On Spectral Bounds for the k-Partitioning of Graphs

• Published in: Theory of computing systems Vol. 36; no. 5; pp. 461 - 478
• Main Authors: Elsässer, Robert, Lücking, Thomas, Monien, Burkhard
• Format: Journal Article
• Language: English
• Published: New York Springer Nature B.V 01.09.2003
• Subjects: Algorithms; Eigenvalues; Graphs
• ISSN: 1432-4350; 1433-0490
• DOI: 10.1007/s00224-003-1083-9

When executing processes on parallel computer systems a major bottle-neck is interprocessor communication. One way to address this problem is to minimize the communication between processes that are mapped to different processors. This translates to the <k<-partitioning problem of the corresponding process graph, where <k< is the number of processors. The classical spectral lower bound of (|<V<|/2<k<)\sum< <k< < < <i<=1<[lambda]< <i< < for the <k<-section width of a graph is well known. We show new relations between the structure and the eigenvalues of a graph and present a new method to get tighter lower bounds on the <k<-section width. This method makes use of the level structure defined by the <k<-section. We define a global expansion property and prove that for graphs with the same<k<-section width the spectral lower bound increases with this global expansion. We also present examples of graphs for which our new bounds are tight up to a constant factor.