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

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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.

Cite As Get BibTex

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

Author Details

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

Funding

  • Harb, Elfarouk: Supported in part by NSF grants CCF-2028861 and CCF-1910149.
  • Quanrud, Kent: Supported in part by NSF grant CCF-2129816.
  • Chekuri, Chandra: Supported in part by NSF grants CCF-2028861 and CCF-1910149.

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