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Biased Pareto Optimization for Subset Selection with Dynamic Cost Constraints

Dan-Xuan Liu, Chao Qian

TL;DR

It is proved that BPODC can maintain the best known $(\alpha_f/2)(1-e^{-\alpha_f})$-approximation guarantee when the budget changes, and experiments show that BPODC adapts more effectively and rapidly to budget changes, with a running time that is less than that of the static greedy algorithm.

Abstract

Subset selection with cost constraints aims to select a subset from a ground set to maximize a monotone objective function without exceeding a given budget, which has various applications such as influence maximization and maximum coverage. In real-world scenarios, the budget, representing available resources, may change over time, which requires that algorithms must adapt quickly to new budgets. However, in this dynamic environment, previous algorithms either lack theoretical guarantees or require a long running time. The state-of-the-art algorithm, POMC, is a Pareto optimization approach designed for static problems, lacking consideration for dynamic problems. In this paper, we propose BPODC, enhancing POMC with biased selection and warm-up strategies tailored for dynamic environments. We focus on the ability of BPODC to leverage existing computational results while adapting to budget changes. We prove that BPODC can maintain the best known $(α_f/2)(1-e^{-α_f})$-approximation guarantee when the budget changes. Experiments on influence maximization and maximum coverage show that BPODC adapts more effectively and rapidly to budget changes, with a running time that is less than that of the static greedy algorithm.

Biased Pareto Optimization for Subset Selection with Dynamic Cost Constraints

TL;DR

It is proved that BPODC can maintain the best known -approximation guarantee when the budget changes, and experiments show that BPODC adapts more effectively and rapidly to budget changes, with a running time that is less than that of the static greedy algorithm.

Abstract

Subset selection with cost constraints aims to select a subset from a ground set to maximize a monotone objective function without exceeding a given budget, which has various applications such as influence maximization and maximum coverage. In real-world scenarios, the budget, representing available resources, may change over time, which requires that algorithms must adapt quickly to new budgets. However, in this dynamic environment, previous algorithms either lack theoretical guarantees or require a long running time. The state-of-the-art algorithm, POMC, is a Pareto optimization approach designed for static problems, lacking consideration for dynamic problems. In this paper, we propose BPODC, enhancing POMC with biased selection and warm-up strategies tailored for dynamic environments. We focus on the ability of BPODC to leverage existing computational results while adapting to budget changes. We prove that BPODC can maintain the best known -approximation guarantee when the budget changes. Experiments on influence maximization and maximum coverage show that BPODC adapts more effectively and rapidly to budget changes, with a running time that is less than that of the static greedy algorithm.
Paper Structure (7 sections, 2 theorems, 11 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 7 sections, 2 theorems, 11 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Subset Selection with Cost Constraints
  3. Previous Algorithms
  4. The BPODC Algorithm
  5. Theoretical Analysis
  6. Empirical Study
  7. Conclusion

Key Result

theorem thmcountertheorem

For the static problem in Definition def:static-problem, BPODC using at most $O(nB/(p_{min}\cdot\delta_{\widehat{c}}))$ expected number of iterations finds a subset $X\subseteq V$ with where $f(\widetilde{X})$ is defined in Eq. (eq-optimal).

Figures (4)

  • Figure 1: The cumulative budget changes relative to the initial budget.
  • Figure 2: The average value $\pm$ std for influence maximization under each budget change.
  • Figure 3: The average value $\pm$ std for maximum coverage under each budget change.
  • Figure 4: Additional results for influence maximization under dynamic environment.

Theorems & Definitions (10)

  • definition thmcounterdefinition: Submodularity Ratio GGADBLP:conf/ijcai/Qian0T18
  • definition thmcounterdefinition: Subset Selection with Static Cost Constraints
  • definition thmcounterdefinition: Subset Selection with Dynamic Cost Constraints
  • definition thmcounterdefinition: Influence Maximization
  • definition thmcounterdefinition: Maximum Coverage
  • definition thmcounterdefinition: Domination
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof