Most balanced minimum cuts and partially ordered knapsack

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 ] = ∅, (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 shown to be NP-hard. We give a PTAS for the edge cut variant and for (i). 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 the 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 PTAS is also a PTAS for special types of UPOK instances.