Approximately Bisubmodular Regret Minimization in Billboard and Social Media Advertising

TL;DR

This work proposes a monotone, approximately bisubmodular influence model and develops two algorithmic solutions: Projected Subgradient Method based on the Lovasz extension of the regret function, and an Approximate Bisubmodular Local Search algorithm with provable guarantees.

Abstract

We study the problem of minimizing regret in multi-mode advertisement settings, where an influence provider allocates advertising resources such as social network seeds and billboard slots to multiple advertisers with specified influence demands and payments. Unlike prior work focusing on a single mode of advertising, we consider the interplay between online and offline modes and introduce a novel regret model that captures their interaction effect. This leads to a regret minimization problem that is non-monotone, non-submodular, and NP-hard to approximate within any constant factor. To address this, we propose a monotone, approximately bisubmodular influence model and develop two algorithmic solutions: Projected Subgradient Method based on the Lovász extension of the regret function, and an Approximate Bisubmodular Local Search algorithm with provable guarantees. Experiments on large-scale real-world datasets, including billboard and trajectory data from major U.S. cities, as well as social network graphs, demonstrate that our methods outperform existing baselines in minimizing total regret while satisfying advertiser demands. Our framework is broadly applicable to other resource allocation scenarios beyond advertising.

Figures (5)

• Figure 1: Projected Subgradient Method (PGM) for Regret Minimization Problem
• Figure 2: Approximate Bisubmodular Local Search (ABLS) for Regret Minimization Problem
• Figure 3: Regret on varying $\alpha$, when $\lambda = 5\%, \mathcal{|A|} = 20$ for Uniform (a-e), Trivalency (f-j), Weighted Cascade (k-o)Settings
• Figure 4: Time on varying $\alpha$, $(a)$ Uniform, $(b)$ Weighted, $(c,d,e)$ Trivalency probability Setting and Satisfied Adv. on Varying $\alpha$,$(f)$ Uniform, $(g)$ Weighted, $(i,j,k)$ Trivalency probability Setting
• Figure 5: Regret on varying $\alpha$, when $\lambda = 1\%, \mathcal{|A|} = 100$$(a-e)$, when $\lambda = 20\%, \mathcal{|A|} = 5$$(f-k)$, for Trivalency Settings

Theorems & Definitions (21)

• definition 1: $\varepsilon$-Approximately Bisubmodular
• definition 2: Lovász Extension
• definition 3: Influence of Billboard Slots
• theorem 1
• definition 4: Independent Cascade Model
• theorem 2
• definition 5: Influence Model
• definition 6: Interaction Effect
• proposition 1
• proposition 2