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Compact Representations of Cuts

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Hartvigsen, David
Copyright Year	2001
Abstract	Consider the ${n}\choose{2}$ (or $O(n^{2})$) min-cut problems on a graph with n nodes and nonnegative edge weights. Gomory and Hu [J. Soc. Indust. Appl. Math., 9 (1961), pp. 551--570] showed (essentially) that there are at most n-1 different min-cuts. They also described a compact structure (the flow equivalent tree) of size O(n) with the following property: for any pair of nodes, the value of a min-cut can be obtained from this structure. Furthermore, they showed how this structure can be found by solving only n-1 min-cut problems. This paper contains generalizations of these results. For example, consider a k-terminal cut problem on a graph: for a given set of k nodes, delete a minimum weight set of edges (called a k-cut) so that each of the k nodes is in a different component. There are ${n}\choose{k}$ (or O(nk)) such problems. Hassin [Math. Oper. Res., 13 (1988), pp. 535--542] showed (essentially) that there are at most ${n-1}\choose{k-1}$ (or O(nk-1)) different min k-cuts. We describe a compact structure of size O(nk-1) with the following property: for any k nodes, the value of a min k-cut can be obtained from this structure. We also show how this structure can be found by solving only ${n-1}\choose{k-1}$ k-terminal cut problems. This work builds upon the results of Hassin [Math. Oper. Res., 13 (1988), pp. 535--542], [Oper. Res. Lett., 9 (1990), pp. 315--318], and [Lecture Notes in Comput. Sci. 450, 1990, pp. 228--299].
Starting Page	49
Ending Page	66
Page Count	18
File Format	PDF
ISSN	08954801
DOI	10.1137/S0895480196312334
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	1
Volume Number	14
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-08-01
Access Restriction	Subscribed
Subject Keyword	flow tree k-cut min-cut
Content Type	Text
Resource Type	Article
Subject	Mathematics

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