Most balanced minimum cuts

Abstract

We consider the problem of finding most balanced cuts among minimum s t -edge cuts and minimum s t -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 ] = 0̸ , (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 s t -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.