1 Introduction
It has been reported that one of the most effective ways of running promotions on social media is the use of coupon campaigns on the largest social media site such as Facebook and Twitter. Different from conventional online advertising, the offer of a coupon on social network could trigger a large amount of likes and shares on your product and build brand awareness rapidly. This motivates us to study the coupon allocation problem, e.g., allocating coupons to a few users so as to maximize the expected cascade. More specifically, we assume that the company would like to promote a product through offering coupons with different values to a subset of users. Whether a user would accept the coupon and further become the seed node is probabilistic, which depends on the value of the coupon. Our goal is to select a set of users and allocate one proper coupon to each of them, such that the expected cascade is maximized.
We notice that our problem is closely related to influence maximization problem (IM). Although IM has been extensively studied in the literature [Kempe et al.2003, Chen et al.2013, Leskovec et al.2007, Cohen et al.2014, Chalermsook et al.2015]
, most of existing studies assume a “budgeted, deterministic and submodular” setting, where there is a predefined budget for selecting seed nodes, any node is guaranteed to be activated as a seed node once it is selected, and the utility function is assumed to be submodular in terms of seed nodes. However, these assumptions may not always hold in reality. Firstly, the probability that a user is activated as a seed node depends on many factors, including the actual amount of rewards received from the company
[Yang et al.2016]. Secondly, the utility function may not always satisfy the submodular property, e.g., some user can only be influenced if majority of her friends are activated [Kuhnle et al.2017].In this paper, we study the coupon allocation problem that allows a general utility function. In particular, we assume an approximate submodular function where measures the distance between a given function and its closest submodular function (a detailed definition can be found in Section 3.1). We formulate the coupon allocation problem as an approximate submodular maximization problem subject to a partition matroid and knapsack constraints. Existing techniques relying on the submodularity of the utility function, such as greedy algorithm, can not work directly on a nonsubmodular function. We propose a novel approximate algorithm with approximation ratio with . This is, to the best of our knowledge, the strongest theoretical result available for approximate submodular maximization problem subject to a partition matroid and knapsack constraints. Although we restrict our attention to coupon allocation in this paper, our results apply to a broad range of nonsubmodular maximization problems.
The contributions of this paper can be summarized as follows: (1) We are the first to study the coupon allocation problem that allows a general utility function. Our objective is to determine the best coupon allocation so as to maximize the expected cascade subject to (partition) matroid and knapsack constraints. (2) We develop an efficient algorithm with approximation ratio that depends on . When (the utility function is submodular), our result converges to which is the best possible for submodular maximization subject to matroid constraints. This research contributes fundamentally to the development of approximate solutions for any problems that fall into the family of approximate submodular maximization.
Some important notations are listed in Table 1.
Notation  Meaning 

the ground set that contains all uc pairs  
all uc pairs whose user is  
the expected cascade of  
a decision matrix  
the multilinear extension of  
the concave extension of  
a submodular function used to bound  
the coordinate wise maximum of and  
2 Related Work
IM has been extensively studied in the literature [Kempe et al.2003, Chen et al.2013, Leskovec et al.2007, Cohen et al.2014, Chalermsook et al.2015], their objective is to find a set of influential customers so as to maximize the expected cascade. However, our work differ from all existing studies in several major aspects. Traditional IM assumes any node is guaranteed to be activated once it is selected, we relax this assumption by allowing users to response differently to different coupon values. Recently, [Yang et al.2016] study discount allocation problem in social networks. However, they assume a submodular utility function and continuous setting of discount value, our model allows a general utility function. We formulate our problem as an approximate submodular maximization problem subject to matroid and knapsack constraints. Existing approaches, such as greedy algorithm [Horel and Singer2016], can not apply directly to matroid and knapsack constraints. We propose a novel algorithm that provides the first bounded approximate solutions to this problem. It was worth noting that our approximation ratio converges to when . This is the best theoretical result available for approximate submodular maximization subject to a partition matroid constraint.
3 Preliminaries
3.1 Submodular Function and Its Continuous Extensions
A submodular function is a set function , where denotes the power set of , which satisfies a natural “diminishing returns” property: the marginal gain from adding an element to a set is at least as high as the marginal gain from adding the same element to a superset of . Formally, a submodular function satisfies the follow property: For every with and every , we have that We say a submodular function is monotone if whenever .
3.1.1 Continuous Extensions
Consider any vector
such that each . The multilinear extension of is defined as , and the concave extension of is defined as .3.1.2 approximate Submodular Function
An approximate submodular function is a set function that satisfies the following condition: there exists a submodular function such that for any , we have .
3.2 Coupon Adoption and Propagation Model
Assume there are users and types of coupons . For simplicity of presentation, we will directly use to denote the coupon value of . Given the set of users and possible coupon values , define as the solution space, adding a usercoupon (uc) pair to our solution translates to offering coupon to user . When any coupon is allocated to any user , we assume that with probability , adopts and becomes a seed node, thus incurs a cost . We further assume that coupons cannot be combined, if a user received multiple coupons, her adoption decision only depends on the coupon with the highest value.
After a set of users become seed nodes, they start to influence other users. Assume is the seed set, the expected cascade of , which is the expected number of influenced users given seed set , is denoted as . In this work, we assume that is a approximate submodular function: there exists a submodular function such that for any , we have . Our propagation model subsumes many classic propagation models, including Independent Cascade Model and Linear Threshold Model [Kempe et al.2003] as special cases, as the propagation functions defined in their models are submodular and monotone, i.e., .
4 Problem Statement
Our objective is to identify the best coupon allocation policy, not necessarily deterministic, to maximize the expected cascade. Given a coupon allocation , we use to denote the largest coupon value allocated to user under , then the probability that a subset of users successfully become the seed set is
As introduced earlier, we use to denote the expected cascade under seed set , then the expected cascade under allocation is .
We first prove that is also approximate submodular.
Lemma 1.
If is approximate submodular, is also approximate submodular: there exists a submodular function such that .
Proof.
Based on a similar proof provided in [Soma and Yoshida2015], we can show that if is submodular of , is submodular of . Because
we have . We finish the proof by defining . ∎
The expected cost of a coupon allocation is
Coupon Allocation Policy
We denote by a coupon allocation policy, where is the probability that is adopted. Then the expected cascade under is . The expected cost of is .
Definition 1 (Feasible Coupon Allocation Policy).
We say a policy is feasible if and only the following conditions are satisfied:

(Attention Constraint) For every user , we denote . , e.g., every user receives at most one coupon.

(Budget Constraint) , e.g., the expected value of the coupons redeemed is at most .
Our objective is to identify a feasible coupon allocation policy that maximizes the expected cascade. We present the formal definition of our problem in P.A.
P.A subject to:
We extend our model in Section 6 by incorporating one more constraint, e.g., a feasible policy can not allocate coupons to more than users. This constraint captures the fact that the company often has limited budgeted on coupon producing and distribution.
5 An Approximate Solution
We first introduce a new problem P.B as follows.
P.B: Maximize subject to:
In the above formulation, is a decision matrix and is a concave extension of .
(1) 
We first prove that P.B is a relaxed version of P.A.
Lemma 2.
Assume is the optimal solution to P.A and is the optimal solution to P.B, we have .
Proof.
Given the optimal policy , for every uc pair , we define as the probability that is offered by , e.g., . We first prove that is a feasible solution to P.B. Because is a feasible policy, we have

, e.g., every user receives at most one coupon under . It follows that , thus condition (C1) is satisfied.

, e.g., the expected cost of any coupon allocation under is no larger than . Because is the expected cost of , we have , thus condition (C2) is satisfied.

. Because is the probability that is offered by , we have , thus condition (C3) is satisfied.
On the other hand, by setting for every in (1), we have . ∎
Assume is the concave extension of , we next prove that is an approximate of .
Lemma 3.
Proof.
Given any , assume the value of is achieved at , we have . ∎
5.1 Algorithm Design
In this section, we present a greedy algorithm that achieves a bounded approximation ratio. Our general idea is to first find a fractional solution with a bounded approximation ratio and then round it to an integral solution.
5.1.1 Continuous Greedy
In [Vondrák2008] they develop a continuous greedy algorithm based on the multilinear extension in order to maximize a submodular monotone function over a matroid constraint. We extend their results to nonsumbodular maximization subject to a partition matroid and knapsack constraints. We use to denote a matrix with entry one and all other entries zero. Given two matrices and , let denote the coordinatewise maximum. Define . A detailed description of our algorithm is listed in Algorithm 1.
Lemma 4.
Let denote the solution returned from Algorithm 1, we have .
Proof.
As proved in [Calinescu et al.2011], if is a submodular function, . Let denote the optimal solution to problem P.B, assume is our solution at round , then for every round , we have
It follows that
(2)  
The first inequality is due to Lemma 3. Let denote the optimal solution to problem P.C in Algorithm 1 at round , then we have
(3)  
The first inequality is due to (2) and the second inequality is because is the optimal solution to problem P.C. According to Line 7, the increased value of our solution at round is at least
(4)  
The last inequality is due to .
Inequality (5) is due to . Define , according to (6), we have , thus
It follows that
( can be removed when choosing small enough [Vondrák2008] ) ∎
5.1.2 Rounding
In the rest of this paper, we use to denote for short. After obtaining , we use swap rounding [Chekuri et al.2010] to round the fractional solution to integral solutions.
Theorem 1.
Our rounding approach returns a feasible solution to problem P.A, and
where .
Proof.
We first prove the feasibility of our rounding approach. First, our policy trivially satisfied Cc in P.A.1. Since the solution returned from swap rounding satisfies matroid constraint, our solution satisfies Ca in P.A.1. Because each uc pair is selected with probability and is a feasible solution to problem P.B, the expected cost of is (Cb in P.A is satisfied).
We next prove that . First, because is a submodular function, we have . Because , we have
(7) 
6 Extension: Incorporating Distribution Cost
We now discuss one extension of our analysis thus far. The difference between the extended model and the previous model is that in the new model, we add one more constraint to Definition 1. This additional constraint models the fact that the company only has limited resource to produce and distribute coupons to different users. Assume the cost of allocating a coupon to user is , we say a policy is feasible if and only the following conditions are satisfied:

(Attention Constraint) .

(Coupon Cost Constraint) .

(Coupon Distribution Budget Constraint) .
Similar to the previous model, we formulate our problem P.A.1 as follows.
P.A.1 subject to:
Its relaxation P.B.1 can be formulated as follows.
P.B.1 Maximize subject to:
We follow a similar idea used in the previous section to design our algorithm, e.g., compute a fractional solution first and then round it to integral solution. However, to tackle the new challenge brought by the additional constraint, we need a brand new solution to ensure the feasibility of the final solution.
We first introduce a new problem P.A.2 as follows. P.A.2 is a restricted version of P.A.1 where the coupon distribution budget is scaled down by a factor of .
P.A.2 subject to:
We next introduce a relaxed version of problem P.A.2.
P.B.2 Maximize subject to:
6.0.1 Continuous Greedy Algorithm
6.0.2 Rounding
We next design a brand new rounding approach that satisfies all constraints. In the rest of this paper, we use to denote the solution returned from Algorithm 2, let denote the optimal solution to problem P.B.2 and denote the optimal solution to problem P.B.1. Our approach consists of two major parts: a rounding stage (Step 1) and a conflict resolution stage (Step 2).
Step 1: Given , we first introduce a dummy coupon with value and for each user , define . For each user , we assign exactly one coupon to her, coupon to user with probability . The returned solution is denoted as . We notice that may not be a feasible solution, e.g., the coupon distribution cost of could be larger than .
Step 2: Consider uc pairs from in nondecreasing order for their distribution cost, let the current uc pair survive if it does not violate C4, else we discard it. We add all survived uc pairs to .
We next prove that is feasible and its expected cascade is close to the optimal solution.
Theorem 2.
is feasible and
where and .
Proof.
We first prove the feasibility of . First, our policy trivially satisfied Cc in P.A.1. Consider the random set returned from Step 1, it allocates at most one coupon to each user (Ca in P.A.1 is satisfied) and the expected cost of is (Cb in P.A.1 is satisfied). Since is a subset of , it also satisfied Ca and Cb. Moreover, according to Step 2, satisfied Cd in P.A.1.
We next prove by putting together the following three lemmas.
Lemma 5.
.
Proof of Lemma 5. First, it is easy to verify that is a feasible solution to P.B.2, thus we have . Assume the value of is achieved at , we have .
Lemma 6.
.
Proof of Lemma 6. It was shown in [Vondrák2008] that “Step 1” can be used to round solutions in the basic partition matroid polytope without losing in terms of the objective function. Because is a submodular function, we have . The rest of the proof is now analogous to the proof of Theorem 1.
Lemma 7.
.
Proof of Lemma 7. According to Step 1, given the fractional solution , we decide the coupon allocation for each user independently, that is the coupon allocated to each user is independent from each other. This nice property enables us to follow a similar proof in [Vondrák et al.2011] to prove that for each , , that is for any coupon that is having survived after Step 1, the probability that it still survives after Step 2 is at least . According to Theorem 5 in [Bansal et al.2010], we have .
∎
We emphasize that Theorem 2 is of independent interest and may find applications in any approximate submodular maximization problem subject to knapsack and a basic partition matroid constraints.
7 Conclusion
In this paper, we study coupon allocation problem in social networks. Our framework allows a general utility function and more complicated constraints. Therefore, existing techniques relying on the submodularity of the utility function can not apply to our problem directly. We propose a novel approximate algorithm with approximation ratio depending on . Although we limit our attention to coupon allocation problem in this paper, our results apply to a broad range of approximate submodular maximization problems.
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