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A Dynamic Algorithm for Weighted Submodular Cover Problem

Kiarash Banihashem, Samira Goudarzi, MohammadTaghi Hajiaghayi, Peyman Jabbarzade, Morteza Monemizadeh

TL;DR

This work studies dynamic, weighted submodular cover under insertions and deletions of ground elements. It develops a randomized, multi-run algorithm that maintains an approximately optimal bicriteria solution: $f(S) \

Abstract

We initiate the study of the submodular cover problem in dynamic setting where the elements of the ground set are inserted and deleted. In the classical submodular cover problem, we are given a monotone submodular function $f : 2^{V} \to \mathbb{R}^{\ge 0}$ and the goal is to obtain a set $S \subseteq V$ that minimizes the cost subject to the constraint $f(S) = f(V)$. This is a classical problem in computer science and generalizes the Set Cover problem, 2-Set Cover, and dominating set problem among others. We consider this problem in a dynamic setting where there are updates to our set $V$, in the form of insertions and deletions of elements from a ground set $\mathcal{V}$, and the goal is to maintain an approximately optimal solution with low query complexity per update. For this problem, we propose a randomized algorithm that, in expectation, obtains a $(1-O(ε), O(ε^{-1}))$-bicriteria approximation using polylogarithmic query complexity per update.

A Dynamic Algorithm for Weighted Submodular Cover Problem

TL;DR

This work studies dynamic, weighted submodular cover under insertions and deletions of ground elements. It develops a randomized, multi-run algorithm that maintains an approximately optimal bicriteria solution: $f(S) \

Abstract

We initiate the study of the submodular cover problem in dynamic setting where the elements of the ground set are inserted and deleted. In the classical submodular cover problem, we are given a monotone submodular function and the goal is to obtain a set that minimizes the cost subject to the constraint . This is a classical problem in computer science and generalizes the Set Cover problem, 2-Set Cover, and dominating set problem among others. We consider this problem in a dynamic setting where there are updates to our set , in the form of insertions and deletions of elements from a ground set , and the goal is to maintain an approximately optimal solution with low query complexity per update. For this problem, we propose a randomized algorithm that, in expectation, obtains a -bicriteria approximation using polylogarithmic query complexity per update.
Paper Structure (23 sections, 24 theorems, 61 equations, 5 algorithms)

This paper contains 23 sections, 24 theorems, 61 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Submodular cover
  5. Submodular maximization
  6. Deletion robust algorithms
  7. Preliminaries
  8. Dynamic Algorithm
  9. Setting
  10. Overview of the Algorithms
  11. Parallel Runs
  12. Data Structure Construction
  13. Insertion
  14. Deletion
  15. Choice of Sample Size
  16. ...and 8 more sections

Key Result

Theorem 1.2

Define the weight ratio of $\mathcal{V}$ as $\rho := \frac{\max_{v \in \mathcal{V}} w(v)}{\min_{v \in \mathcal{V}} w(v)},$ and set $n := |\mathcal{V}|$. For any $\epsilon > 0$, there is an algorithm for the dynamic submodular cover problem that maintains an expected $(1-O(\epsilon), O(\epsilon^{-1})

Theorems & Definitions (52)

  • Definition 1.1: Dynamic Submodular Cover problem
  • Theorem 1.2
  • Definition 4.1: Suitable sample size
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • Theorem 5.3
  • proof
  • Definition 1.1
  • ...and 42 more