold even for unit intervals. For unit intervals in the List Model, we obtain a deterministic 8-competitive algorithm, a randomized 2e-competitive algorithm and lower bounds of 2 on the deterministic competitive ratio and on the randomized competitive ratio. Finally, we employ our framework to obtain an offline e-approximation algorithm for max coloring of perfect graphs, improving and simplifying a recent result of Pemmaraju and Raman.
UR - http://www.scopus.com/inward/record.url?scp=49949118325&partnerID=8YFLogxK
U2 - 10.1007/978-3-540-77918-6_12
DO - 10.1007/978-3-540-77918-6_12
M3 - Conference contribution
AN - SCOPUS:49949118325
SN - 3540779175
SN - 9783540779179
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 142
EP - 155
BT - Approximation and Online Algorithms - 5th International Workshop, WAOA 2007, Revised Papers
T2 - 5th International Workshop on Approximation and Online Algorithms, WAOA 2007
Y2 - 11 October 2007 through 12 October 2007
ER -