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Convergence to Lexicographically Optimal Base in a (Contra)Polymatroid and Applications to Densest Subgraph and Tree Packing

Authors Elfarouk Harb, Kent Quanrud, Chandra Chekuri



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Elfarouk Harb
  • University of Illinois at Urbana Champaign, IL, USA
Kent Quanrud
  • Purdue University, West Lafayette, IN, USA
Chandra Chekuri
  • University of Illinois at Urbana Champaign, IL, USA

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Elfarouk Harb, Kent Quanrud, and Chandra Chekuri. Convergence to Lexicographically Optimal Base in a (Contra)Polymatroid and Applications to Densest Subgraph and Tree Packing. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 56:1-56:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.56

Abstract

Boob et al. [Boob et al., 2020] 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 [Chandra Chekuri et al., 2022] 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’s quadratic program for finding a lexicographically optimal base in a (contra) polymatroid [Satoru Fujishige, 1980], and a noisy version of the Frank-Wolfe method from convex optimization [Frank and Wolfe, 1956; Jaggi, 2013]. 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. [Harb et al., 2022]. A second contribution of the paper is to understand Thorup’s work on ideal tree packing and greedy tree packing [Thorup, 2007; Thorup, 2008] 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 optimization.

Subject Classification

ACM Subject Classification
  • Networks → Network algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • Polymatroid
  • lexicographically optimum base
  • densest subgraph
  • tree packing

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