TY - GEN

T1 - New bounds for the nearly equitable edge coloring problem

AU - Xie, Xuzhen

AU - Yagiura, Mutsunori

AU - Ono, Takao

AU - Hirata, Tomio

AU - Zwick, Uri

PY - 2007

Y1 - 2007

N2 - An edge coloring of a multigraph is nearly equitable if, among the edges incident to each vertex, the numbers of edges colored with any two colors differ by at most two. It has been proved that this problem can be solved in O(m 2/k) time, where m and k are the numbers of edges and given colors, respectively. In this paper, we present a recursive algorithm that runs in O (mn log (m/(kn) + 1)) time, where n is the number of vertices. This algorithm improves the best-known worstcase time complexity. When k = O(1), the time complexity of all known algorithms is O(m2), which implies that this time complexity remains to be the best for more than twenty years since 1982 when Hilton and de Werra gave a constructive proof for the existence of a nearly equitable edge coloring for any graph. Our result is the first that improves this time complexity when m/n grows to infinity; e.g., m = nθ for an arbitrary constant θ > 1. We also propose a very simple randomized algorithm that runs in O (m3/2n1/2/k 1/2) time with probability at least 1 -1/c for any constant c > 1, whose worst-case time complexity is O(m2/k).

AB - An edge coloring of a multigraph is nearly equitable if, among the edges incident to each vertex, the numbers of edges colored with any two colors differ by at most two. It has been proved that this problem can be solved in O(m 2/k) time, where m and k are the numbers of edges and given colors, respectively. In this paper, we present a recursive algorithm that runs in O (mn log (m/(kn) + 1)) time, where n is the number of vertices. This algorithm improves the best-known worstcase time complexity. When k = O(1), the time complexity of all known algorithms is O(m2), which implies that this time complexity remains to be the best for more than twenty years since 1982 when Hilton and de Werra gave a constructive proof for the existence of a nearly equitable edge coloring for any graph. Our result is the first that improves this time complexity when m/n grows to infinity; e.g., m = nθ for an arbitrary constant θ > 1. We also propose a very simple randomized algorithm that runs in O (m3/2n1/2/k 1/2) time with probability at least 1 -1/c for any constant c > 1, whose worst-case time complexity is O(m2/k).

UR - http://www.scopus.com/inward/record.url?scp=38149127083&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-77120-3_26

DO - 10.1007/978-3-540-77120-3_26

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AN - SCOPUS:38149127083

SN - 9783540771180

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 280

EP - 291

BT - Algorithms and Computation - 18th International Symposium, ISAAC 2007, Proceedings

PB - Springer Verlag

T2 - 18th International Symposium on Algorithms and Computation, ISAAC 2007

Y2 - 17 December 2007 through 19 December 2007

ER -