Abstract
We consider path multicoloring problems, in which one is given a collection of paths defined on a graph and is asked to color some or all of them so as to optimize certain objective functions. Typical objectives are: (a) the minimization of the average, over all edges, of the maximum-multiplicity color when the number of colors is given (MinAvgMult-PMC), (b) the minimization of the number of colors when the maximum multiplicity for each edge is given (Min-PMC), or (c) the maximization of the number of colored paths when both the number of colors and a maximum multiplicity constraint for each edge are given (Max-PMC). Such problems also capture edge multicoloring variants (such as MinAvgMult-EMC, Min-EMC, and MaxEMC) as special cases and find numerous applications in resource allocation, most notably in optical and wireless networks, and in communication task scheduling.
Our contribution is two-fold: On the one hand, we give an exact polynomial-time algorithm for Min-PMC on spider networks with even admissible color multiplicities on each edge. On the other hand, we present an approximation algorithm for MinAvgMult-PMC in star networks, with a ratio strictly better than 2; our algorithm uses an appropriate path orientation. We also show that any algorithm which is based on path orientation cannot achieve an approximation ratio better than \(\frac{7}{6}\). Our results apply to the corresponding edge multicoloring problems as well.
Research supported in part by the ALGONOW project, co-financed by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF) Research Funding Program: THALES, Investing in knowledge society through the European Social Fund.
E. Bampas—Partial support by the ANR project DISPLEXITY (ANR-11-BS02-014).
C. Karousatou—This work was done while Christina Karousatou was with the School of Electrical and Computer Engineering, National Technical University of Athens, Greece.
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Notes
- 1.
A spider graph is a tree with at most one node of degree strictly greater than \(2\). A caterpillar is a tree in which all nodes of degree \(2\) or more lie on the same simple path.
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Bampas, E., Karousatou, C., Pagourtzis, A., Potika, K. (2015). Scheduling Connections via Path and Edge Multicoloring. In: Papavassiliou, S., Ruehrup, S. (eds) Ad-hoc, Mobile, and Wireless Networks. ADHOC-NOW 2015. Lecture Notes in Computer Science(), vol 9143. Springer, Cham. https://doi.org/10.1007/978-3-319-19662-6_3
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