SlowGuess/NeurIPS_2023_OCR huggingface dataset:
1.  $u^{*} \notin O$ : In this case, we can use similar reasoning as Lemma 2 to get
$$
\mathrm {O P T} \leq 2 f (A) + \alpha f (u ^ {*}) \leq 2 f (T) + \alpha f (u ^ {*})
$$
where the second inequality is due to  $A \subseteq T$  according to Lines 11-12.
2.  $u^{*} \in O$  and  $u^{*}$  accepts the price in Line 11: In this case, we have  $T = A \cup \{u^{*}\}$  and hence can also use similar reasoning as Lemma 2 to get
$$
\mathrm {O P T} - f (T) \leq \sum_ {u \in O \backslash T} f (u \mid A) \leq f (A) + \alpha f (u ^ {*}) \leq f (T) + \alpha f (u ^ {*}),
$$
which yields  $\mathrm{OPT} \leq 2f(T) + \alpha f(u^{*})$ .
3.  $u^{*} \in O$  and and  $u^{*}$  rejects the price in Line 11: In this case, we must have  $T = A$ ,  $c(u^{*}) > B - p(A)$  and hence
$$
c \left(O \backslash \left\{u ^ {*} \right\}\right) \leq p (A) \leq \frac {B \cdot f (A)}{f (\emptyset) + \alpha f \left(u ^ {*}\right)} \leq \frac {B f (A)}{\alpha f \left(u ^ {*}\right)} \tag {8}
$$
where the second inequality is due to Lemma 1. Note that all the sellers in  $O \setminus \{u^{*}\}$  has been processed by Line 1 and each seller  $u \in O \setminus \{u^{*}\} \setminus A$  must reject the offered price and hence satisfy  $c(u) > p(u)$ . So we can use this and Eqn. (8) to get
$$
\begin{array}{l} f (O \setminus \{u ^ {*} \}) - f (A) \leq \sum_ {u \in O \setminus \{u ^ {*} \} \backslash A} f (u \mid A _ {u}) \\ = \sum_ {u \in O \backslash \{u ^ {*} \} \backslash A} p (u) \cdot \frac {f \left(A _ {u}\right) + \alpha f \left(u ^ {*}\right)}{B} \leq \sum_ {u \in O \backslash \{u ^ {*} \} \backslash A} c (u) \cdot \frac {f (A) + \alpha f \left(u ^ {*}\right)}{B} \\ \leq \frac {f (A)}{\alpha f \left(u ^ {*}\right)} \left(f (A) + \alpha f \left(u ^ {*}\right)\right) \leq \frac {1}{\alpha} f (A) + f \left(u ^ {*}\right) \tag {9} \\ \end{array}
$$
Therefore, when  $\alpha > 2$ , we have
$$
\mathrm {O P T} \leq f (O \setminus \{u ^ {*} \}) + f (u ^ {*}) \leq \left(1 + \frac {1}{\alpha}\right) f (A) + 2 f (u ^ {*}) \leq 2 f (T) + \alpha f (u ^ {*}) \tag {10}
$$
According to Lines 6-13, the algorithm returns  $T$  or  $\{u^{*}\}$  with probability of  $\frac{2}{2 + \alpha}$  or  $\frac{\alpha}{2 + \alpha}$ , respectively. So we get
$$
\begin{array}{l} \mathbb {E} [ f (S) ] = \Pr [ S = T ] \cdot f (T) + \Pr [ S = \{u ^ {*} \} ] \cdot f (u ^ {*}) (11) \\ = \frac {2 f (T)}{2 + \alpha} + \frac {\alpha f \left(u ^ {*}\right)}{2 + \alpha} \geq \frac {\mathrm {O P T}}{2 + \alpha} (12) \\ \end{array}
$$
Combining Lemmas 3-4, we immediately get:
Theorem 1 With  $\alpha = \frac{\sqrt{13} + 1}{2} \approx 2.3$ , the TripleEagleRan mechanism can return a set  $S$  satisfying  $\mathrm{OPT} \leq \frac{\sqrt{13} + 5}{2}\mathbb{E}[f(S)]$  by offering one price to each seller in  $\mathcal{N}$  and incurring at most  $\mathcal{O}(n)$  value oracle queries.
Remark: As mentioned in Sec. 2, we have adopted the assumption made in [9, 13] that  $\forall v\in \mathcal{N}$  ..  $c(v)\leq B$  for the simplicity of description. This assumption can be easily removed according to the following discussion. Note that each buyer is guaranteed to be offered a price no more than  $B$  in TripleEagleRan. Therefore, before running TripleEagleRan, we only need to identify a valid  $u^{*} = \arg \max_{v:v\in \mathcal{N}\wedge c(v)\leq B}f(v)$  . This can be done by first sorting all the sellers according to the non-increasing order of their values, and then offer  $B$  to them one by one until seeing the first seller accepting  $B$  and then that seller is clearly a valid  $u^{*}$  . After that, we can delete all the sellers rejecting  $B$  and then run TripleEagleRan. Clearly, using such a method, the complexity of our mechanism on the number of value oracle queries is still  $\mathcal{O}(n)$  , while only one additional price query (for  $u^{*}$  ) is incurred. This method can also be used for all our mechanisms described in the sequel.
4 Deterministic BFM for Monotone Submodular Valuations
If we directly use the LSDPricing algorithm to process all the sellers in  $\mathcal{N}$ , it is possible that the valuation of the sellers who accept the offered prices is no more than  $f(u^{*})$ , while  $u^{*}$  rejects its price and hence cannot be selected as a winner, resulting in a poor approximation ratio (especially when  $f(u^{*})$  is large). To address this issue, we have used randomization in TripleEagleRan. In this section, we introduce TripleEagleDet (i.e., Algorithm 3), a deterministic BFM that circumvents the aforementioned issue using an idea of "two-phase pricing", as elaborated in the following.
In the first pricing phase (Lines 2-6), Algorithm 3 constructs a set  $K$  by using the pricing rule in Line 3 to processes the sellers in  $\mathcal{N} \setminus \{u^{*}\}$ ; this pricing rule is a relaxation of the pricing rule in the LSDPricing algorithm, because the valuation of the already processed sellers is not used to lower the price for the next seller to be processed.
The first pricing phase stops as soon as  $f(K) \geq f(u^{*})$  (Line 6). If  $f(K)$  is still less than  $f(u^{*})$  when all the sellers in  $\mathcal{N} \setminus \{u^{*}\}$  have been processed in the first pricing phase, then Algorithm 3 sets the winner set  $S = \{u^{*}\}$  (Line 8). In the case that the first pricing phase generates a set  $K$  satisfying  $f(K) \geq f(u^{*})$ , Algorithm 3 enters the second pricing phase in Line 10, where we use LSDPricing to process all the sellers not processed in the first phase (including  $u^{*}$ ). Intuitively, since  $f(K) \geq f(u^{*})$
Algorithm 3: TripleEagleDet  $(\alpha)$
1  $K\gets \emptyset ;C\gets \mathcal{N};$    
2 foreach  $u\in \mathcal{N}\backslash \{u^{*}\}$  do   
3  $\begin{array}{r}p(u)\leftarrow \frac{Bf(u|K)}{\alpha f(u^{*})};\mathrm{Show~}u\mathrm{~the~price~}p(u);\\ \mathrm{if~}u\mathrm{accepts}p(u)\mathrm{then}K\leftarrow K\cup \{u\} ;\\ C\leftarrow C\backslash \{u\} ;\\ \mathrm{if~}f(K)\geq f(u^{*})\mathrm{then~break}; \end{array}$    
4   
5   
6 if  $f(K)\geq f(u^{*})$  then break;   
7 if  $f(K) <   f(u^{*})$  then   
8  $p(u^{*})\gets B;S\gets \{u^{*}\} ;$    
9 else   
10 A LSDPricing(K,C,af(u*));   
11 S the largest suffix of A satisfying  $\sum_{u\in S}p(u)\leq B;$    
12 return S
and  $K$  is a prefix of  $A$  in the second pricing phase (Line 10), we can regard  $K$  as a substitute of  $u^{*}$  and hence no longer need to worry about the issue explained in the beginning of this section.
We analyze the approximation ratio of Algorithm 3 by considering two cases. In the first case, we have  $f(K) < f(u^{*})$  or  $p(A) \leq B$ , which implies that  $S = \{u^{*}\}$  or  $S = A$  due to Line 8 and Line 11. In this case, we can directly use the two-phase pricing rules to bound the total valuation of the sellers in  $O$  who rejected the offered prices and hence get Lemma 5:
Lemma 5 When Algorithm 3 finishes, if  $f(K) < f(u^{*})$  or  $p(A) \leq B$ , then we have OPT  $\leq (2 + \alpha)f(S)$ .
In the second case (i.e.,  $f(K) \geq f(u^{*})$  and  $p(A) > B$ ), we must have that  $S$  is generated by Line 11 and  $S \subset A$ . In this case, bounding the performance of Algorithm 3 is more complicated and is given by Lemma 6:
Lemma 6 Suppose that  $\alpha \in (2, \frac{\sqrt{17} + 1}{2})$ . When Algorithm 3 finishes, if  $f(K) \geq f(u^{*})$  and  $p(A) > B$ , then we have
$$
\mathrm {O P T} \leq \max  \left(3 + \frac {3 \alpha + 4}{\alpha^ {2} + 2 \alpha}, 2 + \frac {\alpha (2 + \alpha)}{\alpha^ {2} + \alpha - 4}, 2 + \frac {\alpha (4 + \alpha)}{\alpha^ {2} + \alpha - 2}\right) \cdot f (S) \tag {13}
$$
We roughly explain the idea for proving Lemma 6 as follows. Consider the set  $A$  generated by Line 10 (recall that  $K$  is prefix of  $A$ ). If the smallest suffix  $H$  of  $A$  satisfying  $p(H) > B$  does not intersect  $K$ , then we can use similar reasoning as Lemma 3 to prove OPT  $\leq \left(3 + \frac{3\alpha + 4}{\alpha^2 + 2\alpha}\right)f(S)$ . Under the other case (i.e.,  $H \cap K \neq \emptyset$ ),  $H$  must have a prefix generated using the relaxed pricing rule of Line 3, which possibly degrades the valuation of  $H$  and hence  $S$  (note that  $S$  is a suffix of  $H$ ). Fortunately, since  $f(K)$  is at most  $2f(u^{*})$  due to Line 6, we can prove that the approximation ratio of  $f(S)$  is only slightly weaker than that in the case of  $H \cap K = \emptyset$ , i.e., OPT  $\leq \max \left(2 + \frac{\alpha(2 + \alpha)}{\alpha^2 + \alpha - 4}, 2 + \frac{\alpha(4 + \alpha)}{\alpha^2 + \alpha - 2}\right) \cdot f(S)$ . Combining Lemmas 5-6, we can immediately get the performance bounds of TripleEagleDet, as shown by Theorem 2, which reveals that TripleEagleDet has a better approximation ratio than the deterministic BFM with a provable ratio of 4.75 in [9]. Note that Balkanski et al. [9] show that their BFM cannot achieve a ratio better than 4.5 even if their analysis for the 4.75-ratio is not tight.
Theorem 2 With  $\alpha = \sqrt{6} \approx 2.45$ , the TripleEagleDet mechanism can return a set  $S$  satisfying  $\mathrm{OPT} \leq (2 + \sqrt{6})f(S)$  by offering one price to each seller in  $\mathcal{N}$  and incurring  $2n$  value oracle queries.
5 BFM for Non-Monotone Submodular Valuations
In this section, we extend our TripleEagleRan algorithm to handle a non-monotone submodular valuation function  $f(\cdot)$ , as shown by the TripleEagleNm algorithm (i.e., Algorithm 4).
Algorithm 4: TripleEagleNm(α)
1  $A_{1}\gets \emptyset ;A_{2}\gets \emptyset$    
2 foreach  $u\in \mathcal{N}\backslash \{u^{*}\}$  do   
3  $j\gets \arg \max_{i\in \{1,2\}}f(u\mid A_i)$    
4  $p(u)\gets \frac{B\cdot f(u|A_j)}{f(A_j) + \alpha f(u^*)};$  Show u the price p(u);   
5 if u accepts p(u) then   
6  $\begin{array}{r}\lfloor A_j\leftarrow A_j\cup \{u\} ; \end{array}$    
7 if max{f(A1),f(A2)}≥f(u*)then   
8 Let  $u = u^{*}$  and process u using Lines 3-6;   
9  $A^{*}\gets \arg \max_{X\in \{A_1,A_2\}}f(X);$    
10 S -the largest suffix of  $A^{*}$  satisfying  $\sum_{u\in S}p(u)\leq B$    
11 else   
12 Sample a random number Z from the uniform distribution U[0,1];   
13 if  $Z\leq \frac{\alpha}{3 + \alpha}$  then   
14  $p(u^{*})\gets B;S\gets \{u^{*}\} ;$    
15 else   
16 q- arg maxi{1,2}f(u\* | Ai);   
17 Set  $p(u^{*}) = B - p(A_{q})$  and show u* the price p(u\*);   
18 if  $f(u^{*}\mid A_q)\geq 0$  and u* accepts  $p(u^{*})$  then  $A_{q}\gets A_{q}\cup \{u^{*}\}$  .   
19 T- arg maxX∈{A1,A2}f(X);S-T;   
20 return S
The TripleEagleNm algorithm is similar in spirit to the TripleEagleRan algorithm except that it maintains two sets of candidate winners:  $A_{1}$  and  $A_{2}$ . In Lines 1-6 of Algorithm 4, each seller  $u \in \mathcal{N} \setminus \{u^{*}\}$  is greedily routed to  $A_{j}(j \in \{1,2\})$  such that  $f(u \mid A_{j})$  is maximized, and then  $u$  is offered a price generated using the same rule as Line 2 of Algorithm 2. After all the sellers in  $\mathcal{N} \setminus \{u^{*}\}$  are processed, Lines 12-19 of Algorithm 4 adopt the operations similar to those in Lines 6-13 of Algorithm 1, except that  $u^{*}$  is offered a "best-effort" price of  $B - p(A_{q})$  after it is greedily routed to  $A_{q}$  by Line 16. We show the performance bounds of Algorithm 4 in Theorem 3:
Theorem 3 With  $\alpha = 3$ , the TripleEagleNm mechanism can return a set  $S$  satisfying OPT  $\leq 12\mathbb{E}[f(S)]$  by offering one price to each seller in  $\mathcal{N}$  and incurring at most  $3n$  value oracle queries.
6 Performance Evaluation
We use the influence maximization application mentioned in Sec. 1 to compare the following BFMs: (1) Our TripleEagleRan (TER) and TripleEagleDet (TED) algorithms, (2) The Iterative-Pruning (IP) algorithm in [9] with a ratio of 4.75, and (3) The Random-TM (RTM) algorithm in [24] with a ratio of 5. All the algorithms are implemented using  $\mathrm{C + + }$  and are run on a Linux server with Intel Xeon Gold 6126 @ 2.60GHz CPU and 128GB memory. Each implemented randomized algorithm is independently executed 50 times, and the average result is reported. We use three real social network datasets: (1) Flixster [5] with 28,843 nodes and 272,786 edges; (2) Epinions [29] with 75,879 nodes and 508,837 edges; and (3)Slashdot [29] with 82,168 nodes 948,464 edges. We adopt the well-known Independence Cascade (IC) model [27] for the influence spread function  $f(\cdot)$ , and follow [14] to set the activation probability  $p_{u,v}$  of each edge  $(u,v)$  to  $1 / N_{in}(v)$ , where  $N_{in}(v)$  denotes the set of in-neighbors of  $v$ . The cost  $c(u)$  of each node is generated uniformly at random from the interval [0, 1]. Since evaluating  $f(\cdot)$  is known to be #P-hard [14], we adopt the approach in [11] to generate one million random reverse-reachable sets for approximately evaluating  $f(\cdot)$ . It is known that the influence spread function through such an approximation is still monotone and submodular [11].
The experimental results in Fig. 1 reveal that, for all three datasets, our BFMs outperform the baselines on utility (i.e., the valuation of returned solution). Specifically, TER and TED have similar utilities that outperform RTM (resp. IP) with the performance gain ranging from  $55\% - 122\%$  (resp.  $4\% - 14\%$ ). Moreover, TER and TED incur much fewer value oracle queries than RTM (resp. IP) in more than one (resp. three to four) orders of magnitude. It can also be seen that, although TER and RTM are
(b) Flixster
(a) Flixster
(c) Epinions
(e)Slashdot
Figure 1: Comparing the BFMs for the influence maximization application
(d) Epinions
(f)Slashdot
both randomized mechanisms, RTM incurs large variations while TER is almost deterministic in Fig. 1. This can be explained by the fact that RTM always returns a random solution by its design, while the randomization of TER is only triggered in some special cases (see Sec. 3). Finally, we have also compared our BFMs with the existing ones using a crowdsourcing application similar to that in [8, 16, 20, 22], where non-monotone submodular valuations are also considered. The experimental results are qualitatively similar to Fig. 1 and can be found in the Appendix due to the page limits.
We have proposed TripleEagle, a novel algorithmic framework for designing budget feasible mechanisms with submodular valuations, based on which we have presented several simple, fast, effective and practical BFMs that achieve obvious strategyproofness, low complexity, and better approximation ratios than the-state-of-the-art studies. The efficiency and effectiveness of our approach has been strongly corroborated by our experiments on influence maximization and crowdsourcing.
Kai Han's work is partially supported by the National Natural Science Foundation of China (NSFC) under Grant No. 62172384.
[1] Amazon Mechanical Turk. https://www.mturk.com.
[2] Georgios Amanatidis, Pieter Kleer, and Guido Schäfer. Budget-feasible mechanism design for non-monotone submodular objectives: Offline and online. In ACM Conference on Economics and Computation (EC), pages 901-919, 2019.
[3] Georgios Amanatidis, Pieter Kleer, and Guido Schäfer. Budget-feasible mechanism design for non-monotone submodular objectives: Offline and online. Mathematics of Operations Research, 47(3):2286-2309, 2022.
[23] Pooya Jalaly and Éva Tardos. Simple and efficient budget feasible mechanisms for monotone submodular valuations. In The 14th International Conference on Web and Internet Economics (WINE), pages 246-263, 2018.
[24] Pooya Jalaly and Éva Tardos. Simple and efficient budget feasible mechanisms for monotone submodular valuations. ACM Transactions on Economics and Computation (TEAC), 9(1):1-20, 2021.
[25] Radu Jurca and Boi Faltings. Mechanisms for making crowds truthful. Journal of Artificial Intelligence Research (JAIR), 34:209-253, 2009.
[26] John H. Kagel, Ronald M Harstad, and Dan Levin. Information impact and allocation rules in auctions with affiliated private values: A laboratory study. *Econometrica: Journal of the Econometric Society*, pages 1275–1304, 1987.
[27] David Kempe, Jon Kleinberg, and Éva Tardos. Maximizing the spread of influence through a social network. In ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), pages 137-146, 2003.
[28] Ariel Kulik, Hadas Shachnai, and Tami Tamir. Approximations for monotone and nonmonotone submodular maximization with knapsack constraints. Mathematics of Operations Research, 38 (4):729-739, 2013.
[29] Jure Leskovec and Andrej Krevl. Snap datasets: Stanford large network dataset collection. URL: http://snap.stanford.edu/, 2014.
[30] Shengwu Li. Obviously strategy-proof mechanisms. American Economic Review, 107(11): 3257-3287, 2017.
[31] Paul Milgrom and Ilya Segal. Clock auctions and radio spectrum reallocation. Journal of Political Economy, 128(1):1-31, 2020.
[32] Roger B. Myerson. Optimal auction design. Mathematics of Operations Research, 6(1):58-73, 1981.
[33] Aviad Rubinstein. On the computational complexity of optimal simple mechanisms. In Innovations in Theoretical Computer Science (ITCS), pages 21-28, 2016.
[34] Shai Shalev-Shwartz et al. Online learning and online convex optimization. Foundations and Trends® in Machine Learning, 4(2):107-194, 2012.
[35] Yaron Singer. Budget feasible mechanisms. In IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 765-774, 2010.
[36] Yaron Singer. How to win friends and influence people, truthfully: influence maximization mechanisms for social networks. In ACM International Conference on Web Search and Data Mining (WSDM), pages 733-742, 2012.
[37] Yaron Singer and Manas Mittal. Pricing mechanisms for crowdsourcing markets. In International World Wide Web Conferences (WWW), pages 1157-1166, 2013.
[38] Jennifer Wortman Vaughan. Making better use of the crowd: How crowdsourcing can advance machine learning research. Journal of Machine Learning Research (JMLR), 18(1):7026-7071, 2017.
Proof: It is evident that  $f(X) \geq \sum_{v \in X} f(v \mid A_v) = f(A_{u_s} \cup X) - f(A_{u_s})$  due to submodularity. Besides, for each  $v \in X$ , we have  $f(A_v) \geq f(A_{u_s})$  and  $c(v) \leq p(v)$  (because  $v$  accepts the offered price), so we get
$$
c (v) \leq p (v) = \frac {B f (v \mid A _ {v})}{f (A _ {v}) + \alpha f (u ^ {*})} \leq \frac {B f (v \mid A _ {v})}{f \left(A _ {u _ {s}}\right) + \alpha f (u ^ {*})}
$$
$$
\sum_ {v \in X} f (v \mid A _ {v}) \geq \sum_ {u \in X} p (v) \cdot \frac {f \left(A _ {u _ {s}}\right) + \alpha f \left(u ^ {*}\right)}{B} = p (X) \cdot \frac {f \left(A _ {u _ {s}}\right) + \alpha f \left(u ^ {*}\right)}{B}
$$
Proof: After Line 3 is executed, all the elements in  $\mathcal{N}$  must have been processed by the LSDPricing algorithm. As each element in  $O\setminus A$  rejects the offered price, we have
$$
\forall u \in O \backslash A: c (u) > p (u) = \frac {B f (u \mid A _ {u})}{f \left(A _ {u}\right) + \alpha f \left(u ^ {*}\right)} \geq \frac {B f (u \mid A _ {u})}{f (A) + \alpha f \left(u ^ {*}\right)} \tag {14}
$$
Using this and submodularity, we get
$$
\begin{array}{l} \mathrm {O P T} - f (A) \leq \sum_ {u \in O \backslash A} f (u \mid A) \leq \sum_ {u \in O \backslash A} f (u \mid A _ {u}) \\ \leq \sum_ {u \in O \backslash A} c (u) \cdot \frac {f (A) + \alpha f \left(u ^ {*}\right)}{B} \leq f (A) + \alpha f \left(u ^ {*}\right) \tag {15} \\ \end{array}
$$
The lemma then follows by re-arranging the above inequality.
C Proof of Lemma 5
Proof: Let  $O_1, O_2$  be the set of sellers in  $O$  who reject the offered prices in Line 4 of Algorithm 3 and in the LSDPricing procedure (Line 10 of Algorithm 3), respectively. Let  $K_u$  denotes the set of elements already in  $K$  when  $u$  is offered a price in Line 3. Under the case of  $f(K) < f(u^*)$ , we must have  $S = \{u^*\}$  (due to Line 8),  $O_1 = O \setminus \{u^*\} \setminus K$  and
$$
f (O \setminus \{u ^ {*} \}) - f (K) \leq \sum_ {u \in O _ {1}} f (u \mid K _ {u}) = \sum_ {u \in O _ {1}} p (u) \cdot \frac {\alpha f (u ^ {*})}{B} \leq \sum_ {u \in O _ {1}} c (u) \frac {\alpha f (u ^ {*})}{B} \leq \alpha f (u ^ {*})
$$
$$
\mathrm {O P T} \leq f (K) + \alpha f \left(u ^ {*}\right) + f \left(u ^ {*}\right) \leq (2 + \alpha) f \left(u ^ {*}\right) = (2 + \alpha) f (S) \tag {16}
$$
On the other hand, if  $f(K) \geq f(u^{*})$  and  $p(A) \leq B$ , then we have  $O \setminus A = O_{1} \cup O_{2}$ . Note that  $K$  is a prefix of  $A$  under this case. So we can use submodularity to get
$$
\begin{array}{l} \mathrm {O P T} - f (A) \leq \sum_ {u \in O _ {1}} f (u \mid K _ {u}) + \sum_ {u \in O _ {2}} f (u \mid A _ {u}) \\ \leq \sum_ {u \in O _ {1}} c (u) \frac {\alpha f \left(u ^ {*}\right)}{B} + \sum_ {u \in O _ {2}} c (u) \frac {f \left(A _ {u}\right) + \alpha f \left(u ^ {*}\right)}{B} \\ \leq c \left(O _ {1}\right) \cdot \frac {\alpha f \left(u ^ {*}\right)}{B} + c \left(O _ {2}\right) \cdot \frac {f (A) + \alpha f \left(u ^ {*}\right)}{B} \\ \leq c (O) \cdot \frac {f (A) + \alpha f \left(u ^ {*}\right)}{B} \\ \leq f (A) + \alpha f \left(u ^ {*}\right), \tag {17} \\ \end{array}
$$
which yields  $f(A) \leq 2f(A) + \alpha f(u^{*}) \leq (2 + \alpha)f(A)$  by using  $f(A) \geq f(K) \geq f(u^{*})$ . Note that  $S = A$  under this case (due to Line 11), so the lemma follows.
D Proof of Lemma 6
Proof: By similar reasoning with Lemma 5, we have
$$
\operatorname {O P T} \leq 2 f (A) + \alpha f \left(u ^ {*}\right) \tag {18}
$$
Let  $H$  be the smallest suffix of  $A$  satisfying  $p(H) > B$ , so  $H \setminus S$  contains exactly one element (denoted by  $w$ ) which is added into  $A$  right before the elements in  $S$ . For convenience, we define  $\delta(X) = \sum_{u \in X} f(u \mid A_u)$  for any regular subset  $X$  of  $A$ . Suppose that  $f(A \setminus H) = \mu f(u^*)$ ,  $f(K) = \lambda f(u^*)$ ,  $\delta(K \cap H) = \tau f(u^*)$  and  $p(K \cap H) = \rho B$ . Clearly, we have  $p(H \setminus K) \geq (1 - \rho)B$ ,  $\lambda \geq 1$  and  $\tau \leq \lambda \leq 2$ . We consider the following cases:
1.  $K \cap H = \emptyset$ : In this case, all the sellers in  $H$  are selected using the pricing rule of Algorithm 2. So we can use similar reasoning as Lemma 3 to get
$$
\mathrm {O P T} \leq \left(4 + \frac {2 - \alpha + 2 / x}{x - 1 / x}\right) f (S) \tag {19}
$$
where  $x = \mu + \alpha$ . Note that  $f(K) \geq f(u^{*})$  and  $K \subseteq A \setminus H$ , so  $\mu \geq 1$  and  $x \geq \alpha + 1$ . The R.H.S. of Eqn (19) is no more than  $4f(S)$  when  $x \geq \frac{2}{\alpha - 2}$  or  $\alpha + 1 \geq \frac{2}{\alpha - 2}$ . When  $\alpha + 1 < \frac{2}{\alpha - 2}$  (i.e.,  $\alpha < \frac{\sqrt{17} + 1}{2}$ ) and  $x \in [\alpha + 1, \frac{2}{\alpha - 2})$ , the R.H.S. of Eqn (19) is maximized when  $x = \alpha + 1$ . Therefore, for any  $\alpha \in (2, \frac{\sqrt{17} + 1}{2})$  we have
$$
\mathrm {O P T} \leq \left(4 + \frac {\alpha - \alpha^ {2} + 4}{\alpha^ {2} + 2 \alpha}\right) f (S) = \left(3 + \frac {3 \alpha + 4}{\alpha^ {2} + 2 \alpha}\right) f (S) \tag {20}
$$
2.  $|K \cap H| > 1$ : In this case, we must have  $|K \cap S| \geq 1$  because  $|H \setminus S| = 1$  by definition. According to the construction of  $K$ , this implies  $f(K \setminus S) < f(u^{*})$  and hence
$$
f (A) \leq f (K \setminus S) + f (S) \leq f \left(u ^ {*}\right) + f (S) \tag {21}
$$
Note that each seller  $u \in K$  satisfies  $c(u) \leq p(u) = \frac{Bf(u|A_u)}{\alpha f(u^*)}$ , so we have
$$
\tau f \left(u ^ {*}\right) = \delta (K \cap H) \geq p (K \cap H) \frac {\alpha f \left(u ^ {*}\right)}{B} \geq \rho \alpha f \left(u ^ {*}\right) \tag {22}
$$
This implies  $\rho \leq \tau / \alpha$  and hence  $p(H \setminus K) \geq (1 - \tau / \alpha)B$ . By similar reasoning as Lemma 1, we have
$$
\delta (H \backslash K) \geq (1 - \rho) (f (K) + \alpha \cdot f (u ^ {*})) \geq (1 - \tau / \alpha) (\lambda + \alpha) \cdot f (u ^ {*}) \tag {23}
$$
$$
\begin{array}{l} \delta (H) = \delta (K \cap H) + \delta (H \setminus K) \\ \geq \left[ \tau + (1 - \tau / \alpha) (\lambda + \alpha) \right] \cdot f \left(u ^ {*}\right) = (\alpha + \lambda - \tau \lambda / \alpha) f \left(u ^ {*}\right) \\ \geq \left(\alpha + \lambda - \lambda^ {2} / \alpha\right) f \left(u ^ {*}\right), \tag {24} \\ \end{array}
$$