Elsevier

Discrete Mathematics

Volume 296, Issue 1, 28 June 2005, Pages 25-41
Discrete Mathematics

On the complexity of cell flipping in permutation diagrams and multiprocessor scheduling problems

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Abstract

Permutation diagrams have been used in circuit design to model a set of single point nets crossing a channel, where the minimum number of layers needed to realize the diagram equals the clique number ω(G) of its permutation graph, the value of which can be calculated in O(nlogn) time. We consider a generalization of this model motivated by “standard cell” technology in which the numbers on each side of the channel are partitioned into consecutive subsequences, or cells, each of which can be left unchanged or flipped (i.e., reversed). We ask, for what choice of flippings will the resulting clique number be minimum or maximum. We show that when one side of the channel is fixed (no flipping), an optimal flipping for the other side can be found in O(nlogn) time for the maximum clique number, and that when both sides are free this can be solved in O(n2) time. We also prove NP-completeness of finding a flipping that gives a minimum clique number, even when one side of the channel is fixed, and even when the size of the cells is restricted to be less than a small constant. Moreover, since the complement of a permutation graph is also a permutation graph, the same complexity results hold for the stable set (independence) number. In the process of the NP-completeness proof we also prove NP-completeness of a restricted variant of a scheduling problem. This new NP-completeness result may be of independent interest.

Keywords

Permutation graphs
Dynamic programming
Cell flipping
VLSI layout
Clique number
Stable set number
Independent set number
Multiprocessor scheduling

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A preliminary version of this paper has appeared in the Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science (STACS), 1998 [7]. In this version we also settle the main open problem from [7], see Section 4.