@inproceedings{20dfa612b5594081adedcc17a3112735,
title = "Convergence to Lexicographically Optimal Base in a (Contra)Polymatroid and Applications to Densest Subgraph and Tree Packing",
abstract = "Boob et al. [7] described an iterative peeling algorithm called Greedy++ for the Densest Subgraph Problem (DSG) and conjectured that it converges to an optimum solution. Chekuri, Qaunrud and Torres [10] extended the algorithm to 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{\textquoteright}s quadratic program for finding a lexicographically optimal base in a (contra) polymatroid [18], and a noisy version of the Frank-Wolfe method from convex optimization [17, 25]. This yields 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. [24]. A second contribution of the paper is to understand Thorup{\textquoteright}s work on ideal tree packing and greedy tree packing [46, 47] 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{\textquoteright}s result and convex optimization.",
keywords = "Polymatroid, densest subgraph, lexicographically optimum base, tree packing",
author = "Elfarouk Harb and Kent Quanrud and Chandra Chekuri",
note = "Publisher Copyright: {\textcopyright} Elfarouk Harb, Kent Quanrud, and Chandra Chekuri;; 31st Annual European Symposium on Algorithms, ESA 2023 ; Conference date: 04-09-2023 Through 06-09-2023",
year = "2023",
month = sep,
doi = "10.4230/LIPIcs.ESA.2023.56",
language = "English (US)",
series = "Leibniz International Proceedings in Informatics, LIPIcs",
publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",
editor = "{Li Gortz}, Inge and Martin Farach-Colton and Puglisi, {Simon J.} and Grzegorz Herman",
booktitle = "31st Annual European Symposium on Algorithms, ESA 2023",
}