The principal lattice of partitions of a submodular function

Abstract

This paper studies the partitions on which a function (μ − λ)(Π)≡∑Ni∈Π(μ−λ)(Ni) reaches a minimum when μ is a submodular function and λ takes values between −∞ and ∞. For a given value of λ such partitions form a lattice which contains more than one element only for a finite set of “critical values” of λ. The collection of all such partitions for all values of λ forms a lattice, which we call the principal lattice of partitions of μ. This development has strong parallels with that of the principal partition of μ. We present efficient algorithms for the construction of this lattice for a general submodular function. We also bring out its applications to “electrical network analysis by decomposition” and present more efficient algorithms for the cases relevant to such analysis.