A polynomial-time nearly-optimal algorithm for an edge coloring problem in outerplanar graphs

AbstractThis paper considers several parallel machine scheduling problems with controllable processing times, in which the goal is to minimize the makespan. Preemption is allowed. The processing times of the jobs can be compressed by some extra resources. Three resource use models are considered. If the jobs are released at the same time, the problems under all the three models can be solved in a polynomial time. The authors give the polynomial algorithm. When the jobs are not released at the same time, if all the resources are given at time zero, or the remaining resources in the front stages can be used to the next stages, the offline problems can be solved in a polynomial time, but the online problems have no optimal algorithm. If the jobs have different release dates, and the remaining resources in the front stages can not be used in the next stages, both the offline and online problems can be solved in a polynomial time.

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A CHANNEL ASSIGNMENT PROBLEM IN MULTIHOP WIRELESS NETWORKS AND GRAPH THEORY

Journal of Circuits System and Computers ◽

10.1142/s0218126604001398 ◽

2004 ◽

Vol 13 (02) ◽

pp. 375-385 ◽

Cited By ~ 4

Author(s):

HIROSHI TAMURA ◽

KAORU WATANABE ◽

MASAKAZU SENGOKU ◽

SHOJI SHINODA

Keyword(s):

Wireless Networks ◽

Graph Theory ◽

Channel Assignment ◽

Assignment Problem ◽

Edge Coloring ◽

Multihop Wireless Networks ◽

Personal Communication ◽

Multihop Wireless ◽

Coloring Problem ◽

Communication Devices

Multihop wireless networks consist of mobile terminals with personal communication devices. Each terminal can receive a message and then send it to another terminal. In these networks, it is important to assign channels for communications to each terminal efficiently. There are some studies on this assignment problem using a conventional edge coloring in graph theory. In this paper, we propose a new edge coloring problem in graph and network theory on this assignment problem and we discuss the computational complexity of the problem. This edge coloring problem takes the degree of interference into consideration. Therefore, we can reuse the channels more efficiently compared with the conventional method.

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A reduced upper bound for an edge-coloring problem from relation algebra

Algebra Universalis ◽

10.1007/s00012-019-0592-6 ◽

2019 ◽

Vol 80 (2) ◽

Author(s):

Jeremy F. Alm ◽

David A. Andrews

Keyword(s):

Upper Bound ◽

Edge Coloring ◽

Relation Algebra ◽

Coloring Problem

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WEIGHTED-EDGE-COLORING OF k-DEGENERATE GRAPHS AND BIN-PACKING

Journal of Interconnection Networks ◽

10.1142/s0219265911002861 ◽

2011 ◽

Vol 12 (01n02) ◽

pp. 109-124

Author(s):

FLORIAN HUC

Keyword(s):

Bin Packing ◽

Edge Coloring ◽

Weighted Graph ◽

Packing Problem ◽

Bipartite Graphs ◽

Specific Class ◽

Weighted Graphs ◽

Outerplanar Graphs ◽

Underlying Graph ◽

Multiple Edges

The weighted-edge-coloring problem of an edge-weighted graph whose weights are between 0 and 1, consists in finding a coloring using as few colors as possible and satisfying the following constraints: the sum of weights of edges with the same color and incident to the same vertex must be at most 1. In 1991, Chung and Ross conjectured that if G is bipartite, then [Formula: see text] colors are always sufficient to weighted-edge-color (G,w), where [Formula: see text] is the maximum of the sums of the weights of the edges incident to a vertex. We prove this is true for edge-weighted graphs with multiple edges whose underlying graph is a tree. We further generalise this conjecture to non-bipartite graphs and prove the generalised conjecture for simple edge-weighted outerplanar graphs. Finally, we introduce a list version of this coloring together with the list-bin-packing problem, which allows us to obtain new results concerning the original coloring for a specific class of graphs, namely the k-weight-degenerate weighted graph.

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