Boob et al. [1] described an iterative peeling algorithm called Greedy++ for the Densest Subgraph Problem (DSG) and conjectured that it converges to an optimum solution. Chekuri, Quanrud, and Torres [2] extended the algorithm to general supermodular density problems (of which DSG is a special case) and proved that the resulting algorithm Super-Greedy++ (and hence also Greedy++) converges. In this paper, we revisit the convergence proof and provide a different perspective. This is done via a connection to Fujishige's quadratic program for finding a lexicographically optimal base in a (contra)polymatroid [3], and a noisy version of the Frank-Wolfe method from convex optimisation [4,5]. This gives us a simpler convergence proof, and also shows a stronger property that Super-Greedy++ converges to the optimal dense decomposition vector, answering a question raised in Harb et al. [6]. A second contribution of the paper is to understand Thorup's work on ideal tree packing and greedy tree packing [7,8] via the Frank-Wolfe algorithm applied to find a lexicographically optimum base in the graphic matroid. This yields a simpler and transparent proof. The two results appear disparate but are unified via Fujishige's result and convex optimisation.

翻译：Boob等人[1]描述了一种名为Greedy++的迭代剥离算法，用于解决稠密子图问题（DSG），并推测该算法收敛于最优解。Chekuri、Quanrud与Torres[2]将该算法推广至一般超模密度问题（DSG为其特例），并证明了由此得到的算法Super-Greedy++（因而也包括Greedy++）具有收敛性。本文重新审视了这一收敛性证明，并提供了不同视角。这通过以下两点实现：其一，与Fujishige用于求解（反）拟阵中字典序最优基的二次规划[3]建立联系；其二，引入来自凸优化的带噪声Frank-Wolfe方法[4,5]。这为我们提供了更简洁的收敛性证明，并揭示了更强性质——Super-Greedy++收敛于最优稠密分解向量，从而回答了Harb等人[6]提出的问题。本文的第二项贡献在于，通过将Frank-Wolfe算法应用于图拟阵中字典序最优基的求解，重新诠释了Thorup关于理想树打包与贪心树打包的工作[7,8]。这给出了更简单透明的证明。这两项成果看似无关，却通过Fujishige的结论与凸优化得到了统一。