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Optimal edge ranking of trees in polynomial time

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Abstract

An edge ranking of a graph is a labeling of the edges using positive integers such that all paths between two edges with the same label contain an intermediate edge with a higher label. An edge ranking isoptimal if the highest label used is as small as possible. The edge-ranking problem has applications in scheduling the manufacture of complex multipart products; it is equivalent to finding the minimum height edge-separator tree. In this paper we give the first polynomial-time algorithm to find anoptimal edge ranking of a tree, placing the problem inP. An interesting feature of the algorithm is an unusual greedy procedure that allows us to narrow an exponential search space down to a polynomial search space containing an optimal solution. AnNC algorithm is presented that finds an optimal edge ranking for trees of constant degree. We also prove that a natural decision problem emerging from our sequential algorithm isP-complete.

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Communicated by N. Megiddo.

The research of P. de la Torre was partially supported by NSF Grant CCR-9010445. R. Greenlaw's research was partially supported by NSF Grant CCR-9209184. The research of A. A. Schäffer was partially supported by NSF Grant CCR-9010534.

Subsequent to the acceptance of this paper, Zhou and Nishizeki found faster algorithms for optimal edge ranking of trees, first reducing the time toO(n2) [22] and then toO(n logn) [23].

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de la Torre, P., Greenlaw, R. & Schäffer, A.A. Optimal edge ranking of trees in polynomial time. Algorithmica 13, 592–618 (1995). https://doi.org/10.1007/BF01189071

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  • DOI: https://doi.org/10.1007/BF01189071

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