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

05/04/2023
by   Elfarouk Harb, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/13/2017

Greedy spanners are optimal in doubling metrics

We show that the greedy spanner algorithm constructs a (1+ϵ)-spanner of ...
research
08/17/2018

LP Relaxation and Tree Packing for Minimum k-cuts

Karger used spanning tree packings to derive a near linear-time randomiz...
research
01/10/2018

Online Maximum Matching with Recourse

We study the online maximum matching problem with recourse in a model in...
research
08/06/2021

Adaptive Simulated Annealing with Greedy Search for the Circle Bin Packing Problem

We introduce a new bin packing problem, termed the circle bin packing pr...
research
01/24/2021

A greedy algorithm for dropping digits (Functional Pearl)

Consider the puzzle: given a number, remove k digits such that the resul...
research
11/08/2022

Improved Pattern-Avoidance Bounds for Greedy BSTs via Matrix Decomposition

Greedy BST (or simply Greedy) is an online self-adjusting binary search ...
research
01/20/2020

Adaptive Large Neighborhood Search for Circle Bin Packing Problem

We address a new variant of packing problem called the circle bin packin...

Please sign up or login with your details

Forgot password? Click here to reset