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

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)...

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Bibliographic Details
Published in:arXiv.org 2023-05
Main Authors: Harb, Elfarouk, Quanrud, Kent, Chekuri, Chandra
Format: Article
Language:English
Subjects:
Algorithms
Convergence
Greedy algorithms
Optimization
Quadratic programming
Online Access:Get full text
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Summary: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.
ISSN:2331-8422