Efficient Approximation Algorithms for Adaptive Target Profit Maximization
EEfficient Approximation Algorithms for AdaptiveTarget Profit Maximization
Keke Huang
School of Comp. Sci. and Engg.Nanyang Technological [email protected]
Jing Tang ∗ Dept. of Ind.Syst. Engg. and Mgmt.National University of [email protected]
Xiaokui Xiao
School of ComputingNational University of [email protected]
Aixin Sun
School of Comp. Sci. and Engg.Nanyang Technological [email protected]
Andrew Lim
Dept. of Ind.Syst. Engg. and Mgmt.National University of [email protected]
Abstract —Given a social network G , the profit maximization(PM) problem asks for a set of seed nodes to maximize the profit,i.e., revenue of influence spread less the cost of seed selection. Thetarget profit maximization (TPM) problem, which generalizesthe PM problem, aims to select a subset of seed nodes froma target user set T to maximize the profit. Existing algorithmsfor PM mostly consider the nonadaptive setting, where all seednodes are selected in one batch without any knowledge on howthey may influence other users. In this paper, we study TPMin adaptive setting, where the seed users are selected throughmultiple batches, such that the selection of a batch exploits theknowledge of actual influence in the previous batches. To acquirean overall understanding, we study the adaptive TPM problemunder both the oracle model and the noise model , and proposeADG and A DD ATP algorithms to address them with strongtheoretical guarantees, respectively. In addition, to better handlethe sampling errors under the noise model, we propose the idea of hybrid error based on which we design a novel algorithm HATPthat boosts the efficiency of A DD ATP significantly. We conductextensive experiments on real social networks to evaluate theperformance, and the experimental results strongly confirm thesuperiorities and effectiveness of our solutions.
Index Terms —target profit maximization, social networks,approximation algorithms
I. I
NTRODUCTION
Online social networks (OSNs) such as Facebook and Twit-ter have witnessed their prosperous developments in recentyears. Many companies have taken OSNs as the major adver-tising channels to promote their products via word-of-moutheffect. Those rapid proliferations have motivated substantialresearch on viral marketing strategies for maximal profits.Specifically, market strategy makers seek for a proper set ofinfluential individuals and invest in each of them (e.g., cash-back rewards, coupons, or discounts) to exert their influenceon advertising, aiming to maximize the expected profit. Thisproblem is commonly studied as the profit maximization (PM) problem in the literature [2], [19], [23], [26], [27], [32]. PM
A short version of the paper will appear in the 36th IEEE InternationalConference on Data Engineering (ICDE ’20), April 20–24, 2020, Dallas,Texas, USA. ∗ Corresponding author: Jing Tang. problem asks for a set of seed users S from network G at thecost of c ( S ) so as to maximize the total expected profit, i.e.,the expected spread of S less the total investment cost c ( S ) .However, this vanilla PM problem overlooks one fact thateven though the advertisers have the full knowledge about thewhole network, they are likely to only have access to a fractionof users [4], which is quite common in marketing applications.For example, companies advertise new products to users intheir subscription mailing list, or new shop owners provide freesamples to the popularities or celebrities who visit their storeon site, to name a few. To circumvent this potential issue, weextend the vanilla PM problem into a more generalized versionand propose the target profit maximization (TPM) problem.Specifically, given a social network G = ( V, E ) and a targetuser set T ⊆ V with nonnegative expected profit where eachuser u ∈ T is associated with a cost c ( u ) , the TPM problemaims to select a subset S ⊆ T to maximize the expectedprofit. The target user set T can be either made up by currentaccessible users in the social network or selected by currentseed selection algorithms. In particular, if the target set T contains all users in the social network, i.e., T = V , TPMthen degenerates to PM.Existing work on profit maximization mostly focuses on the nonadaptive setting [2], [19], [23], [26], [27], [32], where allseed nodes are selected in one batch without any knowledgeon how they may influence other users. As a consequence ofthe nonadaptiveness, the maximal possible profit might not beachieved. On the contrary, if we could observe the real-timefeedback from the market and response with a smarter seedselection procedure, we could make further improvement onthe final profit due to the adaptivity gap [2], [12], [13], [21].Motivated by this fact, we propose an adaptive strategy on seedselection for profit maximization and formulate the problemas the adaptive TPM problem. In a nutshell, as long as thereexists a profitable target seed set, adaptive TPM would tryto derive a subset to maximize the profit by exploiting theadaptivity advantage.To the best of our knowledge, we are the first to consider a r X i v : . [ c s . S I] O c t rofit maximization in adaptive setting and aim to provideinspirational insights for future research. To acquire an overallunderstanding, we first study this problem in oracle model where the profit of any node set can be obtained in O (1) time. We then consider it in a more practical noise model where the expected profits (or expected spreads) can onlybe estimated through sampling. Eventually, we design anefficient algorithm with nontrivial approximation guaranteesto address the adaptive TPM problem. In summary, our majorcontributions are as follows. • Proposal of adaptive target profit maximization.
Weare the first to consider profit maximization in adaptivesetting and propose the adaptive target profit maximizationproblem. By utilizing the knowledge from the marketfeedback, we study adaptive strategies to select moreeffective seed nodes to maximize the expected profit. • Theoretical analyses on oracle and noise model.
To gaina comprehensive understanding, we study adaptive TPMin both oracle model and noise model . Specifically, in oracle model where the profit of any node set is accessiblein O (1) time, we propose ADG algorithm and provethat it could achieve an -approximation. In the noisemodel where any expected profit (or expected spread)only can be estimated, we extend ADG into A DD ATPby considering additive sampling error and derive thecorresponding approximation guarantee. • Practical algorithm with optimized efficiency.
To fur-ther optimize the efficiency of A DD ATP in the noisemodel, we propose HATP algorithm by adopting the ideaof hybrid error . Specifically, additive error could incurprohibitive computation overhead (see Section IV-A). Totackle this issue, we propose hybrid error which combinesboth relative error and additive error, fitting nicely tothe scenario of nodes with diverse expected spreads.Based on this novel technique, we design HATP thatboosts the efficiency of A DD ATP significantly. Finally, itsapproximation guarantee and time complexity are derived.II. P
RELIMINARIES
This section presents the formal definition of the adaptivetarget profit maximization problem. To demonstrate the influ-ence propagation process, we take the independent cascade(IC) model [16] as an illustration, which is one of theextensively studied propagation models on this topic. Table Isummarizes the notations that are frequently used.
A. Influence Propagation and Realization
Let G = ( V, E ) be a social network with a node set V and a directed edge set E where n = | V | and m = | E | . Foreach edge (cid:104) u, v (cid:105) ∈ E , u is called an incoming neighbor of v and v is an outgoing neighbor of u . Each edge e ∈ E isassociated with a probability p ( e ) ∈ (0 , . For simplicity, sucha social network is referred to as probabilistic social graph .Given an initial node set S ⊆ V , the influence propagationprocess started by S under the independent cascade (IC) model TABLE IF
REQUENTLY USED NOTATIONS
Notation Description G = ( V, E ) a social network with node set V and edge set En, m the numbers of nodes and edges in G , respectively T the set of target seed nodes k the number of elements in T , i.e., k = | T | S i the seed set selected in the i -th iteration T i the seed set candidate in the i -th iteration G i the i -th residual graph n i , m i the numbers of nodes and edges in G i I G i ( u i ) the number of nodes activated by node u i on G i Cov R ( S ) the number of RR-sets in R that overlap S E [ I ( S )] the expected spread of seed set Sφ, Φ , Ω a specific realization, a random realization, and the real-ization space π f , π r , π opt a front greedy policy, a rear greedy policy, and an optimalpolicy is stochastic as follows. During time interval t , all nodesin S are activated and other nodes in V \ S are inactive.At time interval t i , each node activated during time interval t i − has one chance to activate its inactive outgoing neighborswith the probability associated with that edge. This influencepropagation process continues until no more inactive nodescan be activated. After the propagation process terminates, let I G ( S ) be the number of nodes activated in G . We call S asthe seed set and I G ( S ) as the spread of S on G . Note that I G ( S ) is a random variable with respect to the propagationprocess. For simplicity, we use I ( S ) to represent I G ( S ) byomitting the subscript G unless otherwise specified.The above process can also be interpreted by utilizingthe concept of realization (aka possible world). Given aprobabilistic social graph G , one realization, denoted as φ ,is a residual graph of G constructed by removing each edge e ∈ E with probability − p ( e ) . Let Ω be the set of all possiblerealizations and we have | Ω | = 2 m where m = | E | . Let Φ bea random realization randomly sampled from Ω , denoted as Φ ∼ Ω . Given a specific realization φ ∈ Ω and any seed set S ⊆ V , the number of nodes that can be reached by S under φ is the spread of S , denoted as I φ ( S ) . With this regard, theexpected spread of S , denoted as E [ I ( S )] , can be expressedas follows. E [ I ( S )] := E Φ ∼ Ω [ I Φ ( S )] = (cid:88) φ ∈ Ω I φ ( S ) · p ( φ ) , (1)where p ( φ ) is the probability of φ sampled from Ω . B. Adaptive Target Profit Maximization
Given a social network G = ( V, E ) , we consider a targetuser set T ⊆ V and each user u ∈ T is associated with acost c ( u ) . For any set S ⊆ T , we denote ρ ( S ) as the expectedprofit of S , which is defined as ρ ( S ) := E [ I ( S )] − c ( S ) , (2) .4 v v v v v v v (a) A social graph G v v v v v v v (b) v as the first seed v v v v (c) The residual graph G v v v v (d) v as the second seedFig. 1. An adaptive profit maximization process. where E [ I ( S )] is the expected spread of S and c ( S ) = (cid:80) u ∈ S c ( u ) . As ρ ( · ) is a positive linear combination of amonotone submodular function (i.e., E [ I ( · )] ) and a modularfunction (i.e., c [ · ] ), we know that ρ ( · ) is still submodular,which, however, may not be monotone. We assume that the ex-pected profit of the target set T is nonnegative, i.e., ρ ( T ) ≥ .Then, the target profit maximization (TPM) problem aims toselect a subset S ⊆ T to maximize expected profit.Conventionally, seed nodes are nonadaptively selected with-out any knowledge of realization happened in the actualinfluence propagation process. Contrarily, the adaptive strategyselects each node based on the real-time feedback from therealization observed, which is proved to be more effective [2],[12], [13]. Specifically, the adaptive strategy selects node oneby one in an adaptive manner as follows. Let G = G and T = T at the beginning. It first selects a node u from thetarget set T based on G , and then observes how node u activates other nodes. As follows, it removes those activatednodes from G and T , resulting in the residual graph G andset T respectively. Subsequently, it then selects the next node u from T based on the residual graph G . This process willbe repeated until a solution S is produced.For example, Fig. 1(a) gives a probabilistic social graph G under the IC model. Suppose that the target seed set is T = { v , v , v } and each node in T has a cost of . . Onecan verify that the optimal nonadaptive solution is T with anexpected profit of ρ G ( T ) = E [ I G ( T )] − c ( T ) = 6 . − . . . As with the adaptive seeding strategy, it wouldselect node v as the first seed node, as shown in Fig. 1(b).Specifically, nodes in gray double-cycle are seeds and nodesin blank double-cycle are activated by the seeds. Bold full-line arrow (resp. bold dashed-line arrow) indicates a successful(resp. failed) step of influence. Observe that v and v areactivated by v and thus G is updated to G by removing { v , v , v } , as shown in Fig. 1(c). Then, node v is selectedas the second seed and v and v will be activated, as shown inFig. 1(d). Eventually, { v , v } are selected with a total profitof − . However, under this realization, the solution { v , v , v } selected by the nonadaptive algorithm produces aspread of , which results in the final profit of − . . .Thus, the adaptive strategy generates more profits.In this paper, we aim to study algorithms that obtain seedselection strategies (also known as policies ) for adaptive targetprofit maximization. For convenience, we abuse notation anduse the terms “policy” and “algorithm” interchangeably in thepaper. Let S φ ( π ) be the seed set selected by the policy π under realization φ . Then, the profit of policy π under realization φ is ρ φ ( S φ ( π )) = I φ ( S φ ( π )) − c ( S φ ( π )) . Now, we give theexpected profit of policy over all possible realizations. Definition 1 (Expected Profit of Policy) . The expected profit Λ( π ) of policy π is defined as Λ( π ) := E Φ ∼ Ω [ ρ Φ ( S φ ( π ))] = (cid:88) φ ∈ Ω ρ φ ( S φ ( π )) · p ( φ ) , (3) where p ( φ ) is the probability of φ sampled from Ω . Based on the definition of expected profit of policy, adaptivetarget profit maximization is defined as follows.
Definition 2 (Adaptive TPM) . Given a probabilistic graph G = ( V, E ) and a target user set T ⊆ V with nonnegativeexpected profit where each node u ∈ T is associated witha cost c ( u ) for seed selection, the adaptive target profitmaximization problem asks for a policy π ∗ that maximizesthe expected profit over all possible realizations, i.e., π ∗ := arg max π Λ( π ) . (4)III. A DAPTIVE T ARGET P ROFIT M AXIMIZATION
In this section, we first introduce the double greedy methodwhich is widely adopted to address the profit maximizationproblem in the literature. Then we extend double greedy to itsadaptive version based on which we develop ADG algorithmand A DD ATP algorithm to address the adaptive target profitmaximization problem under the oracle model and noise model respectively. As what follows, we then present the analyses ofthe approximation guarantees of ADG and A DD ATP.
A. Double Greedy
Profit maximization is an application of unconstrained sub-modular maximization (USM) for which Buchbinder et al. [7]propose a double greedy algorithm as shown in Algorithm 1.The double greedy algorithm is widely used to address theprofit maximization problem in the literature [20], [26], [30].Algorithm 1 presents the pseudocode of double greedy. Forany submodular function f ( · ) , double greedy works as follows.It maintains two sets S and T initialized with empty set andground set V , respectively. It then checks each node u ∈ V one by one (in arbitrary sequence) to decide on selecting orabandoning u . To this end, it calculates the marginal gain ofselecting it conditioned on current seed set S , denoted as z + ,and the marginal gain of abandoning it conditioned on currentcandidate set T , denoted as z − . If z + ≥ z − , node u will lgorithm 1: Double Greedy [7] Initialize S ← ∅ , T ← V ; foreach node u ∈ V do z + ← f ( u | S ) ; z − ← − f ( u | T \ { u } ) ; if z + ≥ z − then S ← S ∪ { u } ; else T ← T \ { u i } ; return S (= T ) ;be selected and added to S . Otherwise, u is removed from T . When all nodes in V have been checked, the resultant S and T are equal and either of them is returned. The rationalebehind double greedy is quite straightforward, i.e., the node isselected only if it could bring larger marginal gain by keepingit than that by removing it. B. Adaptive Double Greedy under the Oracle Model1)
Description of ADG:
Under the oracle model, weassume that the expected spread of any node set is accessiblein O (1) time. Based on this assumption, we design adaptivedouble greedy ( ADG ) . The decision on seed selection of ADGrelies on the conditional expected marginal profit. Definition 3 (Conditional Expected Marginal Profit) . Given anode u and a node set S , the conditional expected marginalprofit of u conditioned on S on graph G is defined as ∆ G ( u | S ) := ρ G ( S ∪ { u } ) − ρ G ( S ) . (5) Then, we have ∆ G ( u | S ) = E [ I G ( u | S )] − c ( u ) if u (cid:54)∈ S ,and ∆ G ( u | S ) = 0 otherwise. Algorithm 2 presents the pseudocode of ADG. First, twosets S and T are initialized with empty set and the target set T respectively (Line 1), where T contains k nodes. In the i -thiteration, if the examining node u i is activated already, it justchecks the next node (Lines 3–5). Otherwise, it calculates theconditional expected marginal profit ∆ G i ( u i | S i − ) (Line 6),denoted as front profit ρ f . Meanwhile, it also calculates − ∆ G i ( u i | T i − \ { u i } ) (Line 7), denoted as rear profit ρ r . If ρ f ≥ ρ r , it would (i) insert u i into S i as one seed node, (ii)observe the set of nodes A ( u i ) activated by u i , and (iii) update G i into G i +1 by removing all nodes in A ( u i ) . Otherwise, u i isremoved from T i − . This process is repeated until all k nodesin T are examined and then the resultant S k is returned. Notethat when ADG terminates, S k and T k contain exactly thesame nodes, i.e., S k = T k .As shown, the decision on seed selection in ADG is fullydetermined by the value of ρ f and ρ r . In a nutshell, for eachnode u ∈ T , u will be selected only if selecting u could bringmore marginal profit than abandoning it. Approximation Guarantee:
In this section, we wouldexplore the approximation guarantee that ADG could achieve.First, we show the relation between the front profit ρ f and therear profit ρ r in Algorithm 2. Algorithm 2:
ADG
Input:
Social graph G , a target seed set T with k nodes Output:
Selected seed node set S k Initialize S ← ∅ , T ← T , G ← G ; for i ← to k do if u i is activated then T i ← T i − \ { u i } , S i ← S i − , G i +1 ← G i ; continue ; ρ f ← ∆ G i ( u i | S i − ) ; ρ r ← − ∆ G i ( u i | T i − \ { u i } ) ; if ρ f ≥ ρ r then S i ← S i − ∪ { u i } , T i ← T i − ; Observe the node set A ( u i ) activated by u i ; Update G i into G i +1 by removing A ( u i ) ; else T i ← T i − \ { u i } , S i ← S i − , G i +1 ← G i ; return S k ; Lemma 1. If u i is inactive, we have ρ f + ρ r ≥ .Proof of Lemma 1. Given a residual graph G i and the cor-responding node u i , ρ f = E [ I G i ( u i | S i − )] − c ( u i ) and ρ r = c ( u i ) − E [ I G i ( u i | T i − \ { u i } )] . Then we have ρ f + ρ r = E [ I G i ( u i | S i − )] − E [ I G i ( u i | T i − \ { u i } )] . Since S i − ⊆ T i − \ { u i } and u i ∈ T \ ( T i − \ { u i } ) , based onthe property of submodularity, we have ρ f + ρ r ≥ , whichcompletes the proof.To facilitate the analysis that follows, we define the notionsof policy truncation , policy concatenation , and policy intersec-tion , which are conceptual operations performed by a policy.Note that these policy operations are used for our theoreticalanalysis only, and they do not affect the actual implementationof our algorithms. Definition 4 (Policy Truncation [12]) . For any adaptive seed-ing policy π , the policy truncation π [ i ] denotes an adaptivepolicy that performs exactly the same as π , except that π [ i ] onlyevaluates the first i ( i ≤ n ) nodes in a given node sequence. Definition 5 (Policy Concatenation [12]) . For any two adap-tive seeding policies π and π (cid:48) , the policy concatenation π ⊕ π (cid:48) denotes an adaptive policy that first executes the policy π , andthen executes π (cid:48) as if from a fresh start without any knowledgeabout π . Definition 6 (Policy Intersection) . For any two adaptiveseeding policies π and π (cid:48) , the policy intersection π ⊗ π (cid:48) denotesan adaptive policy that executes the intersection part of policy π and π (cid:48) . Note that under any given realization φ , policy concatena-tion shows that S φ ( π ⊕ π (cid:48) ) = S φ ( π ) ∪ S φ ( π (cid:48) ) and policyintersection shows that S φ ( π ⊗ π (cid:48) ) = S φ ( π ) ∩ S φ ( π (cid:48) ) . Let π opt be the optimal policy for adaptive TPM problem. Let π f (resp. π r ) be the front greedy (resp. rear greedy ) policyof ADG that executes as same as ADG and selects a set S i resp. T i ) of nodes after the i -th iteration, i.e., S ( π f[ i ] ) = S i (resp. S ( π r[ i ] ) = T i ). Define policy π ◦ = ( π opt ⊕ π f ) ⊗ π r and its truncation as π ◦ [ i ] = ( π opt ⊕ π f[ i ] ) ⊗ π r[ i ] . Let S ◦ bethe solution obtained by π ◦ , and S ◦ i be the temporal seed setobtained by π ◦ [ i ] , i.e., S ◦ i = S ( π ◦ [ i ] ) = ( S ◦ ∪ S i ) ∩ T i . Then, wehave Λ( π f[0] ) = 0 , Λ( π r[0] ) = ρ ( T ) ≥ , Λ( π ◦ [0] ) = Λ( π opt ) ,and Λ( π f[ k ] ) = Λ( π r[ k ] ) = Λ( π ◦ [ k ] ) , where k = | T | . First ofall, we bound the profit achievement of policies π f and π r onresidual graph G i in the following lemma. Lemma 2.
For the i -th iteration of ADG , we have ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) ≤ ρ G i ( S i ) − ρ G i ( S i − ) + ρ G i ( T i ) − ρ G i ( T i − ) . (6)The formal proofs of most theoretical results are givenin Appendix A. Based on Lemma 2, we further derive thefollowing lemma for the expected profit of policies π f , π r ,and policy π ◦ . Lemma 3.
For the i -th iteration of ADG , we have Λ( π ◦ [ i − ) − Λ( π ◦ [ i ] ) ≤ Λ( π f[ i ] ) − Λ( π f[ i − )+Λ( π r[ i ] ) − Λ( π r[ i − ) . Lemma 3 establishes the relation on expected profit ofpolicy between ADG and the optimal policy π opt for eachiteration. Based on Lemma 3, the approximation guarantee ofADG is derived as follows. Theorem 1.
ADG achieves the approximation ratio of / .Proof of Theorem 1. From Lemma 3, we have Λ( π ◦ [0] ) − Λ( π ◦ [ k ] ) = (cid:88) ki =0 (cid:0) Λ( π ◦ [ i − ) − Λ( π ◦ [ i ] ) (cid:1) ≤ (cid:88) ki =0 (cid:0) Λ( π f[ i ] ) − Λ( π f[ i − ) + Λ( π r[ i ] ) − Λ( π r[ i − ) (cid:1) = Λ( π f[ k ] ) − Λ( π f[0] ) + Λ( π r[ k ] ) − Λ( π r[0] ) . Recalling that Λ( π f[0] ) = 0 , Λ( π r[0] ) = ρ ( T ) ≥ , Λ( π ◦ [0] ) =Λ( π opt ) , and Λ( π f[ k ] ) = Λ( π r[ k ] ) = Λ( π ◦ [ k ] ) , we obtain that Λ( π f[ k ] ) = Λ( π r[ k ] ) ≥ Λ( π opt ) / . Remark.
At the first glance, the equations and transformationsin this paper might look similar to those in the existinginfluence maximization papers, which is because we adoptthe commonly used notations and definitions in the literature.However, we note that influence maximization is based onmonotone submodular optimization with a cardinality con-straint, whereas profit maximization is based on unconstrained(i.e., nonmonotone) submodular optimization. Due to thefundamental difference between the two problems, our algo-rithms and mathematical analysis (including the equations andtransformations) actually differ considerably from those in theexisting influence maximization papers. Specifically, a simplegreedy algorithm is used to address influence maximization,while we devise an adaptive double greedy algorithm tailoredfor adaptive target profit maximization.
Algorithm 3: A DD ATP
Input:
Social graph G , a target seed set T with k nodes,the initial error ζ Output:
Selected seed node set S k Initialize S ← ∅ , T ← T ; for i ← to k do if u i is activated then T i ← T i − \ { u i } , S i ← S i − , G i +1 ← G i ; continue ; ζ i ← ζ , δ i ← / ( kn ) ; // ζ ≥ /n while true do θ ← ζ i ln δ i ; Generate θ RR sets as R and R , respectively; ˜ ρ f ← Cov R ( u i | S i − ) · n i θ − c ( u i ) ; ˜ ρ r ← − Cov R ( u i | T i − \ { u i } ) · n i θ + c ( u i ) ; if C or C then if ˜ ρ f ≥ ˜ ρ r then S i ← S i − ∪ { u i } , T i ← T i − ; Observe the node set A ( u i ) activated by u i ; Update G i into G i +1 by removing A ( u i ) ; else T i ← T i − \ { u i } , S i ← S i − , G i +1 ← G i ; break ; ζ i ← ζ i / √ , δ i ← δ i / ; return S k ; C. Adaptive Double Greedy under the Noise Model
As well-known, computing the exact expected spread of anynode set is relative error and additive error . Considering thatwe need to estimate the expected marginal spreads of k node sets during the whole process, it can be quite intricateutilizing relative error. To explain, for those with small ex-pected marginal spreads, only a trivial amount of estimationerror would be allowed, which is rather difficult to achieve byexisting methods for spread estimation. Motivated by this, weadopt the additive error instead and propose the A DD ATP algorithm. Algorithm 3 presents the details of A DD ATP. Description of A DD ATP:
The design principle ofA DD ATP follows that of ADG except that ρ f and ρ r inA DD ATP are not accessible but to be estimated. The esti-mation reliability is guaranteed by
Hoeffding Inequality [14].Specifically, A DD ATP first initializes S with ∅ and T withtarget set T (Line 1). In each iteration on residual graph G i , itfirst checks if the current node u i is activated. If u i is activated,it skips current node and starts the next iteration immediately.Otherwise, A DD ATP initializes the additive error parameter ζ i = ζ and the probability parameter δ i = 1 / ( kn ) (Line 6) Algorithms with additive error for adaptive targeted profit maximization or the following evaluations. Notice that a proper initializationvalue ζ could speed up the performance empirically becausesome nodes can be decided within relatively small numberof samples. To estimate the expected marginal spreads, A D - D ATP adopts the commonly used reverse influence sampling(RIS) [6] technique, referred to as
RR sets . Specifically,A DD ATP generates θ RR sets into R and R to calculate ˜ ρ f (Line 10) and ˜ ρ r (Line 11) respectively, where θ is determinedby ζ i and δ i , ˜ ρ f and ˜ ρ r are the estimations of E [ I G i ( u i | S i − )] − c ( u i ) and c ( u i ) − E [ I G i ( u i | T i − \ { u i } )] . ByHoeffding Inequality [14], we have ρ f ∈ [˜ ρ f − n i ζ i , ˜ ρ f + n i ζ i ] and ρ r ∈ [˜ ρ r − n i ζ i , ˜ ρ r + n i ζ i ] with high probability. Then, thestopping conditions C and C are checked (Line 12), where C and C are defined as C : ( | ˜ ρ f − ˜ ρ r | ≥ n i ζ i ) ∨ (˜ ρ f ≤ − n i ζ i ) ∨ (˜ ρ r ≤ − n i ζ i ) ,C : n i ζ i ≤ . If the stopping condition C is met, it indicates that the currentestimations ˜ ρ f and ˜ ρ r are accurate enough to help make theright decision with high probability. If C is observed instead,it indicates that (i) the expected spread of current node is tooclose to the judgement bar to distinguish, and (ii) the profit lossis insignificant if wrong decision is made. The purpose of thiscondition is to avoid the unnecessarily prohibitive samplingoverhead for sufficient accurate of spread estimations. Whenone of the stopping conditions is met, the decision is madeaccordingly (Lines 13–17). Otherwise, ζ i (resp. δ ) is dividedby √ (resp. ) (Line 19) to generate more samples for moreaccurate estimations. This process terminates when all k nodesin T are checked through. Theoretical Analysis of A DD ATP:
In what follows, weanalyze the approximation guarantee and time complexity ofA DD ATP.
Approximation Guarantee.
First, we present the
HoeffdingInequality [14] based on which the estimation is reliable withhigh probability.
Lemma 4 (Hoeffding Inequality [14]) . Let X i be an indepen-dent bounded random variable such that for each ≤ i ≤ θ , X i ∈ [ a i , b i ] . Let X = θ (cid:80) θi =1 X i . Given ζ ∈ (0 , , then Pr[ | X − E [ X ] | ≥ ζ ] ≤ − θ ζ (cid:80) θi =1( bi − ai )2 . (7)Similarly, to obtain the approximation ratio of A DD ATP,we need to establish the relation between Λ( π ◦ [ i − ) − Λ( π ◦ [ i ] ) and Λ( π f[ i ] ) − Λ( π f[ i − ) + Λ( π r[ i ] ) − Λ( π r[ i − ) as the one inLemma 3. We note that ADG is a deterministic algorithmwhile A DD ATP is a randomized algorithm. To explain, ADGhas access to the expected spread of any node set under theoracle model. Contrarily, under the noise model, A DD ATPadopts the randomized reverse influence sampling (RIS) [6] technique to estimate the expected spread, which makes π f , π r , and π ◦ all random polices. Consequently, we need to tacklethe internal randomness of A DD ATP.
Lemma 5.
For the i -th iteration of A DD ATP , we have E A (cid:2) ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) (cid:3) − (2 + 2 /k ) ≤ E A (cid:2) ρ G i ( S i ) − ρ G i ( S i − ) + ρ G i ( T i ) − ρ G i ( T i − ) (cid:3) . (8)Thus, for the expected profit of policy π f , π r , and π ◦ in the i -th iteration, we have following lemma. Lemma 6.
For the i -th iteration of A DD ATP , we have E A (cid:2) Λ( π ◦ [ i − ) − Λ( π ◦ [ i ] ) (cid:3) − (2 + 2 /k ) ≤ E A (cid:2) Λ( π f[ i ] ) − Λ( π f[ i − ) + Λ( π r[ i ] ) − Λ( π r[ i − ) (cid:3) , (9) where the expectation E A [ · ] is over the internal randomnessof A DD ATP and k = | T | . Note that there are additional terms /k and in (9).Specifically, /k is the compensation factor on the profit losswhen A DD ATP makes the wrong choice on u i due to thefailed spread estimation. is the upper bound of the profitloss incurred when A DD ATP terminates due to the stoppingcondition C .Based on Lemma 6, we could derive the approximation ratioof A DD ATP as follows.
Theorem 2. A DD ATP could achieve the expected profit atleast Λ( π opt ) − (2 k +2)3 , where Λ( π opt ) is the expected profit ofoptimal policy π opt and k is the number of nodes in T .Proof of Theorem 2. According to Lemma 6, accumulatingboth sides of (9) from i = 1 to k gives E A (cid:2) Λ( π ◦ [0] ) − Λ( π ◦ [ k ] ) (cid:3) − (cid:88) ki =1 (2 + 2 /k ) ≤ E A (cid:2) Λ( π f[ k ] ) − Λ( π f[0] ) + Λ( π r[ k ] ) − Λ( π r[0] ) (cid:3) . Rearranging it completes the proof.
Discussion.
On particular social graphs, our algorithm couldachieve an expected ratio of (1 − ε ) / . The main idea is todynamically set the threshold of n i ζ i for stopping condition C instead of fixing to . Specifically, in each iteration, wehave (at most) a profit loss of /k due to a failure estimationof marginal expected spread and an extra profit loss of ifstopping condition C , i.e., n i ζ i ≤ , occurs (see the proof ofLemma 5). Thus, we have (at most) a total profit loss of k for all k iterations. However, we note that it is unlikely that all k iterations meet C . Thus, we could try to bound the actualprofit loss within ε Λ( π opt ) by adjusting the error thresholddynamically as follows. For the i -th iteration, let η i be thesettled threshold of n i ζ i , i.e., C : n i ζ i ≤ η i , and ρ i be theaccumulated profit. Let ˜ η i be the indicator whether C occurssuch that ˜ η i = η i if C occurs and ˜ η i = 0 if C occurs. In the ( i + 1) -th iteration, we set η i +1 = ( ερ i − (cid:80) ij =1 ˜ η j − / aslong as ερ i ≥ (cid:80) ij =1 ˜ η j + 2 , which ensures that (cid:80) i +1 j =1 ˜ η j +2 ≤ ερ i . With this dynamic strategy, A DD ATP could achievean approximation ratio of (1 − ε ) / . Time Complexity.
As indicated in Algorithm 3, there are O ( ζ i ln δ i ) random RR sets generated in the i -th iteration.There are at most (cid:100) log( n i ζ ) (cid:101) rounds for the i -th iterationnd ζ ∈ [1 /n, , thus δ i is bounded by O ( kn ) . Then the totalnumber of RR sets is at most O ( ln nζ i ) . According to Lemma 4in [29], the expected time for generating a random RR set on G i , denoted as EPT , is
EPT ≤ m i n i E [ I G i ( { v ◦ i } )] where v ◦ i isthe node with the largest expected spread on G i . By Wald’sequation [31], the expected time complexity of A DD ATP is O (cid:16) k (cid:88) i =1 (cid:0) ln nζ i · m i E [ I ( { v ◦ i } )] n i (cid:1)(cid:17) = O (cid:16) kmn E [ I ( { v ◦ } )] ln n (cid:17) , since n i ζ i ≥ √ / and E [ I ( { v ◦ i } )] ≤ E [ I ( { v ◦ } )] for any i . Theorem 3.
The expected time complexity of A DD ATP is O ( kmn E [ I ( { v ◦ } )] ln n ) where v ◦ is the node with the largestexpected spread on G . IV. O
PTIMIZATION WITH H YBRID E RROR
In this section, we aim to optimize A DD ATP and proposeHATP in terms of efficiency. We first analyze the rationale ofoptimization and then conduct theoretical analysis on HATPtowards its approximation guarantee and time complexity.
A. Rationale of Optimization A DD ATP in Section III considers the additive error onspread estimation, which may suffer from efficiency issues.Recall that for nodes with expected spreads close to thejudgement bar, A DD ATP uses the stopping condition n i ζ i ≤ to avoid unnecessary computation overhead. However, when ζ i = O (1 /n i ) , the number of RR sets required is O ( n i ln n ) ,which would incur prohibitive computation overhead. Totackle this issue, we propose to estimate the expected spreadvia a hybrid error , i.e., the combination of the additive errorand relative error. The rationale of hybrid error is that for thosenodes with large expected marginal spread, their estimationerrors are easily to be bounded within relative error, whilefor those nodes with small expected marginal spread, theirestimation errors are easily to be bounded within additive error.Thus, the expected marginal spread for every node can beefficiently estimated utilizing hybrid error. This estimation re-liability using hybrid error is guaranteed by Relative+AdditiveConcentration Bound as follows.
Lemma 7 (Relative+Additive Concentration Bound) . Let X − E [ X ] , · · · , X θ − E [ X θ ] be a martingale differencesequence such that X i ∈ [0 , for each i . Let X = θ (cid:80) θi =1 X i and µ = E [ X ] . Given ε, ζ ∈ (0 , , then Pr[ X ≥ (1 + ε ) µ + ζ ] ≤ e − θεζ (1+ ε/ , (10) Pr[ X ≤ (1 − ε ) µ − ζ ] ≤ e − θεζ . (11) Proof.
According to the martingale concentration [28], wehave Pr (cid:2) X ≤ (1 − ε ) µ − ζ (cid:3) ≤ e − ( εµ + ζ )2 θ µ ≤ e − (2 √ εµζ )2 θ µ =e − εζθ . Similarly, we have Pr[ X ≥ (1 + ε ) µ + ζ ] ≤ e − h ( λ ) ,where h ( λ ) = ( λ θ )2( λ − ζ ) /ε +2 λ/ and λ = εµ + ζ . Let d h ( λ )d λ = (cid:0) λ (( λ − ζ ) /ε + λ/ − (1 /ε + 1 / λ (cid:1) θ (cid:0) ( λ − ζ ) /ε + λ/ (cid:1) (cid:44) . Algorithm 4:
HATP
Input:
Social graph G , a size- k set T , the initial error ε and ζ , the threshold ε Output:
Selected seed node set S k Initialize S ← ∅ , T ← T ; for i ← to k do if u i is activated then T i ← T i − \ { u i } , S i ← S i − , G i +1 ← G i ; continue ; ε i ← ε , ζ i ← ζ , δ i ← / ( kn ) ; // ε ≥ ε, ζ ≥ /n while True do θ ← (1+ ε i / ε i ζ i ln( δ i ) ; Generate θ RR sets as R and R , respectively; f est ← Cov R ( u i | S i − ) · n i θ ; r est ← Cov R ( u i | T i − \ { u i } ) · n i θ ; if C (cid:48) or C (cid:48) then if f est + r est ≥ c ( u i ) then S i ← S i − ∪ { u i } , T i ← T i − ; Observe the node set A ( u i ) activated by u i ; Update G i into G i +1 by removing A ( u i ) ; else T i ← T i − \ { u i } , S i ← S i − , G i +1 ← G i ; break ; if ε i < = ε t and n i ζ i > then ζ i ← ζ i / ; else if ε i > ε and n i ζ i < = 1 then ε i ← ε i / ; else if f est ≥ n i ζ i then ε i ← ε i / ; else if f est < = n i ζ i then ζ i ← ζ i / ; else ε i ← ε i / √ , ζ i ← ζ i / √ ; δ i ← δ i / ; return S k ;Thus, h ( λ ) achieves its minimum at λ = ζε (1 /ε +1 / such that h ( λ ) = εβθ (1+ ε/ . This completes the proof.Based on the hybrid error, we propose HATP algorithm,as shown in Algorithm 4. B. Description of
HATPThe design principle of HATP is similar to that of A DD ATPexcept that HATP adopts hybrid error instead of additiveerror . The major differences between HATP and A DD ATPlie in two aspects. First, the stopping conditions C and C (Line 12) have been updated accordingly in HATP as follows. C (cid:48) : (cid:0) f est + r est − n i ζ i ε ≥ c ( u i ) (cid:1) ∨ (cid:0) r est − n i ζ i ε ≥ c ( u i ) (cid:1) ∨ (cid:0) f est + r est +2 n i ζ i − ε ≤ c ( u i ) (cid:1) ∨ (cid:0) f est + n i ζ i − ε ≤ c ( u i ) (cid:1) ,C (cid:48) : ( ε i ≤ ε ) ∧ ( n i ζ i ≤ . Second, the error parameters ε i and ζ i are adjusted adaptivelyin HATP (Lines 20–23) instead of decreasing with a fixed ratioin A DD ATP. Specifically, if the current addition error n i ζ i Algorithms with hybrid error for adaptive targeted profit maximization eaches the threshold or it is the one magnitude smaller thanestimated f est , we can infer that the expected marginal spreadof u i is much larger than the additive error. In such case, therelative error ε i is halved (Line 20). Similarly, if the currentrelative error ε i reaches the threshold or the additive error islarger than estimated f est , we should halve the additive error n i ζ i (Line 19). Otherwise, both relative and additive errors aredeceased by a factor of √ (Line 23). This adaptive adjustmentcould boost the efficiency of HATP significantly. C. Theoretical Analysis of
HATP
Approximation Guarantee.
To derive the approximationguarantee of HATP, we need a similar equation like (9) tobridge our solution with the optimal solution in each iteration.Toward this end, we have the following lemma.
Lemma 8.
For the i -th iteration of HATP , we have E A (cid:2) Λ( π ◦ [ i − ) − Λ( π ◦ [ i ] ) (cid:3) − εc ( u i ))1 − ε − k ≤ E A (cid:2) Λ( π f[ i ] ) − Λ( π f[ i − ) + Λ( π r[ i ] ) − Λ( π r[ i − ) (cid:3) where the expectation E A is over the internal randomness of HATP and ε is the threshold of relative error. Lemma 8 establishes the relation between policies π f , π r and π ◦ for each iteration, based on which, we have followingtheorem on the approximation of HATP. Theorem 4.
HATP achieves the expected profit at least Λ( π opt ) − k + εc ( T )) / (1 − ε ) − for any ε ∈ (0 , , where c ( T ) isthe cost of T . Time Complexity.
In the i -th iteration, there are O ( εζ i ln δ i ) random RR sets, where δ i = O ( εn ) . Thus, the expected timecomplexity of HATP is O (cid:16) k (cid:88) i =1 (cid:0) ln nε εζ i · m i E [ I ( { v ◦ i } )] n i (cid:1)(cid:17) = O (cid:16) km E [ I ( { v ◦ } )] ε ln nε (cid:17) . Theorem 5.
The expected time complexity of
HATP is O ( km E [ I ( { v ◦ } )] ε ln nε ) where v ◦ is the node with the largestexpected spread on G . Note that HATP is approximately O ( εn ) more efficient thanA DD ATP. Usually, ε = O (1) , e.g., ε = 0 . , in the literature,HATP achieves a factor of O ( n ) improvement on efficiency.V. R ELATED W ORK
As introduced in Section III-A, profit maximization problemis an application of unconstrained submodular maximization(USM) problem [7], [10]. Thus in this section, we mainlydiscuss the related work on submodular maximization and profit maximization . Submodular Maximization.
Submodular maximization hasattracted considerable interest [3]–[5], [7], [10] in the pastdecades. There are a plethora of applications of submodularmaximization in real world, such as maximum facility loca-tion [1], Max-Cut [11] and influence maximization (IM) [16]. Compared with the profit maximization (PM) problem, the IMproblem is the most relevant work. Spread function defined inthe IM problem is submodular and monotone under the inde-pendent cascade (IC) and linear threshold (LT) models [16].However, profit function is submodular but not necessarilymonotone, by which profit maximization is unconstrainedsubmodular maximization (USM) [7], [10]. As pointed out byprevious work [10], [20], there is no efficient approximationalgorithms for general USM problem without any additionalassumptions. For nonnegative USM, Feige et al. [10] provethat an uniformly random selected method could achievean -approximation (resp. -approximation) if the submod-ular function is nonsymmetric (resp. symmetric). As whatfollows, Buchbinder et al. [7] propose deterministic doublegreedy and randomized double greedy methods, achieving -approximation and -approximation for USM respectivelyunder the assumption that submodular function on the groundset is nonnegative. Profit Maximization.
The profit maximization (PM) problemhas been a hot topic in academia recently. The existingwork all focuses on PM problem in the nonadaptive setting,i.e., nonadaptive PM. Tong et al. [30] consider the couponallocation in the profit maximization problem. By utilizingthe randomized double greedy [7], they design algorithmsto address the proposed simulation-based profit maximization and realization-based profit maximization and claim to achieve -approximation with high probability. Liu et al. [18] alsoconsider the coupon allocation in profit maximization undera new diffusion model named independent cascade modelwith coupons and valuations . To address this problem, theypropose PMCA algorithm based on the local search algo-rithm [10]. PMCA is claimed to achieve an -approximationupon the assumption that the submodular function is non-negative for every subset. However, this assumption is toostringent. Moreover, the time complexity of PMCA is aslarge as O (log( n ) mn /ε ) , due to which PMCA does notwork in practice. Tang et al. [26] utilize the deterministic andrandomized double greedy algorithms [7] to address the profitmaximization problem. With the assumption that submodularfunction on the ground set is nonnegative, they prove the - and -approximation guarantees respectively. Furthermore,they design an novel method and relax this assumption to amuch weaker one. However, they do not analyze the samplingerrors in spread estimation, which makes the proposed algo-rithms heuristic. VI. E XPERIMENTS
In this section, we evaluate the performance of our proposedalgorithms through extensive experiments. We measure theefficiency and effectiveness in real online social networks. Ourexperiments are deployed on a Linux machine with an InterXeon 2.6GHz CPU and 64GB RAM.
ABLE IID
ATASET DETAILS . (K = 10 , M = 10 ) Dataset n m
Type Avg. deg
NetHEPT 15.2K 31.4K undirected 4.18Epinions 132K 841K directed 13.4DBLP 655K 1.99M undirected 6.08LiveJournal 4.85M 69.0M directed 28.5
A. Experimental Setting
Datasets.
Four online social networks are used in our ex-periments, namely NetHEPT, Epinions, DBLP, LiveJournal,as presented in Table II. Among them, NetHEPT [8] is theacademic collaboration networks of “High Energy Physics-Theory” area. The rest three datasets are real-life socialnetworks available in [17]. In particular, LiveJournal containsmillions of nodes and edges. For fair comparison, we randomlygenerate 20 possible realizations for each dataset, and reportthe average performance of each tested algorithm on the 20possible realizations.
Algorithms.
First, we evaluate the two proposed adaptivealgorithms HATP and A DD ATP. We also adopt the randomset (RS) algorithm [10] and extend it into an adaptive version,i.e., adaptive random set (ARS). Specifically, ARS selectseach seed node candidate with probability of . withoutreference to its quality. If one node is selected, it then observesand removes all the nodes activated by this node from thegraph. (The removed nodes are not examined and selected byARS.) This process is repeated until all nodes in the targetset have been decided. To verify the advantage of adaptivealgorithms over nonadaptive algorithms, we tailor HATP intoa nonadaptive version, referred to as HNTP , to address thenonadaptive TPM problem. Meanwhile, we also include twoextra nonadaptive algorithms proposed in the latest work fornonadaptive profit maximization problem [26], i.e., nonadap-tive simple greedy (NSG) and nonadaptive double greedy (NDG). Note that NSG and NDG are nonadaptive algorithmswhere all the seed nodes are selected in one batch before wedeploy these nodes into the viral marketing campaign. Also,the analysis in [26] ignores the sampling errors in spreadestimation; in contrast, our A DD ATP and HATP algorithmstake such errors into account. For relatively fair comparison,we set the sample size of NSG and NDG as the largest numberof samples generated in HATP for one iteration in all settings.Recall that by evaluating the efficacy of adaptive algorithmsA DD ATP and HATP, we could verify the advantage of ourproposed adaptive policies over nonadaptive policies on targetprofit maximization.
Parameter settings.
To conduct a comprehensive evaluation,we design two different procedures to obtain a suitable targetset T and the corresponding cost of each user in T . First,we follow the setting in [2], [19] where the cost of each user Algorithms with hybrid error for nonadaptive targeted profit maximization in T is based on the expected spread of T . We use one ofthe state of the arts [28] for influence maximization to obtainthe top- k influential users as the target seed set. We vary thetarget size k as k = { , , , , , } . To determinethe cost of each user u ∈ T , we (i) estimate the lower boundof T ’s expected spread E [ I ( T )] as E l [ I ( T )] , and (ii) ensurethat c ( T ) = E l [ I ( T )] . Under this condition, we design twocost settings, i.e., degree-proportional cost setting and uniform cost setting. In the degree-proportional setting, the cost ofeach node is proportional to its out-degree. In the uniformcost setting, the cost of each node is equal.Second, we follow the setting in the latest work [26] wherethe cost of each user in the graph G is predefined before wechoose the target set T . Let λ = c ( V ) /n be the ratio of cost tonode number where n = | V | and c ( V ) is the cost of V . To geta proper size of T , we vary λ as λ = { , , , } .Then the cost of each node in V also follows the degree-proportional cost setting and uniform cost setting respectively.We then adopt the two proposed methods, namely NDG andNSG to identify the target set T .In a nutshell, in the first setting, we choose a target set T first and then assign a cost to each node in T based on theexpected spread of T , while in the second setting, we assigna cost to each node first and then find a target set T basedon the cost assignment. For the two settings, we set the inputparameters n i ζ = 64 , ε = 0 . , and its threshold ε = 0 . in HATP and HNTP. Following the common setting in theliterature [15], [22], [24], [25], [29]: for each dataset, we setthe edge probability p ( (cid:104) u, v (cid:105) ) = v , where indeg v is thein-degree of node v . B. Comparison of Profit
Degree-proportional Cost.
Fig. 2 reports the profits achievedby the six tested algorithms under the degree-proportional costsetting. In addition, the dark-blue line with cross mark ( ××× )named
Baseline represents the estimated profit of the targetset T . We observe that all six tested algorithms could im-prove the profit of baseline significantly. In particular, HATPshows superior advantage over the other three nonadaptivealgorithms. Specifically, profit achieved by HATP is around – larger than those of nonadaptive algorithms onaverage, where the improvement percentage is as high as on the Epinions dataset. This verifies the effectiveness of oursolutions. Meanwhile, A DD ATP achieves comparable profitwith HATP on the NetHEPT dataset. However, A DD ATP runs out of memory on other larger datasets. (The filled triangle ( (cid:78) )represents the largest value of k that A DD ATP can run.)As with the nonadaptive algorithms, i.e., HNTP, NSG,and NDG, they obtain quite comparable profits on the fourdatasets. In particular, we observe that the profit of NSG isslightly higher than the profits of HNTP and NDG on datasetsDBLP and LiveJournal, which implies the minor advantage ofsimple greedy over double greedy on large datasets. We alsoobserve that ARS achieves the lowest profits among the sixalgorithms, since ARS selects each node with probability of . without reference to its quality. ATP NSG NDGHNTP BaselineA DD ATP ARS number of seeds kprofit (10 ) (a) NetHEPT number of seeds kprofit (10 ) (b) Epinions number of seeds kprofit (10 ) (c) DBLP number of seeds kprofit (10 ) (d) LiveJournalFig. 2. Profit in degree-proportional cost. number of seeds kprofit (10 ) (a) NetHEPT number of seeds kprofit (10 ) (b) Epinions number of seeds kprofit (10 ) (c) DBLP number of seeds kprofit (10 ) (d) LiveJournalFig. 3. Profit in uniform cost. Uniform Cost.
Fig. 3 shows the profit results under theuniform cost setting. At the first glance, the results followthe trend of Fig. 2. However, there are two major distinctionson the profit results between the two cost settings. First, algo-rithms achieve around more profits in the uniform costsetting than they do in the degree-proportional cost setting.This can be explained as follows. In the degree-proportionalcost setting, the cost of each user is proportional to its out-degree which is highly correlated with its expected spread.In this regard, each user’s cost is roughly proportional to itsspread, which largely limits the profits the influential nodescould contribute in some degree. Contrarily, in the uniformcost setting where all users are assigned with the same cost,influential nodes can be more easily picked out to exert theirinfluence and bring more profits.The other notable distinction is that the gap of profitbetween adaptive algorithms and nonadaptive algorithms be-comes smaller in the uniform cost setting. As aforementioned,those profitable nodes are easier to be identified under thissetting. As a consequence, the overlap of selected seed setsbetween adaptive and nonadaptive algorithms expands, whichweakens the adaptivity advantage slightly.
Random Cost.
Fig. 4(a) shows the results under the randomcost setting where the cost of each node is randomly assignedsuch that c ( T ) = E l [ I ( T )] . Due to the space limitations, weonly present the profit results on dataset Epinions. We observethat HATP again achieves the highest profits, with around more profits than the other three nonadaptive algorithms.We also observe that (i) the advantage of adaptive algorithmsover nonadaptive algorithms becomes less significant under therandom cost setting, and (ii) the profits achieved by differentalgorithms under the random cost setting are around number of seeds kprofit (10 ) (a) Profits under the random cost relative error ε profit (10 ) (b) Sensitivity of relative errorFig. 4. Profits on Epinions. (resp. ) more than those under the degree-proportional(resp. uniform) cost setting respectively. The reason is thatunder the random cost setting, user costs have no correlationwith their spreads. In such cases, profitable nodes are easierto be identified by both adaptive and nonadaptive algorithmsin expectation, which lessens the adaptivity advantage. Mean-while, those profitable nodes will influence the same numberof nodes but have relatively less costs, which could improvethe final profit. Sensitivity Test of ε . Recall that HATP involves a keyparameter of ε , which represents the approximation guaranteeof HATP (Section IV-B). Specifically, we vary the value of ε as { . , . , . , . , . } under k = 500 and the degree-proportional cost setting on the Epinions dataset. Fig. 4(b)presents the profit achieved by HATP with different ε . Asshown, the profits remain nearly steady for all settings, whichdemonstrates the robustness of HATP on the setting of ε . ATP NSG NDGHNTPA DD ATP -1
10 25 50 100 200 500 number of seeds krunning time (sec) (a) NetHEPT -1
10 25 50 100 200 500 number of seeds krunning time (sec) (b) Epinions
10 25 50 100 200 500 number of seeds krunning time (sec) (c) DBLP
10 25 50 100 200 500 number of seeds krunning time (sec) (d) LiveJournalFig. 5. Running time in degree-proportional cost. -1
10 25 50 100 200 500 number of seeds krunning time (sec) (a) NetHEPT -1
10 25 50 100 200 500 number of seeds krunning time (sec) (b) Epinions
10 25 50 100 200 500 number of seeds krunning time (sec) (c) DBLP
10 25 50 100 200 500 number of seeds krunning time (sec) (d) LiveJournalFig. 6. Running time in uniform cost.
C. Comparison of Running Time
Degree-proportional Cost.
Fig. 5 presents the results ofrunning time under the degree-proportional cost setting. Notethat in Fig. 5 and Fig. 6, the value of k in x-axis is roughlyin exponential scale instead of linear scale. Considering thatARS selects node randomly without generating any samples,we do not include its running time (which is vary small) inall figures. Fig. 5(a) shows that A DD ATP runs around times slower than HATP. This confirms that our optimizationtechniques for HATP can significantly reduce running time.Meanwhile, HATP and HNTP run slower than the heuristicalgorithms NSG and NDG. This is because HATP and HNTPregenerate RR sets in all k iterations from scratch to ensurebounded sampling errors, as shown in Algorithm 4. On thecontrary, NSG and NDG complete seed selection on one set ofRR sets as they do not have any guarantee on profit estimation.Therefore, HATP and HNTP generate around k times samplesthan NSG and NDG do.Meanwhile, HNTP runs slightly slower than HATP does.Recall that HNTP is the nonadaptive version of HATP andthere is no any graph update on G after each node selection.Contrarily, HATP would update current graph into residualgraph by removing newly activated nodes in each iteration. Weobserve that generating an RR set on a smaller residual graphis faster than that on the original graph. Therefore, HATPspends less time than HNTP on sampling. Uniform Cost.
Fig. 6 displays the results of running timeunder the uniform cost setting. The result trend keeps con-sistent with that in degree-proportional cost setting. The ma-jor difference between the two is that the running time inFig. 6 is smaller than that in Fig. 5 for each algorithm onthe corresponding setting. This can be explained with the
Ratio of cost to node number λ profit (10 ) HATP NDG (a) Degree-proportional cost
Ratio of cost to node number λ profit (10 ) HATP NDG (b) Uniform costFig. 7. Profits of HATP and NDG on LiveJounral. reasons aforementioned, i.e., profitable nodes in uniform costsetting are easier identified within less samples. Therefore, thecorresponding running time becomes shorter.
D. Comparison of Profit with Predefined Cost
This section explores the profit improvement of HATP overNDG and NSG, following the setting that the cost of each useris predefined before we choose the target set T . Specifically,after the cost of each user is set, we adopt NDG and NSGto derive the target seed set T respectively. As implied inSection VI-B, profit improvement exhibits similar characteris-tics on the four datasets. Therefore, this part of experiment isconducted only on the largest dataset LiveJournal.Fig. 7 presents the profits of HATP and NDG with various λ values under the two cost settings. Note that smaller λ value means larger seed set size k . The overall improvementof HATP over NDG is around and under thedegree-proportional and uniform cost settings respectively. Inparticular, the improvement ratio can be up to . with λ = 200 in Fig. 7(b). The advantage of HATP over NDG inFig. 7(b) gets more notable along the decrease of λ , which Ratio of cost to node number λ profit (10 ) HATP NSG (a) Degree-proportional cost
Ratio of cost to node number λ profit (10 ) HATP NSG (b) Uniform costFig. 8. Profits of HATP and NSG on LiveJounral. scale ratio of sample sizerunning time (10 sec) NSG NDG (a) Running time scale ratio of sample sizeprofit (10 ) NSG NDG (b) ProfitFig. 9. NSG and NDG with various sample sizes on Epinions. reveals that larger the target set size is, more effective theadaptive algorithms become. This can also explain why theadvantage in Fig. 7(b) is more obvious than that in Fig. 7(a),since the target set size under the uniform cost setting is largerwith the same value of λ . Fig. 8 shows the profit improvementof HATP over NSG. The overall improvement of HATP overNSG is or so, less significant compared with the results inFig. 7. This indicates that NSG might be more effective thanNDG on profit maximization. Observe that the improvementis more notable in Fig. 8(b) than that in Fig. 8(a), which againverifies the fact that the advantage of adaptive algorithms overnonadaptive algorithms is more impressive when the target sizegets larger.To further verify the advantage of adaptive algorithms overnonadaptive algorithms, we increase the sample size of NSGand NDG with a multiplicative factor of { , , , , , } on the Epinions dataset with k = 500 under the degree-proportional cost setting. Fig. 9(a) shows that the runningtime of both NSG and NDG increases linearly along withthe sample size. However, as shown in Fig. 9(b), the profitsachieved by NSG and NDG almost remain the same (withnegligible changes) when we increase the sample size, whichis also confirmed in existing work [15], [22], [29] that thespreads cannot be improved after the sample size reachescertain threshold. This indicates that the superiority of ouradaptive algorithms over the nonadaptive algorithms is due tothe adaptive techniques rather than the sample size.VII. C ONCLUSION
This paper studies the adaptive target profit maximization(TPM) problem which aims to identify a subset of nodes fromthe target node set to maximize the expected profit utilizing the adaptive advantage. To acquire an overall understandingon adaptive TPM problem, we investigate this problem in bothoracle model and noise model. We prove that adaptive doublegreedy could address adaptive TPM under the oracle modeland achieve an approximation ratio of . To address it underthe noise model, we design A DD ATP that achieves provableapproximation guarantee. We later optimize A DD ATP intoHATP that has made remarkable improvement on efficiency.HATP is designed based on the concept of hybrid error thatcould efficiently handle different nodes with various expectedmarginal spread by adjusting the relative error and additiveerror adaptively. To evaluate the performance of HATP, weconduct extensive experiments on real social networks, andthe experimental results strongly confirm the superiorities andeffectiveness of our approaches.A
CKNOWLEDGMENT
This research is supported by Singapore National ResearchFoundation under grant NRF-RSS2016-004, by SingaporeMinistry of Education Academic Research Fund Tier 2 undergrant MOE2015-T2-2-069, and by National University ofSingapore under an SUG.A
PPENDIX
A. Proofs of Lemmas and TheoremsProof of Lemma 2.
Consider the case that ρ f ≥ ρ r . Then, wehave S i = S i − ∪ { u i } , T i = T i − , and S ◦ i = S ◦ i − ∪ { u i } .Thus, (6) becomes − ∆ G i ( u i | S ◦ i − ) ≤ ∆ G i ( u i | S i − ) = ρ f .If u i / ∈ S ◦ i − , we have − ∆ G i ( u i | S ◦ i − ) ≤ − ∆ G i ( u i | T i − \{ u i } ) = ρ r ≤ ρ f due to the submodularity of ρ G i ( · ) and thefact that S ◦ i − ⊆ T i − \{ u i } and u i / ∈ T i − \{ u i } . If u i ∈ S ◦ i − ,we have − ∆ G i ( u i | S ◦ i − ) = 0 ≤ ρ f since ρ f + ρ r ≥ byLemma 1. Thus, it holds that − ∆ G i ( u i | S ◦ i − ) ≤ ρ f .The other case of ρ f < ρ r is analogous. Specifically, wehave S i = S i − , T i = T i − \ { u i } , and S ◦ i = S ◦ i − \ { u i } and need to show that ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) ≤ ρ r . Again, if u i / ∈ S ◦ i − , we have ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) = 0 ≤ ρ r since ρ f + ρ r ≥ . If u i ∈ S ◦ i − , we have ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) =∆ G i ( u i | S ◦ i − \ { u i } ) ≤ ∆ G i ( u i | S i − ) = ρ f < ρ r since S i − ⊆ S ◦ i − \ { u i } and u i / ∈ S ◦ i − \ { u i } .Hence, the lemma is proved. Proof of Lemma 3.
For simplicity, let S i := S φ ( π f[ i ] ) , T i := S φ ( π r[ i ] ) , and S ◦ i := S φ ( π ◦ [ i ] ) . By definition, we have Λ( π f[ i ] ) − Λ( π f[ i − ) = (cid:88) φ (cid:0) ρ φ ( S i ) − ρ φ ( S i − ) (cid:1) · p ( φ ) , (12) Λ( π r[ i ] ) − Λ( π r[ i − ) = (cid:88) φ (cid:0) ρ φ ( T i ) − ρ φ ( T i − ) (cid:1) · p ( φ ) , (13) Λ( π ◦ [ i ] ) − Λ( π ◦ [ i − ) = (cid:88) φ (cid:0) ρ φ ( S ◦ i ) − ρ φ ( S ◦ i − ) (cid:1) · p ( φ ) . (14)Let G i be the distribution of i -th residual graph with respectto the ADG policy, where each residual graph G i ∈ G i hasa probability of Pr[ G i ] . Then, the realization space Ω can bedisjointedly partitioned with respect to each G i ∈ G i . For each i , we denote the corresponding set of realizations as Ω( G i ) ,which implies that Pr[ G i ] = (cid:80) φ ∈ Ω( G i ) p ( φ ) . Moreover, forany S and G i , we have ρ G i ( S ) = (cid:88) φ ∈ Ω( G i ) ρ φ ( S ) · p ( φ )Pr[ G i ] . Then, (12)–(14) can be rewritten as Λ( π f[ i ] ) − Λ( π f[ i − ) = (cid:88) G i (cid:0) ρ G i ( S i ) − ρ G i ( S i − ) (cid:1) · Pr[ G i ] , Λ( π r[ i ] ) − Λ( π r[ i − ) = (cid:88) G i (cid:0) ρ G i ( T i ) − ρ G i ( T i − ) (cid:1) · Pr[ G i ] , Λ( π ◦ [ i ] ) − Λ( π ◦ [ i − ) = (cid:88) G i (cid:0) ρ G i ( S ◦ i ) − ρ G i ( S ◦ i − ) (cid:1) · Pr[ G i ] . Putting it all together with Lemma 2 completes the proof.
Proof of Lemma 5.
Let f est and r est be the estimations of E [ I G i ( u i | S i − )] and E [ I G i ( u i | T i − \ { u i } )] . For the i -th iteration of A DD ATP and the j -th round estimations ofexpected marginal spreads (or profits), we define two events E i,j and E i,j as E i,j : | E [ I G i ( u i | S i − )] − f est | ≤ n i ζ i , E i,j : | E [ I G i ( u i | T i − \ { u i } )] − r est | ≤ n i ζ i . Regarding the event E i,j or E i,j , we know that δ i = j − kn and θ = ln(8 /δ i )2 ζ i . Thus, by Lemma 4, we have Pr[ ¬E i,j ∨ ¬E i,j ] ≤ Pr[ ¬E i,j ] + Pr[ ¬E i,j ] ≤ − θζ i = 12 j kn . As a result, we have Pr (cid:104) (cid:95) ∞ j =1 (cid:0) ¬E i,j ∨ ¬E i,j (cid:1)(cid:105) ≤ (cid:88) ∞ j =1 j kn = 1 kn . (15)In what follows, we assume that events E i,j and E i,j happenfor every j . Thus, we have ˜ ρ f − n i ζ i ≤ ρ f ≤ ˜ ρ f + n i ζ i and ˜ ρ r − n i ζ i ≤ ρ r ≤ ˜ ρ r + n i ζ i . We first consider the case that C occurs. If ˜ ρ f ≥ ˜ ρ r , we have ˜ ρ f − ˜ ρ r ≥ n i ζ i or ˜ ρ r ≤ − n i ζ i since ˜ ρ f + ˜ ρ r ≥ (using the same argument in the proof ofLemma 1). For the former, ρ f − ρ r ≥ ˜ ρ f − ˜ ρ r − n i ζ i ≥ ,while for the latter, ρ r ≤ and hence ρ f ≥ ρ r as ρ f + ρ r ≥ by Lemma 1. Thus, ρ f ≥ ρ r always hold if ˜ ρ f ≥ ˜ ρ r . With asimilar analysis, ρ f < ρ r also holds if ˜ ρ f < ˜ ρ r . Then, when C occurs, by Lemma 2, we have ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) ≤ ρ G i ( S i ) − ρ G i ( S i − ) + ρ G i ( T i ) − ρ G i ( T i − ) . Now, we consider the case that C occurs. Again, if ˜ ρ f ≥ ˜ ρ r , we have ρ r ≤ ˜ ρ r + n i ζ i ≤ ˜ ρ f + n i ζ i ≤ ρ f + 2 n i ζ i ≤ ρ f + 2 . On the other hand, one can verify that ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) ≤ max { ρ f , ρ r } ≤ ρ f + 2 = ρ G i ( S i ) − ρ G i ( S i − ) + 2 and ρ G i ( T i ) − ρ G i ( T i − ) = 0 . Similarly, if ˜ ρ f < ˜ ρ r , we have ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) ≤ max { ρ f , ρ r } ≤ ρ r + 2 ≤ ρ G i ( T i ) − ρ G i ( T i − ) + 2 and ρ G i ( S i ) − ρ G i ( S i − ) = 0 . Then, ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) − ≤ ρ G i ( S i ) − ρ G i ( S i − ) + ρ G i ( T i ) − ρ G i ( T i − ) , (16) which also holds for the case that C occurs.On the other hand, suppose that at least one of theevents events E i,j and E i,j does not happen. We know that ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) ≤ max { ρ f , ρ r } ≤ n and ρ G i ( S i ) − ρ G i ( S i − )+ ρ G i ( T i ) − ρ G i ( T i − ) ≥ min { ρ f , ρ r } ≥ − n . Thus,it always holds that ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) − n ≤ ρ G i ( S i ) − ρ G i ( S i − ) + ρ G i ( T i ) − ρ G i ( T i − ) , (17)Combining (15), (16), and (17) completes the proof. Proof of Lemma 6.
We first fix a random seed for A DD ATPsuch that A DD ATP becomes a deterministic algorithm. Usinga similar argument in the proof of Lemma 3, we have Λ( π f[ i ] ) − Λ( π f[ i − ) = (cid:88) G i (cid:0) ρ G i ( S i ) − ρ G i ( S i − ) (cid:1) · Pr[ G i ] , Λ( π r[ i ] ) − Λ( π r[ i − ) = (cid:88) G i (cid:0) ρ G i ( T i ) − ρ G i ( T i − ) (cid:1) · Pr[ G i ] , Λ( π ◦ [ i ] ) − Λ( π ◦ [ i − ) = (cid:88) G i (cid:0) ρ G i ( S ◦ i ) − ρ G i ( S ◦ i − ) (cid:1) · Pr[ G i ] . Then, taking the expectation on both sides over the internalrandomness of the A DD ATP algorithm and combing withLemma 5 complete the proof.
Proof of Lemma 8.
The proof is analogous to those of Lem-mas 5 and 6. The key step is to figure out the bound onprofit loss when C (cid:48) occurs, i.e., n i ζ i ≤ and ε i ≤ ε ,and the estimation is sufficiently accurate (with probability − / ( kn ) ), i.e., f est − n i ζ i ε i − c ( u i ) ≤ ρ f ≤ f est + n i ζ i − ε i − c ( u i ) and c ( u i ) − r est + n i ζ i − ε i ≤ ρ r ≤ c ( u i ) − r est − n i ζ i ε i . If f est + r est ≥ c ( u i ) , one can verify that ρ r ≤ f est − n i ζ i ε − c ( u i ) + n i ζ i + εc ( u i ))1 − ε ≤ ρ f + εc ( u i ))1 − ε . Therefore, we have ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) ≤ max { ρ f , ρ r } ≤ ρ f + εc ( u i ))1 − ε = ρ G i ( S i ) − ρ G i ( S i − )+ εc ( u i ))1 − ε and ρ G i ( T i ) − ρ G i ( T i − ) =0 . Similarly, if f est + r est < c ( u i ) , we have ρ G i ( S ◦ i − ) − ρ G i ( S ◦ i ) ≤ max { ρ f , ρ r } ≤ ρ r + εc ( u i ))1 − ε ≤ ρ G i ( T i ) − ρ G i ( T i − ) + εc ( u i ))1 − ε and ρ G i ( S i ) − ρ G i ( S i − ) = 0 .Following the same arguments in the proofs of Lemmas 5and 6, the lemma is proved.R EFERENCES[1] A. A. Ageev and M. Sviridenko, “An 0.828-approximation algorithmfor the uncapacitated facility location problem,”
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