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.

翻译：如何有效利用分配的预算进行商业住宅的品牌推广与宣传是一个重要问题，尤其是在存在多个广告媒体渠道的情况下。其中主要包含两种流行媒体：广告牌和社交媒体广告。在此背景下自然产生一个问题：如何分配预算以最大化总影响力？虽然已有大量文献研究单个广告媒体中预算的有效使用，但几乎未见关于跨多个广告媒体预算分配的研究。为填补这一空白，本文提出了《广告牌与社交网络广告中的预算分配问题》。我们引入"交互效应"概念来刻画多广告媒体触发产生的额外影响力。基于该概念，我们提出一个新颖的影响力函数$Φ(,)$来刻画总影响力，并证明该函数具有非负性、单调性及非双次模性。我们引入"双次模比率$(γ)$"和"广义曲率$(α)$"分别衡量函数接近双次模的程度以及偏离模性的程度。针对非双次模的影响力函数，我们提出随机贪婪算法与两阶段自适应贪婪算法，其近似保证为$\frac{1}α\left(1-e^ {-γα} \right)$。基于真实数据集的实验表明，所提方法的预算分配方案相比现有方法能产生更大的影响力。