Profit maximization for competitive social advertising

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Abstract

In social advertising, the social platform host may run marketing campaigns for multiple competing clients simultaneously. In this case, each client comes up with a budget and an influence spread requirement. The host runs campaigns by allocating a set of seed nodes for each client. If the influence spread triggered by a seed set meets the requirement, the host can earn the budget from the corresponding client. In this paper, we study the problem of Profit Maximization, considering that different seeds incur different costs. Given all the clients' requirements met, we aim to find the optimal seed allocation with minimum cost. Under the competitive K-LT propagation model, we show the Profit Maximization problem is NP-hard and NP-hard to approximate with any factor. To find a feasible solution, we propose an effective algorithm that iteratively selects a candidate set and obtains an approximate allocation. The experimental results over a real-world dataset validate the effectiveness of the proposed methods.

Introduction

A viral marketing campaign offers incentives, e.g., free products, to a set of influential users (or seeds) in a social network. The marketer may expect a large chain of product adoption (or user activation) by the online influence spread from these seeds. Finding the top-k seeds for generating the largest influence spread is called the influence Maximization (IM) problem, a hot research topic that has attracted ample attention in the past decade [1], [2], [3], [4].

Most existing IM models assume that the social network structure is available to marketers. However, in the real-world, social network hosts will keep social graphs secret for their own benefits or privacy reasons [5]. Lacking the social graph, marketers are not able to select their optimal seed sets. Therefore, a novel research, social advertising arose.

Social advertising is anchored in the trade between the social network host and marketers (or clients). Existing works assume that each client will set a price for each node in the social network. The host, in turn, will accordingly launch a marketing campaign for each client based on the price. Based on the assumption, many different objectives are implemented. For instance, the regret minimization problem [6] minimizes a regret function which leads to a win-win between the host and the clients. The revenue maximization problem [7] defines the revenue which the host wants to maximize. The fair seed allocation problem [5] aims to make each seed set to yield similar influence spread. Other variants with specific constraints are also actively explored [8].

The above works explicitly consider each successful node activation, i.e., once a node is activated, the host can earn the revenue priced for this node. For example, in [7], the authors give an explicit revenue matrix A in which Aij means the money client i will pay if node j is influenced. The objective in [7] is to find a seed set for each client such that the total revenue for the host is maximized, with each client i giving a budget ki on the seed set size. In the real-world scenario, the clients are unable to set a price for each node due to the huge size of the social network. Meanwhile, since only the host has the knowledge about the influence spread ability of each node, it is very hard for clients to decide a reasonable seed set size. Thus, in this paper, we consider another simple and practical trade model. A client makes a proposition to the platform host with a fixed budget and an influence spread requirement (a threshold). The host can earn the budget if he launches a marketing campaign meeting the influence spread requirement, i.e., the number of activated nodes exceeds the threshold. Note also that selecting different seeds for a marketing campaign may incur different costs. Thus we do not constrain the seed set size for each client and the host can flexibly make selections according to the profit and cost. Combining the above considerations, we propose the problem of Profit Maximization. By finding the optimal allocation of seed sets for each client, the Profit Maximization problem aims to earn the maximum profit, i.e., satisfy all clients with minimum cost.

To find seed sets for each client, we need to understand how multiple marketing campaigns evolve simultaneously. In the real-world, multiple marketing campaigns often compete with each other by launching comparable products over the same market, e.g., Microsoft's Surface vs. Apple's iPad. Considering the cost of product-adoption, each user is assumed to be activated by only one of the competing campaigns [9], [10]. Note the case that users can be activated by multiple campaigns does exist, in which case the influence spread model is a simple aggregation of the model that only one campaign is triggered. In such a case, the model construction and solution approach are somehow trivial. Therefore, in this paper, we focus on such competitive environment, in which multiple marketing campaigns that compete with each other are triggered in the same social network. Accordingly, solving the Profit Maximization problem under a competitive environment requires the seed sets to be mutually exclusive. We show in this paper that this problem is NP-hard, and it is even NP-hard to approximate with any factor.

The hardness makes the problem computationally intractable. Thus we turn to find a feasible solution, which can satisfy all the clients and earn a considerable profit. Note the Profit Maximization problem has separable nature, since we need first to select appropriate seed nodes and then find an optimal allocation of these nodes. A naive approach is to select one node each time and then invoke the allocation method, until the generated allocation can satisfy all the clients. However, such a naive approach is not pragmatic. First, it requires to re-compute the influence spread of each seed whenever a new seed is selected under the competitive influence spread model. This cannot be easily done since computing the influence spread of any seed set is #P-hard [11] and alternative estimation methods are time-consuming [12]. Second, the re-computed influence spread may decrease, which may break the satisfaction of some clients' requirements. This implies we need to re-allocate all the seeds each time a new seed is selected.

We want to decrease the time for repetitive executions. Since the profit is the earned budget minus the incurred cost, we can divide the problem into two subproblems, i.e., Candidate Selection with Minimum Cost (CSMC) problem and Candidate Allocation of Maximum Budget (CAMB) problem. The CSMC tries to find a set of candidate nodes having sufficient influence spread with minimum cost. Then we solve the CAMB problem that finds an allocation of these candidates to earn the maximum budgets. However, the two subproblems are still NP-hard. An effective approximation algorithm is developed for CSMC and we show it has an approximation guarantee with a tight multiplicative ratio on the cost. For the CAMB problem, we prove it cannot be approximate with a factor better than 1/2 and we develop the Next-Fit Decreasing algorithm for CAMB with a 4/9-approximation guarantee under a proportional case. The Feasible Allocation (FA) algorithm for CAMB with a 1/5-approximation guarantee is developed for the general case. Combining the two subproblems, we propose the ICSA (Iterative Candidate Selection and Allocation) algorithm which iteratively generates the candidate set and the corresponding allocation until all the clients are satisfied. The final allocation is returned as a feasible solution. It is worthwhile to highlight our contributions as follows.

  • 1.

    We propose the novel Profit Maximization problem for maximizing the host's profit in presence of competitive clients in social advertising.

  • 2.

    We prove the problem is NP-hard and NP-hard to approximate with any factor. We then divide the problem into two subproblems, the CSMC problem and the CAMB problem. For each subproblem, we develop an efficient algorithm with provable approximation guarantees. The final algorithm ICSA is proposed by consisting of the algorithms for the two subproblems, which can return a feasible solution for the PM problem.

  • 3.

    We conduct experiments over a real-world dataset by testing different kinds of relation functions of budget and influence spread requirement. The experimental results show the effectiveness of the proposed methods.

Section snippets

Related works

The Influence Maximization (IM) problem was first proposed and formulated with a probabilistic framework in [13], [14]. Kempe et al. [1] further formulate it as a discrete optimization problem. They propose the independent cascade (IC) model and linear threshold (LT) model to describe the propagation process of influence and prove the IM problem is NP-hard under these two models. Then they come up with a greedy solution for the IM problem with a 11/e approximation guarantee. Though the greedy

Preliminaries

In this section, we first introduce a variant of the linear threshold model (dubbed as K-LT Model) [5] which describes the competitive influence spread process. Then we formally define the PM problem. For convenience, we list the frequent symbols used in this paper in Table 1.

Proposed solution

Since the problem is NP-hard and is NP-hard to approximate with any factor, we concentrate on finding a feasible solution. The ICSA algorithm proposed in this paper is a combination of solving two subproblems, i.e., Candidate Selection with Minimum Cost (CSMC) problem and Candidate Allocation of Maximum Budget (CAMB) problem. Seeking for high earned budget and low incurred cost, the ICSA algorithm may return a feasible allocation with high profit. At a high level, ICSA works with three steps:

  • Step 1:

Dataset

We use two real social networks Epinion and Gowalla that are both public available.2 Epinions has 75K nodes and 508K directed edges. Gowalla has 197K nodes and 950K undirected edges. In experiments, we change each edge from undirected to bi-directed.

Comparison methods

Since there exists no method for the PM problem, we mainly compare the ICSA algorithm proposed in this paper with two different allocation methods, i.e., NFD and FA. To see the effectiveness of the CSMC algorithm, we

Conclusion and future work

In this paper, we propose the novel Profit Maximization problem and show it is NP-hard and NP-hard to approximate with any factor. We divide the problem into two subproblems, i.e., the CSMC problem and CAMB problem respectively. With solid theoretical analysis, we design the ICSA algorithm which approximately solves the two subproblems and returns a feasible solution for the PM problem. The experimental results on real-world datasets attest the effectiveness of the proposed methods. In

Declaration of Competing Interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Profit Maximization for Competitive Social Advertising”.

Acknowledgement

This work is supported by National Natural Science Foundation of China (Grant No: U1866602, 11671355) and National Key R&D Program of China (Grant No: 2019YFB1600700, 2018AAA0101505).

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