Most balanced minimum cuts

We consider the problem of finding most balanced cuts among minimum st-edge cuts and minimum st-vertex cuts, for given vertices s and t, according to different balance criteria. For edge cuts [S,S@?] we seek to maximize min{|S|,|S@?|}. For vertex cuts C of G we consider the objectives of (i) maximizing min{|S|,|T|}, where {S,T} is a partition of V(G)@?C with s@?S, t@?T and [S,T]=0@?, (ii) minimizing the order of the largest component of G-C, and (iii) maximizing the order of the smallest component of G-C. All of these problems are NP-hard. We give a PTAS for the edge cut variant and for (i). These results also hold for directed graphs. We give a 2-approximation for (ii), and show that no non-trivial approximation exists for (iii) unless P=NP. To prove these results we show that we can partition the vertices of G, and define a partial order on the subsets of this partition, such that ideals of the partial order correspond bijectively to minimum st-cuts of G. This shows that the problems are closely related to Uniform Partially Ordered Knapsack (UPOK), a variant of POK where element utilities are equal to element weights. Our algorithm is also a PTAS for special types of UPOK instances.

[1]  Mihalis Yannakakis,et al.  Suboptimal Cuts: Their Enumeration, Weight and Number (Extended Abstract) , 1992, ICALP.

[2]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[3]  Maurice Queyranne,et al.  On the structure of all minimum cuts in a network and applications , 1982, Math. Program..

[4]  Markus Chimani,et al.  On the Minimum Cut of Planarizations , 2007, Electron. Notes Discret. Math..

[5]  David B. Shmoys,et al.  Cut problems and their application to divide-and-conquer , 1996 .

[6]  Yuval Rabani,et al.  ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[7]  Frank Thomson Leighton,et al.  Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms , 1999, JACM.

[8]  Stavros G. Kolliopoulos,et al.  Partially ordered knapsack and applications to scheduling , 2007, Discret. Appl. Math..

[9]  D. S. Johnson,et al.  On Knapsacks, Partitions, and a New Dynamic Programming Technique for Trees , 1983, Math. Oper. Res..

[10]  Sanjeev Khanna,et al.  Hardness of cut problems in directed graphs , 2006, STOC '06.

[11]  David R. Karger,et al.  Global min-cuts in RNC, and other ramifications of a simple min-out algorithm , 1993, SODA '93.

[12]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[13]  Uriel Feige,et al.  Finding small balanced separators , 2006, STOC '06.

[14]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .

[15]  Lisa Fleischer.  Building Chain and Cactus Representations of All Minimum Cuts from Hao-Orlin in the Same Asymptotic Run Time , 1999, J. Algorithms.

[16]  Stefan Felsner,et al.  Recognition Algorithms for Orders of Small Width and Graphs of Small Dilworth Number , 2003, Order.

[17]  Uriel Feige,et al.  On the complexity of finding balanced oneway cuts , 2003, Inf. Process. Lett..