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Approximation Algorithms for Budget Splitting in Multi-Channel Influence Maximization

Dildar Ali, Ansh Jasrotia, Abishek Salaria, Suman Banerjee

Abstract

How to utilize an allocated budget effectively for branding and promotion of a commercial house is an important problem, particularly when multiple advertising media are available. There exist multiple such media, and among them, two popular ones are billboards and social media advertisements. In this context, the question naturally arises: how should a budget be allocated to maximize total influence? Although there is significant literature on the effective use of budgets in individual advertising media, there are hardly any studies examining budget allocation across multiple advertising media. To bridge this gap, this paper introduces the \textsc{Budget Splitting Problem in Billboard and Social Network Advertisement}. We introduce the notion of \emph{interaction effect} to capture the additional influence due to triggers from multiple media of advertising. Using this notion, we propose a noble influence function $Φ(,)$ that captures the total influence and shows that this function is non-negative, monotone, and non-bisubmodular. We introduce \emph{bi-submodularity ratio $(γ)$} and \emph{generalized curvature $(α)$} to measure how close a function is to being bi-submodular and how far a function is from being modular, respectively. We propose the Randomized Greedy and Two-Phase Adaptive Greedy approach, where the influence function is non-bisubmodular and achieves an approximation guarantee of $\frac{1}α\left(1-e^ {-γα} \right)$. We conducted several experiments using real-world datasets and observed that the proposed solution approach's budget splitting leads to a greater influence than existing approaches.

Approximation Algorithms for Budget Splitting in Multi-Channel Influence Maximization

Abstract

How to utilize an allocated budget effectively for branding and promotion of a commercial house is an important problem, particularly when multiple advertising media are available. There exist multiple such media, and among them, two popular ones are billboards and social media advertisements. In this context, the question naturally arises: how should a budget be allocated to maximize total influence? Although there is significant literature on the effective use of budgets in individual advertising media, there are hardly any studies examining budget allocation across multiple advertising media. To bridge this gap, this paper introduces the \textsc{Budget Splitting Problem in Billboard and Social Network Advertisement}. We introduce the notion of \emph{interaction effect} to capture the additional influence due to triggers from multiple media of advertising. Using this notion, we propose a noble influence function that captures the total influence and shows that this function is non-negative, monotone, and non-bisubmodular. We introduce \emph{bi-submodularity ratio } and \emph{generalized curvature } to measure how close a function is to being bi-submodular and how far a function is from being modular, respectively. We propose the Randomized Greedy and Two-Phase Adaptive Greedy approach, where the influence function is non-bisubmodular and achieves an approximation guarantee of . We conducted several experiments using real-world datasets and observed that the proposed solution approach's budget splitting leads to a greater influence than existing approaches.

Paper Structure

This paper contains 32 sections, 10 theorems, 13 equations, 3 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Observation.
  3. Motivation.
  4. Main Contributions.
  5. Organization of the Paper.
  6. Related Work
  7. Influence Maximization
  8. Regret Minimization
  9. Background Problem Definition
  10. Set Function and Its Properties
  11. Billboard Advertisement
  12. Social Network Advertisement
  13. Combine Influence Model
  14. Problem Definition
  15. Proposed Methodology
  16. ...and 17 more sections

Key Result

Lemma 3

The influence function $\mathcal{I}()$ is non-negative, monotone, and submodular ali2023efficientali2024effectiveali2025fairness. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Varying Algorithms Vs. Influence in Trivalency $(a,b,c,d)$, in Weighted Cascade $(e,f,g,h)$, in Uniform probability setting $(i,j,k,\ell)$ in CA Dataset
  • Figure 2: Varying Algorithms Vs. Budget split percentage in Trivalency $(a,b,c,d)$, in Weighted Cascade $(e,f,g,h)$, in Uniform probability setting $(i,j,k,\ell)$ in CA Dataset
  • Figure 3: Varying Algorithms Vs. Budget split percentage in Trivalency $(a,b,c,d)$, Budget Vs. Influence for Trivalency $(e)$, Weighted Cascade $(f)$, Uniform $(g)$, Budget Vs. Time for Trivalency $(h)$, Weighted Cascade $(i)$, Uniform $(j)$ for CA dataset. Budget Vs. Influence for Trivalency $(k)$, Weighted Cascade $(\ell)$, Uniform $(m)$, Budget Vs. Time for Trivalency $(n)$, Weighted Cascade $(o)$, Uniform $(p)$ for USA dataset.

Theorems & Definitions (17)

  • Example 1
  • Definition 2: Influence of Billboard Slots
  • Lemma 3
  • Definition 4: Combine Influence Model
  • Definition 5: Interaction Effect
  • Theorem 6
  • Definition 7: Budget Split Problem in Billboard and Social Network Advertisement
  • Theorem 8
  • Definition 9: Bisubmodularity Ratio
  • Definition 10: Generalized Curvature
  • ...and 7 more