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 $α$-orderings, that unifies these two previously known orderings. We prove a general inequality for $α$-orderings, and our framework recovers the known pendent-pair and flat-pair results as special cases, corresponding to $α= -1$ and $α= 1$, respectively. For each $α\in [-1, 1]$, the last two elements of an $α$-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 $α$ that ensure $α$-ordering to obtain these special pairs.

翻译：对称子模函数的最小化问题可以通过使用基集特殊排序的纯组合算法来解决。Nagamochi和Ibaraki(1992)的最小割算法被Queyranne(1998)推广，他证明了最大邻接序能产生一个悬垂对，该对可用于寻找非平凡极小化子。Nagamochi(2010)后来引入了最小度序，该序产生一个平面对，并导致极端集的识别。尽管这两种算法表面上相似，但它们之间的联系一直不清楚。本文提出了一种称为最小容量序的新排序，并将其扩展为一个单参数排序族——$α$-序，该族统一了这两种已知的排序。我们证明了$α$-序的一个一般不等式，我们的框架恢复了已知的悬垂对和平面对结果作为特例，分别对应于$α= -1$和$α= 1$。对于每个$α\in [-1, 1]$，$α$-序的最后两个元素构成一个可收缩对，即其收缩保持非平凡极小化子存在性的对，这导出了一个收缩算法，该算法能在$O(n^3)$次预言机调用内找到对称子模函数的非平凡极小化子，其中$n$是基集的基数。此外，我们讨论了确保$α$-序获得这些特殊对的$α$的取值范围。