A Unified Approach to Minimizing Symmetric Submodular Functions

Abstract

Symmetric submodular function minimization admits purely combinatorial algorithms using special orderings of the ground set. Extending the minimum-cut algorithm of Nagamochi and Ibaraki (1992), Queyranne (1998) showed that the maximum adjacency ordering yields a pendent pair, which can be used to find a nontrivial minimizer. Nagamochi (2010) later introduced the minimum degree ordering, which yields a flat pair and leads to the identification of extreme sets. Despite the apparent similarity between these two algorithms, their connection remained unclear. In this paper, we introduce yet another ordering called minimum capacity ordering, and extend it to a one-parameter family of orderings, called $\alpha$-orderings, that unifies these two previously known orderings. We prove a general inequality for $\alpha$-orderings, and our framework recovers the known pendent-pair and flat-pair results as special cases, corresponding to $\alpha = -1$ and $\alpha = 1$, respectively. For each $\alpha \in [-1, 1]$, the last two elements of an $\alpha$-ordering form a contractible pair, i.e., a pair whose contraction preserves the existence of a nontrivial minimizer, which leads to a contraction algorithm that finds a nontrivial minimizer of a symmetric submodular function in $O(n^3)$ oracle calls, where $n$ is the cardinality of the ground set. In addition, we discuss the ranges of $\alpha$ that ensure $\alpha$-ordering to obtain these special pairs.