Cut problems and their application to divide-and-conquer
Approximation algorithms for NP-hard problems
Global min-cuts in RNC, and other ramifications of a simple min-out algorithm
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
Suboptimal Cuts: Their Enumeration, Weight and Number (Extended Abstract)
ICALP '92 Proceedings of the 19th International Colloquium on Automata, Languages and Programming
On the complexity of finding balanced oneway cuts
Information Processing Letters
Finding small balanced separators
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Hardness of cut problems in directed graphs
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT
Computational Complexity
Partially ordered knapsack and applications to scheduling
Discrete Applied Mathematics
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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.