TY - JOUR

T1 - On the complexity of cell flipping in permutation diagrams and multiprocessor scheduling problems

AU - Golumbic, Martin Charles

AU - Kaplan, Haim

AU - Verbin, Elad

PY - 2005/6/28

Y1 - 2005/6/28

N2 - 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.

AB - 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.

KW - Cell flipping

KW - Clique number

KW - Dynamic programming

KW - Independent set number

KW - Multiprocessor scheduling

KW - Permutation graphs

KW - Stable set number

KW - VLSI layout

UR - http://www.scopus.com/inward/record.url?scp=19944404494&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2004.08.042

DO - 10.1016/j.disc.2004.08.042

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AN - SCOPUS:19944404494

SN - 0012-365X

VL - 296

SP - 25

EP - 41

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 1

ER -