Fair Representation in Submodular Subset Selection: A Pareto Optimization Approach
Authors: Adriano Fazzone
Yanhao Wang
Francesco Bonchi
Venue: Accepted by TMLR ()

Abstract: Many machine learning applications, such as feature selection, recommendation, and social advertising, require the joint optimization of the global utility and the representativeness for different groups of items or users. To meet such requirements, we propose a novel multi-objective combinatorial optimization problem called Submodular Maximization with Fair Representation (SMFR), which selects subsets from a ground set, subject to a knapsack or matroid constraint, to maximize a submodular (utility) function $f$ as well as a set of $d$ submodular (representativeness) functions $g_1, \dots, g_d$. We show that the maximization of $f$ might conflict with the maximization of $g_1, \dots, g_d$, so that no single solution can optimize all these objectives at the same time. Therefore, we propose a Pareto optimization approach to SMFR, which finds a set of solutions to approximate all Pareto-optimal solutions with different trade-offs between the objectives. Our method converts an instance of SMFR into several submodular cover instances by adjusting the weights of the objective functions; then it computes a set of solutions by running the greedy algorithm on each submodular cover instance. We prove that our method provides approximation guarantees for SMFR under knapsack or matroid constraints. Finally, we demonstrate the effectiveness of SMFR and our proposed approach in two real-world problems: maximum coverage and recommendation.

Published in Transactions on Machine Learning Research (06/2024)
Fair Representation in Submodular Subset Selection: A Pareto Optimization Approach
Adriano Fazzone
ddriano.faaaomeeeeettt..cc
CENTAI Institute, Turin, Italy
Yanhao Wang
ghuung@ddase.cou..c..crr
East China Normal University, Shanghai, China
Francesco Bonchi
fynnesso.honnchheeentte
CENTAI Institute, Turin, Italy
Eurecal, Barcelona, Spain
Reviewed OpenReview: https: openreveepnn..veeu. net/ formm? id= 08l001 Vc827
Abstract
Many machine learning applications, such as feature selection, recommendation, and social advertising, require the joint optimization of the global utility and the representaticeness for different groups of items users. To meet such requirements, we propose a novel multi- objective combinatorial optimization problem called Submodular Mazimiiaaion with Fair Representation (SMFR), which selects subsets from a ground set, subject to a knapsack or matroid constraint, maximize submodular (ufility) function f well as set of d submodular (representativeness) functions .d.....9d. We show that the maximization of might conflict with the maximization of 91.-..9dd Id, so that single solution can optimize all these objectives at the same time. Therefore, we propose Pareto optimization approach to SMFR, which finds set of solutions to approximate all Pareto-optimal solutions with different trade-offs between the objectives. Our method converts an instance of SMFR into several submodular cover instances by adjusting the weights the objective functions; then computes a set of solutions by running the greedy algorithm on each submodular cover instance. We prove that our method provides approximation guarantees for SMFR under knapsack matroid constraints. Finally, demonstrate the effectiveness of SMFR and our proposed approach in two real-world problems: mazimum coverage and recommendation.
1 Introduction
The problem of subset selection aims to pick maximum utility subset S, under given constraint, from ground set V of items. This fundamental problem arises in a wide range of machine learning applications, such as social advertising (Kempe et al. 2003 Aslay et al.) 2015 2017 Tang 2018). recommendation
systems (Ohsaka & Matsuoka, [202 Mehrotra & Vishnoi] 2023) ) data summarization (Lin & Bilmes 2010
Mirzasoleiman et al. (0166), and feature selection (Liu et al., 2013 Bao et al. (022), to name just a few. A common combinatorial structure in such problems is submodularity (Krause & Golovin) 2014), which naturally captures the "diminishing returns" property: adding marginal gain than adding it to larger set. This property not only captures the desirable properties of coverage and diversily of subsets, but also enables the design of efficient approximation algorithms.
item to smaller set produces higher
Among the various combinatorial optimization problems for subset selection in the literature, maximizing monotone submodular function subject a knapsack constraint (SMK) or a matroid constraint (SMM) has attracted lot of attention, as such constraints capture common scenarios in which the selected subset must be limited within budget Krause & Guestrin 2005 Calinescu al.) 2011

---

Published in Transactions on Machine Learning Research (06/2024)
+ +R to measure the utility More formally, given a ground set V of n items, we consider a set function f 2
S(S) of any set S V. We assume that ʃ is normalized, i.e., f(0) = 0, monotone, i.e., S(S) S ((T) for any SCT S V. and submodular, f(S (o)) - f(S) 2 (v)) - f(T) for any S F C V and v REVİZz \.. We also consider a cost function c: V R which assigns a positive cost c(u) to cach item V E V, and we denote c(S) the cost of set S _ V, defined as the sum of costs for all ilems in S, i.e., c(S) Eves c(v). For given budget & E RR,, the set of all feasible solutions subject to the knapsack constraint contains all subsets of V whose costs are at most k, i.e., I& = {S V c(S) S kj. The SMK problem on f is defined as Sj = arg max f(S). Furthermore, a matroid M on ground set V is defined by collection I(M) of subsets of V called the independent sets, that satisfies the following properties: (1) 9 E I(M); (2) for any S C T S V, if T' E ZZ, then S E I(M) holds; (3) for any S,T V, if [S| < [T', there exists e E 7AS S such that SU {u} E I(M). Here, the size of the maximum independent sets in .M is called its rank r(M). Similarly to SMK, the SMM problem on f is defined as S = arg max f()).
In many real-world problems, in addition to the primary objective of maximizing the utility function f. it is often essential to take into account the represenlativeness with respect to different. groups of items or users. For example, consider the influence maximization problem (Tsang et al. 2019 Becker et al. 2020):
Example 1. Let G = (V, be graph that denotes the relationships between set of users V on social network. Each user V E V is also associated with a sensitive attribute A to divide V into multiple protected groups. The influence maximization (IM) problem (Kempe et al.) 2003) aims to select a subset S S V of users as seeds to maximize a (monotone, submodular) influence spread function under an information diffusion (e.g., independent cascade or linear threshold) model. If the information to be spread is related to education and employment opportunities, fair access to information between protected groups (Tsang et al., 2019 Becker et al. 2020) becomes a critical issue. This is often formulated maximizing the influence spread functions specific to all protected groups in a balanced manner that none of the groups is much worse off than the others. Furthermore, we should also impose constraints in different contexts on the seed set S, e.g., to limit the overall budget for the propagation campaign, the total cost of S should be within an upper bound (knapsack constraint), or to ensure diversity, the number of seeds in S from any demographic category cannot exceed an upper limit (matroid constraint).
The above problem, as well as many other subset selection problems with fairness or other representativeness considerations (Krause et al.] 2008 Mirzasoleiman et al. 2016 Wang et al.) 2024). can be formulated as a multi-objective optimization problem of selecting sct S to simultancously maximize monotone submodular utility function /f and set of d monotone submodular representativennss functions "),...,9d, all defined on the same ground set V, subject to a knapsack or matroid constraint I:
max SEZ
9a(S)).
(1)
We call this problem Submodular Mazimization with Fair Representation (SMFR | since it captures the case where the submodular utility function f and all the submodular representativeness functions , Id are maximized at the same time to avoid under-representing any of them.
Our Contributions. To the best of our knowledge, SMFR is a novel optimization problem, never addressed before (sce Section 2 for a detailed discussion of how the related literature differs from SMFR). It is casy to see that SMFR is at least hard as SMK and SMM, which cannot be approximated within factor better than 1 1/c unless P = NP [Feige| 1998). However, SMFR is much more challenging than SMK and SMM due to its multi-objective nature. By providing a counterexample (see Example 2). we show that there might not exist any single solution to an instance of SMFR that achicves an approximation factor greater than 0 to maximize f and .1,....99 Ia at the same time, even when d = 1. Due to the inapproximabillity of SMFR, we consider approaching it by Pareto optimization. Specifically, we call set an (a, (o..@)-aprrxiimat solution for an instance of SMFR if S E I, f(S) 2 aOPT,, where OPT, = maxs f(S), and gi( Z 30PT, for all i .,...,d, d, where OPT,, = maxs m An (a, -)appooxiiee solution S is Pareto optimal if there does not exist an (a', ('',"""'apppooxii -approximate solution S' E I for any a' 2 a, 8 2 B and at least one is strictly larger. Since computing a single Pareto optimal solution to SMFR is still NP-hard, we turn our atlention to identifying a set S of multiple solutions to approximate the Parelo frontder, that is, to find
Note that all objective functions are unordered and equally important Eq.
2

---

Published in Transactions on Machine Learning Research (06/2024)
set S such that for any Pareto-optimal solution, there exists a corresponding solution in S achieving bounded approximation for it. Our framework first uses any existing algorithm for SMK (Sviridenko 2004
Yaroslavtsev et al., 2020; Tang et al. 2021 Feldman et al. 2022 Li et al. 2022) or SMM
FFisher et al. 1978,
Vondrak 2008 Calinescu et al.] 2011 Badanidiyuru & Vondrák 2013 Filmus & Ward 2014 Buchbinder
et al. 2019) to approximate , and each g. Based on the approximations ions, our proposal transforms an instance of SMFR into multiple instances of the submodular cover problem with different weights on y and each g to capture the trade-offss between I and each gr. Then, classic greedy algorithms (Wolsey 1982 Torrico et al., 2021) are used to obtain approximate solution for each submodular cover instance. Finally, all the above-computed solutions that are not "dominated") by any other computed solution are returned as the set S of at most O(2) approximate solutions to SMFR for any & E (0, 1). Theoretically, our framework provides approximation bounds for SMFR under both knapsack and matroid constraints:
When using a d-approximation algorithm for SMK, it provides a set S such that for each (a, 3)- approximate Pareto optimal solution of SMFR, there must exist a corresponding (ōa - E, 2.83 E)- approximate solution of cost. O(k log () in 8, where & E R+ is the budget of the knapsack constraint.
When using d-approximation algorithm for SMM, it also provides a set S such that for each (a, 8)- approximate Pareto optimal solution of SMFR, there must exist a corresponding (&a - e, 83 c)- approximate solution of size Orr ) in S, where r & Z+ is the rank of the matroid constraint.
In the empirical assessment, we evaluate our proposed framework for the problems of mazinum coverage and recommendation using real-world data. The numerical results confirm the effectiveness of our proposal compared to competitive baselines.
Paper Organization. The remainder of this paper is organized as follows. We review the related work in Section 2 Then, we analyze the hardness of SMFR in Section 3 Next, our algorithmic framework for SMFR is presented in Section Subsequently, the experimental setup and results are provided in Section 5] Finally, we conclude the paper and discuss future work in Section D. The proofs of theorems and lemmas and several supplemental experiments are deferred to the appendices due to space limitations.
2 Related Work
Monotone Submodular Maximization with Knapsack or Matroid Constraints. There exists a wide literature maximizing a monotone submodular function subject to a knapsack constraint (SMK) matroid constraint (SMM). For cardinality constraints, a special case of both knapsack and matroid constraints, Nemhauser ct al. 1978) proposed a simple greedy algorithm that runs in O(kn) time and yields the best possible approximation factor 1 1-11/ 1//e unless p P=NP. However, the greedy algorithm can be arbitrarily bad for general knapsack or matroid constraints. Sviridenko (2004) first proposed a greedy algorithm with partial enumerations that achieves the best possible approximation 1 11/e for SMK in O(n²) time. Kulik et al. (2021) and Feldman et al. (2022) improved the time complexity to O(n²) while keeping the same approximation factor. Krause & Guestrin
(2055) proposed an O(n2)-time [ _
0.31-approximmtion cost-cffective greedy algorithm for SMK. Tang ct al. (2021), Kulik et al. (2021 and Feldman el al. (2022) improved the approxination factor of the cost-effective greedy algorithm to 0.405, 00.427, 0.4295), and [0.427, 0.462] independently. Ene & Nguyen (2019a) proposed a near-linear time (1-1/e-
c) e-aapproximation algorithm for SMK based on multilincar relaxation. Yaroslavtsev et al. (222)) proposed a d-approximation Greedy+
algorithm for SMK in O(n²) time. Feldman et al. (2022) further provided an approximation factor of 0.6174 in O(n3) time by enumerating each item as a partial solution and running Greedy+ on each partialsolution solution. Li et al. (2022) recently proposed (ppee)aaproxiimatoon algorithm for SMK in O(" n log !) time. Fisher et al. 1978) first proposed -aaprroximation greedy algorithm for SMM running in ((r)) time. Calinescu et al. (2011) and Vondrak (2008) independently proposed randomized continuous greedy algorithms with rounding for SMM. Both algorithms achieved the best possible (1 (1-14 1/e)- approximation in expectation but had prohibitive O(n²) running time. [Bddnndiiyuru & Vondrák (2013) proposed a faster contimuous greedy algorithm that yielded a (1 - 1/e - (lle-=)-ppppeeee =)-aproxiimation for SMM in
2 solution S will be dominated by another solution T the approximation factors a, of 8 are both no greater than those of T and least one is strictly smaller.
3

---

Published in Transactions on Machine Learning Research (06/2024)
2 O log" =) time. Filmus & Ward (2014) proposed a (1- 10-11/e)--appee apparooxiiitooon algorithm in O( time and (1-1/e) 11-1e)-approximaae
algorithm in ((nrr") time, both randomized and based on non-oblivious local search. Buchhiiddr et al. (0199) proposed the first deterministic algorithm for SMM with an approximation
factor over 1/2 in O(nr*) time. Enc & Nguyen (2019b) also proposed a nearly lincar-time (1 - 1/e - E)-
approximation algorithm for SMM based on multilinear relaxation. Although the above algorithms cannot be applied directly to SMFR, any of them can serve as a subroutine our algorithmic framework for SMFR.
Multi-objective Submodular Maximization. There exist several variants of submodular maximization problems to deal with more than one objective. Next, we discuss multi-objective submodular maximiza- tion problems relevant to SMFR. One basic problem of this kind is to maximize a weighted sum of d > 1 submodular functions 91.....90 Id.- Since the weighted sum of multiple submodular functions is still submod- ular (Krause & Golovin 2014), this problem can be directly resolved with any algorithm for submodular maximization. However, maximizing the weighted sum is often not enough to achieve a fair representation among all objectives, as its (optimal) solution may not have any approximation guarantce for each objective function individually. The problem of maximizing the minimum of d > submodular functions 91..... :9
was studied in (Krause et al.) 2008 Udwani 2018 Anari et al. 2019 Torrico et al. 2021]. This problem differs from SMFR because it does not consider maximizing ʃ and aims to return only a single solution for all functions. Nevertheless, we draw inspiration from the SATURATE framework first proposed by Krause et al. 2008) to address SMFR. Another two relevant problems to SMFR are Submodulor Marimization under Submodular Cover (SMSC) (Ohsaka & Matsuoka 2021), which maximizes one submodular function subject to the value of the other submodular function not being below a threshold, and Balancing utility and foirness in Submodular Marimization (BSM) (Wang et al. 2024), which maximizes a submodular util- ity function subject to that a fairness function in form of the minimum of d > 1 submodular functions is approximately maximized. SMSC and BSM differ from SMFR in the following four aspects: (t) they also return single solution to optimize user-specified trade-off between multiple objectives; (ii) they are spe- cific to cardinality constraints but cannot handle more general knapsack or matroid constraints; (Mi) SMSC is limited to two submodular functions, that is, a special case of d = in SMFR; (iv) BSM requires all objective functions to be decomposable. Thus, SMFR can work more general scenarios than SMSC and BSM. Due to the above differences, the algorithms for SMSC and BSM cannot be used directly for SMFR, and in the experiments they will be compared with our algorithm after adaptations. Very recently, [Tang] & Yuan (0023) proposed randomized subset selection method to maximize a (submodular) overall utility function while the (submodular) utility functions for d groups are all not below a lower bound in expectation. They also considered submodular maximization with group equality, which ensures that the difference in the expected utilities of any two groups does not exceed an upper bound. As they limit their consideration to cardinality constraints and their problem formulations are different from SMFR, their proposed methods do not apply to SMFR. The problem of regret-ratio minimization (Soma & Yoshida, 2017 Feng & Qian, 2021; Wang et al., 2023 for multi-objective submodular maxinization is similar to SMFR in the sense that they also aim to find set of approximate solutions for different trade-offs between multiple objectives. However, they consider denoting the tradc-offs as different non-negative lincar combinations of multiple submodular functions but cannot guarantee any approximation for each objective individually.
2020 Roostapour et al., 2022 utilize a Parcto optimization method by transforming a singlo-objoctive problem into a bi-objective problem and then solving the bi-objective problem to obtain an approximate solution to the original problem. These problems are interesting but orthogonal to our work.
Finally, several subset selection problems, e.g., QQian et al. 2015 2017
3 Hardness of SMFR
In this paper, we focus on the SMFR problem in Eq. [ subject to a knapsack or matroid constraint. Next, we formally analyze the computational hardness of SMFR. Since SMK and SMM are both NP-hard and cannot be approximated within a factor 1 - 1/e + s in polynomial time for any & >> O unless P = NP (Feige] 1998 Khuller et al. 1999), the problem of maximizing f or each 9: individually can only be solved approximately. We provide a trivial example to indicate that the maximization of f and the maximization of each 9i could conflict with each other, and there might not exist any S E I with approximation factors greater than 0 for both of them, even when d= 1.
4

---

Published in Transactions on Machine Learning Research (06/2024)
Example 2. Suppose that d = 1 and the set of feasible solutions I is defined by cardinality constraint
1, ie.., I = {S C V |S| & 1}. Note that cardinality constraint is special case of both knapsack and matroid constraints. For the two functions f and 91, we have OPT, = f((vo)) = 1, OPT,, = (I((e1) = 1, 0 for any j > 1. In the above SMFR instance, there =0 is no set S E I such that S(5) > and 91 (1))) > 0.
(f(oo)) = 0. f((v1)) = O, and f((uz)) =
Given the above result, we are motivated to introduce Pareto optimization, a well-known concept for multi- objective optimization (Qian et al.) 2015 Soma & Yoshida 2017) which provides more than one solution with different (best possible) trade-offs between multiple objectives. We call a set S I an (a, [)-approximmtee solution for an instance of SMFR if S(S) aOPT, and g(()) 2 /SPPT Vi E [d]. An (a, p))approxxii solution S is Pareto optimal if there does not exist an (a', -approxiiiee solution for any a' 2 and 8' 2 B and at least one is strictly larger. Ideally, by enumerating all distinct Parelo optimal solutions (which form the so-called Pareto fromtier), one can obtain all different optimal trade-offs between maximizing f and each J:- However, computing any Parcto optimal solution is still NP-hard. To circumvent the barricr, a feasible approach to SMFR is to find a set S of approximate solutions, in which, for any Pareto optimal solution, at least, one solution close to it is included. This the approach that we follow in our framework.
4 The SMFR-Saturate Framework
To find approximate solutions to an instance of SMFR, we propose to transform it into a series of instances of its corresponding decision problems, that is, to determine whether there exists any (a, -aaappooxiee solution for the SMFR instance. Then, we introduce the SATURATE framework first proposed in (Krause let al. 2008) to approximately solve each instance of the decision problem as Submodular Cover (SC), that is, the problem of finding set St with the minimum cardinality/con such that f(St) L for some L R+. Now, we formally define the decision problem and analyze why the transformation follows.
Definition 1 (SMFR-DEC). Given instance of SMFR and two approximation factors a, 3 [0, 1), find set S E I& such that fS) aDPT, and 9:(S) 2 30PT., for each i E [d], decide that there does not exist any set that can meet the conditions.
Assuming that O and each gPTg, are already known, the above conditions can be equivalently expressed I(8) 2 and 2.15) SOPT, 2 1. Then, using the truncation technique in (Krause et al. 2008), SMFR-DEC is converted to decide whether the objective value of the following problem d 1:
as oOPT,
max F (o, min {1, aOPT,
f(S)
Σ min (1, (((
(2)
SET
/0PT
Note that Fa.g is ill-formulated due to division by zero when a, B or OPT,, DPT,, are equal to 0. To solve this problem, the first term of Fa, is replaced by when a 0 or OPT, = O; the second term of Fa, is replaced by d when 3 = 0 or OPT 9; = 0 for any i i [d].
The above conversion holds because (5(5) d 1 if and only if f(S) > aDPT, and (:) > 30PT g. Vi E [d] In addition, Fa,s is a normalized, monotone, and submodular function because the minimum of a positive real number and a monotone submodular function monotone and submodular (Krause et al., 2008), and the nonnegative linear combination of monotone submodular functions is monotone and submodular KKause & Golovin 2014h. In this way, SMFR-DEG transformed to SC on Fa,g-
Since computing oPT, and OPT g. is NP-hard, we should use any existing algorithm for SMK (Siridenkoo
2004 Yaroslavtscy et al. 2020 Tang et al.
2021
Feldman et al. 2022 Li et al. 2022) or SMM Fisher
el al.
1978 Vondrak 2008 Calinescu et al.
2011
Badanidiyuru & Vondrák] 2013 Filmus & Ward 2014
Buchbinder et al. 2019) to compute their approximations. Suppase that we run an approximation algorithm for SMK or SMM to obtain OPT', < OPT, and OPT 9. OPT. Vi E [d] accordingly. The problem in Eq. 2 is relaxed as follows:
max (s. == min (1, SET
f(S) OOPT, } Σ min {1,
d
(())) }
(3)
/OPT
5

---

end Published in Transactions on Machine Learning Research (06/2024)
Algorithm 1: SMFR-SATURATE
Input: Normalized, monotone, and submodular set functions f, 91...... Ia' 2V + R Cost function e: V + R+ and budget E R+ (for knapsack constraint) or Collection of feasible sets I(M) C 2 and rank r Z (for matroid constraint); Error parameter E E (0,
Result: A set of approximate solutions to SMFR
Initialize S 00; 0;
Run algorithm for SMK or SMM to maximize f. f.g1...., Id subject to the constraint In I(M) to
compute OPT OPT g1t.... OPT
92i
for + 0: 3 S 8 8 + 2 do
Initialize Omar t- I Omin + 0:
while Omaz Omin _ do
Set 4- (amaz + Qmin )/2 and define Fa, (S) according to Eq. 3 + CostiffettivCredyyiigo
Id. c, k, E) (for knapsack constraint) or IterativeGreedy(J. ........... I(M), E) (for matroid constraint);
if F (S) 2 d
then
amin + a and S < S:
else
Omas f o;;
end
Add Samincs to S and remove all S with a' S amin and 8' < from S;
end
return S;
Function CostEffectiveGreedy (f.91..... gd, C, ki E):
Initialize <-0 0;
while 3r E S such that (w)) < k(1 + In 2d+2 do
+ E dSu (o}) < k(1 + 2d+2 }}
8* + arg max (SU - Fa (S) /c(u) and S Su
end return S;
Function IterativeGreedy (f. Dd. I(M), e): d+1 1211: 1 1++++1 do
for 1
flog2
S 0
while EVV E do
I (uev e
8" + arg max F a. (U S (v})
Fx Fu and S + S u {b*};
end
end
return -- - +
d+1 Toga
Sj
where the problem of division by zero is solved in the same way as for Fa, when a, B OPT', OPT equal to O. Next, the following lemmas indicate that SMFR-DEC can still be answered approximately by solving the relaxed problem in Eq. 3.
9; are
Lemma 1.
Any set S E I with Fr. (S) d+1-1111 must be a (δa - 5,3 - apppromiimlee solution to SMER, where d E (0, 1/e] is the approzimation factor of the algorithm used for SMK or SMM.
Lemma 2. If there is set S E I with Ft. (S) d 1, (a, -approoiie solution to SMFR erists.
See Appendices A.1 and A.2 for the proofs of the above two lemmas.

---

Published in Transactions on Machine Learning Research (06/2024)
Based on Lemmas 1 and 2 we propose SMFR-SATURATE in Algorithm I for SMFR. Generally, SMFR- SATURATE follows the same framework to handle the knapsack and matroid constraints but uses different greedy algorithms to obtain approximate solutions to SC on F We first run an algorithm for SMK or SMM on each objective function individually with the same knapsack constraint I or matroid constraint I(M) to calculate OPT,, OPT . OPT "s" Then, we iterale over each value of / from to I with an interval of 5- For cach value of B, we perform a bisection search on a between 0 and 1. Given a pair of a and 8, we formulate an instance of SC on F in Eq. 3
To address SC on Fn constraints, respectively. For a knapsack constraint It, we run the weweefffettieereeeeee algorithm, which starts from S = 0 and adds the most "cost-effective" item v" with the largest. ratio between its marginal gain w.r.t. S and its cost c(u") until no more items can be added with a relaxed knapsack constraint with a budget I=11 + In 2dt2), to find the candidate solution S. For a matroid constraint I(M), we run the IterativeGreedy algorithm, which performs the classic greedy algorithm for SMM (Fisher et al., 1978) iteratively in 1+ floge 1 rounds. In the 1-th round, we start from S = 0 and add the item 8* that satisfies [ E I(M and has the largest marginal gain w.r.t. U until more items can be added to St under the knapsack constraint Z(M). Finally, we return the union of the items selected over all rounds, i.c.,
adopt two different types of greedy algorithms specific to the knapsack and matroid
++ [+[log [+1]
St, as the candidate solution S.
After computing candidate solution S, if F", (S) 2 d d+1-1 1 5. that is, S reaches the "saturation level" w.r.t. a, B according to Lemma | we set S as the current solution Sa,3 and search in the upper half for a better solution with a higher value of a: otherwise, we search in the lower half for a feasible solution. When Omazz - Omin $ 5. we add the solution Sa..in..9 to S, remove all solutions dominated by Samin..8 and move on to the next value of 3. Finally, all non-dominated solutions in 8 are returned for SMFR.
The theoretical guarantees of EMFR-SATURATE for SMFR with knapsack and matroid constraints are analyzed in the following two theorems, respectively.
Theorem 1. For SMFR with a knapsack constraint I,,
runs in O(d((A) + -i log t) lime, where ((A) is the lime complexity of the 5-approzination algorithm for SMK, and provides a set S of solutions with the following properties: (1) [S| = O(E), (2) c(S) = Okl () for each S S, (3) for each (a",3")-app aapprrrnziee Pareto optimal solution S* to SMFR, there must cxist its corresponding solution S E S such that S(S) 2 (da" - s)OPT and (5) 2 (ōß* - e)OPT ge Vi E [d].
Theorem 2. For SMFR with a matroid constraint I(M), SMFR-SATURATE runs in O(d((A) + or log² dy time, where t(A) is the time complezity of the 8 6-approzimation algorithm for SMM, and procides a set S of solutions with the following properties: (1) |S| = O(2), (2) |S| Or = for each S E S, (3) for each (a", ,,") appprozmmate Pareto optimal solution S* to SMFR, there must ewist its corresponding solution S = S such that f(S) 2 (ōa" e)OPT, and g(( 2 (63* e)OPT vi [d].
See Appendices
A.3 and A.4 for the proofs of the above two theorems.
5 Experiments
In this section, we present extensive experimental results to evaluate the performance of our proposed algo- rithm (SMFR-SATURATE) on two benchmark problems, namely Marimum Coverage and Recommendation, using several real-world data sets. We compare SMFR-SATURATE with the following non-trivial basclines.
GREED++MAX (or GREEDY): The original greedy algorithms for single-objective submodular max- imization. For SMK, we adopt the O(n?)-time GREEDY+MAX algorithm by Yaroslavtsev et al. (2020); and for SMM, we adopt the O(ar)-time GREEDY algorithm by Fisher et al. (1978). Both algorithms have the same approximation factor of 1/2.
SATURATE:
The bicriteria. approximation algorithms for the problem of multi-objective submodu- lar mazinization (MOSM) that maximizes the minimum among multiple (submodular) objective functions. As for SMFR, we should maximize the minimum among the d + functions of f and
7

---

Published in Transactions on Machine Learning Research (06/2024)
91..... Id. In particular, SATURATE for MOSM with knapsack and matroid constraints is presented
in (Krause et al. 2008 and (Anari et al. 2019 respectively.
SMSC: A (0.16, -ppppoiiioaaaprroximaiiooo algorithm for the problem of Submodular Maximization under Submodular Cover (SMSC) (Ohsaka & MMtsuoka 2021), which can be used for SMFR only when d = 1 by maximizing f under the submodular cover constraint defined on 91-
BSM-SATURATE: The instance-dependent bicriteria approximation algorithm for balancing utility (i.c., maximizing f) and fairness (i.c., maximizing the minimum of 91.....9d in (Wang et al. 2024)
OPT: Formulating an instance of SMFR as an integer-linear program (TI.P) and using a solver to enumerate its Parcto optimal solutions in the worst-case exponential time. The ILP formulations of SMPR for Marinum Coverage and Recommendatton are deferred to Appendix B.
All algorithms are appropriately adapted to provide solutions without violating the specified constraints. We implemented them in Python 3, and for the OPT algorithm, we applied the Gurob optimizer to solve the ILP formulations of the Mazimum Coverage and Recommendation instances. All algorithms except OPT were accelerated using the lazy-forward strategy (Leskovec et al. 2007), as this strategy cannot be applied to OPT. All experiments were run on MacBook Pro laptop with an Apple M1 Max processor and 32GB memory running MacOS 14. Data and code are available publicly at httts: com aamm//drinnaaz az/ haar-pppprertaattooiinsssttaaaap SSstuuuuuurbubaer-uuuueeerrrrrrr
Approachh,
5.1 Maximum Coverage
Setup. In this subsection, we evaluate the performance of different algorithms for SMFR on the Mazimum Coverage problem using two real-world data sets: Facebook and DBLP. The Facebook data set (Mislove ct al. 2010) is an undirected graph of 1, 216 nodes and 42, 443 edges representing the friendships between Rice University students on Facebook, and the DBLP data set (Dong et al. 2023) is an undirected graph of 9,980 nodes and 6, 966 edges denoting the coauthorships between researchers.
Our sottings for Mazimum Coverage follow those used in the existing literature on submodular maximization
HHllab et al. 2020 Ohsaka & Matsuoka 2021 Wang et al. 2024). Given a graph g = (V, E), the utility (i..., coverage) function is defined as f(S)= == U M(v),, where (b) is the set of nodes consisting of v and its neighbors in G. That is, the coverage of a set C V is mcasured by the number of nodes in the union of the neighborhoods of all nodes in S. To define the representativeness functions 91 .2..... Id. we divide the node set into d communities C1,.... Ca such that U C. V. For each E [d], the function gi is associated with a particular community C, as (i( := = (UWessN(e)nCC a community C; is mcasured by the number of nodes in C, covered by S. For both data sets, the node set V is partitioned into four disjoint groups using the Louvain method (Blondel el al. 2008) for communily detection. We then index the four communities according to their sizes as |C1 2 |C2 2 Ca 2 |Cal.. For the DBLP data set, we follow the scheme of (Jin et al. 2021) to define a knapsack constraint by assigning a cost of 0.2 times its degree to each node and then normalizing all costs by the minimum cost. For the Facebook data set, we define a partition matroid constraint by dividing all nodes into 4 disjoint groups based on a sensitive attribute (i.c., age). We then follow the rule of equal representation (Halabi ct al., 2020 to set the same upper bound & E - for each age group, resulting in a partition matroid of rank r 4k.
That is, the representativeness of a set S for
Results. Figures Ia Ic and 2a 2c present the trade-offi between a and 3 achieved by each algorithm for different instances of SMFR on Marimum Coverage with knapsack and matroid constraints on the DBLP and Facebook data sets, respectively. We fix k: = 40 for the knapsack constraint and I: = (and thus r = 20) for the matroid constraint. We set d = 1,2, and 4 by considering the representativeness functions on the first group C₁, the first two groups C and C22 and all four groups from C, to C4. In cach of these figures, the and y-axes represent the values of and B for all solutions with distinct marker for each algorithm. Furthermore, we also use black line and a red line to denote the optimal Pareto frontier returned by OPT and its approximation returned by SMFR-SATURATE. From the results, we observe that the Pareto frontiers provided by SMFR-SATURRN are equal or very close to the optimal ones. This confirms the offectiveness of
htttp://www.guraai..cc///outttts
8

---

Published in Transactions on Machine Learning Research (06/2024)
Greedy+Max
Saturate
BSM-Saturate
SMSC
OPT
SMFR-Saturate
1.0-
1.0
2.0 0
0.5
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.0
0.6
0.6
0.5
0.5
0.5
+++
0.00
0.20
0.60
1.00
0.30
0.00
0.05
0.10
0.25
gost
gaur
(a) DBLP (d 1, 40)
(b) DBLP (d = 2, = 40)
(c) DBLP (d = 4, 40)
200
140
175
175
120
150-
150
100
125
125
1000
S 100
25
25
20
100
(d) DBLP (d= 1, 2 0.8)
(e) DBLP (d = 2,3 2 0.4)
(f) DBLP (d 4, 2 0.2)
Figure 1: Results for Maximaun Coverage on the DBLP data set, with knapsack constraints.
SATURAT for the SMFR problem. We also find that the GREEDY+MAX and GREEDY algorithms, which focus solely on maximizing f, generally provide solutions with low values of B, indicating significant neglect of representativeness functions. Furthermore, SATURATE, which maximizes the minimum among all representativeness and utility functions and does not allow any trade-off between f and g by design, in some cases (e.g., Figures la Icb, it provides a solution with the highest value of 3 while having a value of a equal to close to that of SMFR-SATURATE and OPT for maximum B. However, it returns inferior solutions dominated by those of SMFR-SSTURATT in other cases. BSM-SATURATE and SMSC provide different trade-offs between f and g by adjusting the threshold value t in their definitions. The trade-offs reported by SMSC are marginally better than those of ESTURATT on the Facebook data set with matroid constraints (Figure 2ap. Conversely, it performs poorly for knapsack constraints (Figure Ia). In fact,, SMSC a special case of SMFR when d = 1, the matroid/knapsace constraint reduced to the cardinality constraint, and the trade-off between f and g is predetermined by т. It is also noted that SMSC cannot work when d >> 1. Although BSM-SATURATE does not have the restriction of d = 1, its trade-offs are never better than those obtained by on the DBLP data set with knapsack constraints (Figures Ta Tch.
and significantly worse for Marimum Coverage
Figures Id and 2d report the effect of the parameter k, which directly decides the solution size, on the performance of each algorithm for different instances of SMFR in the context of Maximum Coverage with knapsack and matroid constraints on the DBLP and Facebook data sets, respectively. In each plot, the x- axis represents the value of in the knapsack or matroid constraint, and the y-axis represents the maximum utility value f(S) among all solutions with certain level of representativeness, i.e., the value of 3 reaches a given threshold, provided by algorithm. We also set d 1,2, and 4 by considering the representativenes. functions on C1, C,&C2, and C1-C4. Only solutions with / 2 0.8 are considered for d = 1, / 2 0.4 for d = 2, and 3 > 0.2 for d = 4. A unique marker and a distinct line color are used for each algorithm. From Figures [ we observe that the solutions provided by SMFR-SATURATE consistently achieve the highest utility value f(5) across all values of in the knapsack constraint. The absence of SMSC and BSM-SATURATE indicates that they fail to provide solutions with an adequate level of representativeness (i.e., the value of B is below the given thresholds), with the only exception shown in Figure Te when k = 100.
9

---

Published in Transactions on Machine Learning Research (06/2024)
Greedy
Saturate
BSM-Saturate
SMSC
OPT
SMFR-Saturate
1.0
0.95
0.95
0.9
0.90
0.90
0.8
0.05
0.00
0.00
0.7
0.75
0.75
0.70
0.6
0.70
0.05
0.60
0.75
0.80
0.85
0.90
0.95
1.00
0.75
0.00
0.55
0.90
0.96
1.00
0.00
0.60
0.65
0.70
0.75
0.50
0.85 0.00 0.05 1.00
gest
gait
gest
(a) Facebook (d k 5)
(b) Facebook (d 2,
(c) Facebook (d 1, 5)
1200
1200
1200
11000
11.00
1100
1000
800
800
700
700
20
30
40
50
10
20
30
10
20
30
50
(d) Facebook (d= 0.8)
(e) Facebook (d 2, 2 0.4)
(f) Facebook (d = 4, 2 0.2)
Figure: 2: Results for Muximmum Coverage on the Facebook data set, with matroid constraints.
Furthermore, although SATURATE provides valid solutions in all cases, the gap in the utility value f(S) between SMFRRSATURATE and SATURATE widens as the knapsack restriction becomes less stringent (i.e., increasing k), for all values of the number of representativeness functions d. Figures 2d 2 show that across all values of k, the solutions provided by SMFR-SATCRATE always achieve utility values f(S) higher than those of BSM-SATURATE and SATURATE. Unlike the case of knapsack constraints, the gap in the utility value S(S) among all methods decreases as the matroid constraiit becomes less stringent (i.e. increasing k), for all values of the number of representativeness functions d. In the case of d = 1, SMSC and SMFR- SATURATE exhibit the same performance, as shown in Figure 2d The above results confirm that when the trade-off level between and g is pre-specified, one can still find a corresponding solution from those of EMFR-SATURATE that is comparable to or better than those provided by other basclines.
5.2
Recommendation
Setup. In this subsection, evaluate the performance of different algorithms for SMPR on the Recommen- dation problem using another two real-wool world data sets: X- Wines (de Azambuja et al. 2023) and MovieLeng The X- Wines data set consists of 150 000 ratings from 10 561 users on 1 107 wines, where each rating takes a value in the range [1.0, 1.5.... ... 5.0]. Moreover, each wine in the data set is associated with one more food types that pair with the wine itself; we group these food types into four categories: "meat", ", "fish", "pasla" " and "cheese". The MovieLens data set consists of 100 000 ratings from 600 users on 90000 movies, where each rating takes value in the range [0.5, 1.O....,5.00 Each movie in the data set is associatod with one or more genres, with lotal of 20 genres.
Our experimental settings similar to those adopted in (Osaaa & Matsuoka 2021). In the following, we use the term "item" to refer to either a wine in the X- Wines data set or a movie in the MovieLens data set. By performing the nen-tegative matrix factorization (NMF) on the user-item rating matrix with 32 factors, we obtain a 32-dimensional feature vector for each item and user. Denoting by v, E R the feature vector of
httpe://preupllsssoor//sssssess http:
10

---

Published in Transactions on Machine Learning Research (06/2024)
Greedy+Max
Saturate
BSM-Saturate
SMSC
OPT
SMFR-Saturate
1.0
1.0
1.0
0.9
0.9
0.8
0.7
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.4
0.20
0.50
0.40 0.50 0.60
0.70 0.00 0.90
1.00
0.20
0.30
0:40
0.50
0.70 0.80
0.90
0.10
0.20
0.30
0.40
0.50
0.60
gas
gou
(a) X- Wines (d 1, k- 50)
(b) X- Wines (d = 2, 50)
(c) X- Wines (d = 4, 50)
1600
2000
1400
1750
1700
1500
1500
1000
(5)
800
1000
600
77.
400
500
500
V7.
0-
20
40
60
80
100
20
40
60
80
100
20
40
60
80
100
k
k
K
(d) X- Wines (d = 1,82 0.8)
(e) X- Wines (d 2, 2 0.4)
(f) X- Wines (d = 4,8 2 0.2)
Figure 3: Results for Recommendattion on the X- Wines data set, with knapsack constraints.
item i, and by uy E Iº the feature vector of user j, the inner product (v,, vz) between two feature vectors associated with two items measures their similarity. The same holds for users and items as well: (V;, Uy) indicates the level at which a user likes an item. To design the utility function f according to the facility location objective, we select subset T of items with least 54 ratings ([T) = 503 for the X- Wines data set, and [T* 403 for the MovieLens data set), and define f 2 + RR as f(S) = Eter max,es (v., v:). where V is the set of all items in each data set: [V] 107 for the X- W-Wnses data set, and [V] = 9 000 for the MowieLens data set. The function f captures how well the selected subset S can represent all items in T in the sense that for any item E T, there exists an item in S that is highly similar to it. This function, as defined, is known to be monotone and submodular Frieze 1974). To define the representativeness functions 91, 12.92....: Id, we consider using, for the X- Wines data set, the food type categories with which a wine pair, and, for the MovieLens data set, the genres to which a movie belongs. Specifically, for the X- Wines data set, we divide wines into four groups according to their associated food type categories as G1 (meat), G2 (fish), G3 (pasta), and G4 (cheese). Similarly, for the MovieLens data sct, we divide movics into four groups according to their genres G1 (dramas), G2 (comedies), G3 (thrillers), and G. (action movies). Then, each 9: function is associated with particular set of items and is defined as (:( == |SS To be specific, the representativeness of S for G; is measured by the number of items in S selected from G,. For the X- Wines data set, we define a knapsack constraint by assigning to each item (wine) a random integer cost in the range [1, 10). For the MovieLens data set, to define matroid constraint, we partition the movies into groups according to their release dates: [1900, 1950), [1950, 1970), [1970, 1980), [1980, 1990), [1990, 2000). (2000,2010), and [2010, 2019). We also use an equal upper bound k z+ for each group, resulting in a partition matroid of rank 7k.
Results. Figures 3 and A present the performance of each algorithm for different instances of SMFR on Recommendation with knapsack and matroid constraints on the X-Wines and MovieLens data sets, respectively. In general, we observe results similar to those for Mazimum Coverage and further confirm the effectiveness of SMFR-SATURATE for SMFR in different applications. The absence of OPT Figures 4a- is due to the inefficiency of the ILP solver: cannot finish on any SMFR instance for the MovieLenss data set within one hour. We also find that SMFR-SATURATE shows more significant advantages over SMSC
11

---

Published in Transactions on Machine Learning Research (06/2024)
Greedy
Saturate
+
BSM-Saturate
SMSC
OPT
SMFR-Saturate
1.0
1.00
0.9
0.98
0.96
+ H++1 1pp
0.04
0.92
0.0
0.5
0.88
0.86
0.40
0.50
0.60
0.70 gaut
0.20
0.90
1.00
0.40
0.50
0.50
0.70
0.30
0.90
1.00
0.20
0.30
0.40
0.50
0.60
gese
(a) MovicLens (d 1, 25)
(b) MovieLens (d 25)
(c) MovicLens (d 1, = 25)
1.0
0.9
y V...7...
0.9
0.8
5 0.7
0
0.6
0.7
0.5
0.5
0.4
.7..
0.4
0.31
10
20
40
50
10
20
30
40
50
10
20
30
40
50
k
(d) MovieLens (d 1,8 2 0.8)
(e) MovicLLns (d= 2, 0.4)
(I) MovieLens (d= 4, 0.2)
Figure 4: Results for Recommendation on the MovieLens data set, with matroid constraints.
and BSM-SATURATE for the knapsack constraints than for the matroid constraints. In particular, SMSC slightly outperforms SMFR-SATURATE when = 1 on the MovieLens data set, with matroid constraints. This is because the solutions with cardinality constraints are typically very close to those with the partition matroid constraints that we define but differ significantly from those with knapsack constraints. As such, SMSC, which is designed specifically for cardinality constraints, achieve good performance under matroid constraints without adaptations. Again, SMSC is not comparable to SMFR-SATURATE in other cases.
Finally, we omit the remaining experimental results due to space limitations. Please refer to Appendix for those results, which further confirm the effectiveness of SMFR-SATURATE other experimental settings and provide additional evaluations for the efficiency of SMFR-SATURATE and other basclines.
6 Conclusion and Future Work
In this paper, study novel multi-objective combinatorial optimization problem called Submodular Mar- imization with Fair Representation (SMFR), which aims to select subsets from a ground set under specific knapsack or matroid constraint such that a submodular (utility) function f is maximized while d submod- ular (representativeness) functions 91.... :9 are also maximized. We show the hardness of finding optimal solutions SMFR and propose a Pareto optimization approach, SMFR-SATURATR, to enumerating a set of approximate solutions to all Parcto optimal solutions with different tradc-offs between multiple objectives for SMFR. Finally, we demonstrate the effectiveness of SMFR-SATURATE in two classic submodular problems, Mazimum Coverage and Recommendation, using real-world data.
We note that SMFR-SATURATE still has several limitations. For example, it cannot support more general classes of functions in subset. selection problems, such as non-monolone and weakly submodular functions, and more complex constraints, including the intersection of multiple knapsack and matroid constraints and the P-system constraint. We would like to extend SMFR-SATURATE. to support them in future work. Furthermore, it would also be interesting to expand the realm of fair submodular optimization (Halabi al. 2023 Mehrotra & Vishnoi 2023) by considering more novel and practical notions of fairness.
12

---

Published in Transactions on Machine Learning Research (06/2024)
Acknowledgments
Yanhao Wang was supported by the National Natural Science Foundation of China under grant number 62202169 and the start-up funding from ECNU.
References
Nima Anari, Nika Haghtalab, Selfi Naor, Sebastian Pokutta, Mohit Singh, and Alfredo Torrico. Structured robust submodular maximization: Offline and online algorithms. In Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics (AISTATS), pp. 1128-3137, PMLR, 2019.
URL http:///proceedingss mlr www....eesse////e hatasss/vv//aaa aat///aaarrii9 html
Çigdem Aslay, Wei Lu, Francesco Bonchi, Amit Goyal, and Laks V. S. LLkshmanan. Viral marketing meets social advertising: Ad allocation with minimum regret. Proc. VLDB Endow., 8(7):822-88) 833,, 2015. URL
http:://ww.vlone www.:w/llbbc..... http://wwww./.o.oo//////////---- psaa..y..
Çigdem Aslay, Francesco Bonchi, Laks V. S. Takshmanan, and Wei Lu. Revenue maximization in incentivized social advertising. Proc. VLDB Endow., 10(11):1238 1249, 2017. URL http://www. http:::/wwwwwwldb...
vo110/p1238-aslee
Ashwinkumar Badanidiyuru and Jan Vondrák. Fast algorithms for seneimmiiingsubbddiii functions. In Pro- ceedingg of the 2014 Annvual ACM-STA M Symposium on Discrele Algorithms (SODA), pp. 1497-1514. So- ciety for Industrial and Applied Mathematics, 2013. htttpp::/bbda.rrg//01133//// [10
Wei-Xuan Bao, Jun-Yi Hang, and Min-Ling Zhang. Submodular feature selection for partial label learning. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD 122,, pp. 26 34. Association for Computing Machinery, 2022. URL 3539292
1145/3536788
Ruben Becker, Federico Corò, Gianlorenzo D' Angelo, and Hugo Gilbert. Balancing spreads of influence a social network. Proceedings of the AAAI Conference on Artificial Intelligence, 34(00):3--0, 2020. URL
https://doi.. (aal/oii.orggg// wtg/10../000. 1tte://..109//aaa. 1609 aaai v34101.5327
Vincent D. Blondel, Jcan-Loup Guillaume, Renand Lambiotte, and Etienne Lefebvre. Fast unfolding of communities in large networks. J. Stat. Mech.: Theory Exp., 2000(10);P10008, 2008. URL https://dx:
http://00..000//7225566660000000 doi. 0oi.org/10.1088/ 1742 1722-177822000//////// 10100000
Niv Buchbinder, Moran Feldman, and Mohit Garg. Deterministic (1/2 + #) approxxxmatoo for submodular maximization over a matroid. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete
Algorithms (SODA), pp. 241-254. SIAM, 2019. URL https:://ooi wtt1://0.0011111111..1111111111.
Gruia Cälinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing monotone submodular function subject to matroid constraint. STAM J. Compul., 40(6): 2011. httpss///:///d
111/100.137/080733889
Rogério Xavier de Azambuja, A. Jorge Morais, and Vitor Filipe. X-Wines: A wine dataset for recommender systems and machine learning. Big Data Cogn. Comput., 7(1):20, 2023. (RR.
bdcc7010020
Yushun Dong, Jing Ma, Song Wang, Chen Chen, and Jundong Li. Fairness in graph mining: A survey. IEEE Trans. Knowl. Data Eng., 35(11):10583-100 10602, 2023. URL http::t///oi..rrg////// 11t.//1099//.222. 2023. 3265598
Alina Ene and Huy L. Nguyen. A nearly- linear time algorithm for submodular maximization with knapsack constraint. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). pp. 53:1-53:12, Dagstuhl, Germany, 2019a. Schloss Dagstuhl- Ceestuhl-Leibiiz--e http:s///bb..rr///14422////
ZentrizZeentrum Zentrum fuer Informatik. URL
ICALP 2019..0
13

---

Published in Transactions on Machine Learning Research (06/2024)
Alina Ene and Huy L. Nguyen. Towards nearly-linear time algorithms for submodular maximization with matroid constraint. Tn 46th International Colloquium Aulomala, Languages, and Programming (ICALP 2019), pp. 54:1 54:14, Dagstuhl, Germany, 2019b. Schloss Dagstuhl Leibniz-ZZntrum fuer Informatik. URL attps://doi.orgg:
4230/LIPICS. ICALP. 2019.54
Uriel Feige. A threshold of In for approximating set cover. J. ACM, 45(4):63446522 1998. URL https:
7/doi.orge 10l.11115228555228800 285559
Moran Feldman, Zeev Nutov, and Elad Shoham. Practical sudeotou maniiai maximization. Algorithmica, 55(5):1332-1377) 2022. URL httppp///bllloor//00000000000000
Chao Feng and Chao Qian. Multi-objective submodular maximization by regrel ratio minimization with theoretical guarantee. Proceedings of the AAAI Conference on Artificial Intelligence, 35(14):12302 -22310, 2021. URL httpp:://0ol..rg/10..1
Yuval Filmus and Justin Ward. Monotone submodular maximization over matroid via non-oblivions local search. SIAM J. Compul., 43(2):514 542, 2014. URL http::///a..oorg//0111//33// 1117//3990117
Marshall L. Fisher, George L. Nemhauser, and Laurence A. Wolsey. An analysis of approximations for maximizing submodular set functions-II. III. In Michel L. Balinski and Alan J. Hoffman (eds.), Polyhedral Combinatorics: Dedicated to the memory of D.R. Fulkerson, pp. 73-87. Springer, Berlin, Heidelberg, 1978. URL https://doi. http:////00i..rrr1101000////////
Alan M. Frieze. A cost function property for plant location problems. Math. Program.. 7(1):245-245, 1974.
URL httt::::/eel..orr/11.000///////
Marwa Halabi, Slobodan Mitrovic, Ashkan Norouzi-Fard, Jakab Tardos, and Jakub Tarnawski. Fair- ness in streaming submodular maximization: Algorithms and hardness. Advances in Neural Information Processing Systems, 33:13609 13622, 2020. URL httpppppe///rrcociigs.eeeeer wwwp:ooeedddngg..nrrp......o. wwwpurips.ccpppper//eeeeec haahh//
Marwa El Halabi, Federico Fusco, Ashkan Norouzi-Farde Jakab Tardos, and Jakub Tarnawski. Fairness in strcaming submodular maximization over a matroid constraint. In Proceedings of the 40th International Conference on Machine Learning, pp. 0150-9171. PMLR, 2023. URL URL http::/pproocciingg. mlr press/ html
Tianyuan Jin, Yu Yang, Renchi Yang, Jieming Shi, Keke Huang, and Xiaokui Xiao. Unconstrained submod- ular maximization with modular costs: Tight approximation and application to profit maximization. Proc.
VLDB Endow., (1106:1176611788 2021. URL htttp://www..r vldb. orgo wwtpooggggll///../ org/pp/lbb/ro114///756 jin.-jiinn......
David Kempe, Jon Kleinberg, and Éva Tardos. Maximizing the spread of influence through social network. Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD '03), pp. 137-146. Association for Computing Machinery, 2003. URL https:: /ttp::://dooorrr org/
10. 0ttp///57500.966666.
Samir Khuller, Anna Moss, and Joseph (Seffi) Naor. The budgeted mwximumuiiiicacaaoaaaaaaaa problem. Inf. Process. Lett., httpp://oa.oog/00000000001111111 /0T
1999. URL [Bt/:///g////oo.oo/////////0//00
Andreas Krause and Daniel Golovin. Submodular function maximization. In Tractability: Practical Ap- proaches to Hard Problems, pp. 71 104. Cambridge University Press, Cambridge, UK, 2014. URL
org
Andreas Krause and Carlos Guestrin. A note on the budgeted maximization of submodular functions. Technical Report CMU-CALD--5--03,, Carnegie Mellon University, 2005. URL https:///ass. inf ethz. ch//filee//rraueeSSotte,, .dff]
Andreas Krause, H. Brendan McMahan, Carlos Guestrin, and Anupam Gupta. Robust submodular ob- servation selection. J. Mach. Learn. Res., 9033):2761-2881 2008. URL https://jmlr. www.a//rrrr/appr///...
krause08b.
14

---

Published in Transactions on Machine Learning Research (06/2024)
Ariel Kulik, Roy Schwartz, and Hadas Shachnai. A refined analysis of submodular greedy. Oper. Res. Lett., 49(4)5507-514 2021. URL
10t.1/1.1roll.... .rll. 01211.4.000
Jure Leskovee, Andreas Krause, Carlos Guestrin, Christos Faloutsos, Jeanne VanBriesen, and Natalie Glance. Cost-effective outbreak detection in networks. In Proceedings of the 13th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD '07), pp. 420-424 Association for Computing
Machinery, 2007. URL http:::/doo..orgg/
Wenxin Li, Moran Feldman, Ehsan Kazemi, and Amin Karbasi. Submodular maximiza- tion in clean linear time. Advances in Newral Information Processing Systems, 35:17473 17487,
2022.
URL https://proceeddingg neurips..... wwt..p...cpppppprrrrr.o...... http:/psscpppppeeeeeeeeeeeeeeeee fttpss//pppeee//222/// haahh/
html
Hui Lin and Jeff Bilmes. Multi-document summarization via budgeted maximization of submodular func- tions. In Human Language Technologies: The 2010 Annual Conference of the North American Chapter of the Association for Computational Linguisties, pp. 912-920. Association for Computational Linguistics, 2010. URL aattaa://cllanhooooooooooo//////
Yuzong Liu, Kai Wei, Katrin Kirchholf, Yisong Song, and Jeff A. Bilmes. Submodular feature selection for high-dimensional acoustic score spaces. In IEEE International Conference on Acoustics, Speech and 1109 (.CASSS..011.
Signal Processing (ICASSP), pp. 7184 7188. IEEE, 2013. URL http::://oo....... 6639057
Anay Mehrotra and Nisheeth K. Vishnoi. Maximizing submodular functions for recommendation in the pres- ence of biases. In Proceedings of the ACM Web Conference 2023 (WWW '23), pp. 3625 2633-663.. Association for Computing Machinery, 2023. URL
00155/5543507..25 35831955
Baharan Mirzasoleiman, Ashwinkumar Badanidiyuru, and Amin Karbasi. Fast constrained submodular maximization: Personalized data summarization. In Proceedings of The 33rd International Conference Machine Learning (ICML), pp. 1358-1366 PMLR, 2016. URL http:///proceedings mlr www.mmmww.......eemmmmmmm pwwps//ss////////
Aarzasoleiman66..m htmll
Alan Mislove, Bimal Viswanath, Krishna P. Gummadi, and Peter Druschel. You are who you know: Inferring user profiles in online social networks. In Proceedings of the Third ACM International Conference on Web Search and Data Mining ( (WSDM '10), pp. 251 260. Association for Computing Machinery, 2010. URI
https://doi.org 111.1..1111111..11111 1718519
George L. Nemhauser, Laurence A. Wolsey, and Marshall L. Fisher. An analysis of approximations for maximizing submodular set functions I. Math. Program., 14:265-294 1978. URL https://doi /ttt::://doooor/100..
1007/BF01588971
Naoto Ohsaka and Tatsuya Matsuoka. Approximation algorithm for submodular maximization under sub- modular cover. In Proceedings of the Thirty- Conference on Uncertainty in Artificial Intelligence (UAT), pp. 792-811 801. PMLR, 2021. URL httpss//pproceedingss mlm. wtt.:mmw...........mmmmmmmm mwt..ppres//ul.//osaaaaa...
Chao Qian, Yang Yu, and Zhi-Hua Zhou. Subset selection by pareto optimization. Advances in Neural Information Processing Systems, 28:1774 1782, 2015. URL httpss//pproceeiings www.ips.curpaaaapepeee
Chao Qian, Jing-Chenn ChngCChegg&a Shi, Yang Yu, Ke Tang, and Zui-Hua Zhou. Subset selection under noise. Advances in Neurol Information Processing Systems, 1043560-3570 2017. URL https://proceeddng.s hew.:oooeddnng....uu...... neurips.
Chao Qian, Chao Bian, and Chao Feng. Subset selection by pareto optimization with recombination.
Proceedings of the AAAI Conference on Artificial Intelligence, 34(03):2408-22) 2415, 2020. URL https: 0//oi.orgg/10.160//aaaaa v34103.5621
15

---

Published in Transactions on Machine Learning Research (06/2024)
Vahid Roostapour, Aneta Neumann, Frank Neumann, and Tobias Friedrich. Pareto optimization for subset selection with dynamic cost constraints. Artij. Intell, 302:103597 2022. URI. https://doi.org org/10....
1016/j.artint.222 2021 203111
Tasuku Soma and Yuichi Yoshida. Regret ratio minimization in multi-objective submodular function max- imization. Proceedings of the AAAI Conference Artificial Intelligence, 31(1):905-911, 2017. URL
https: /etps://doi.orgge 1tte:/10.100///// 1609/ v3111. 10652
Maxim Sviridenko. note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lell., 32(1):41-433 43, 2004. URL http://tpp////...ooo///////////0
Jing Tang, Xueyan Tang, Andrew Lim, Kai Han, Chongshou Li, and Junsong Yuan. Revisiting modified greedy algorithm for monotone submodular maximization with a knapsack constraint. Proc. ACM Meas. Anal. Comput. Syst., 5(1):08:1-08:22, 2021. URL btttps://dd..orgg//// 1145/3447386
Shaojie Tang. When social advertising meets viral marketing: Sequencing social advertisements for influence maximization. Proceedings of the AAAI Conference on Artificial Intelligence, 32(1):176 183, 2018. URL
https://doi. org/ 10. 1609/aaai. aaai v3211. 11306
Shaojie Tang and Jing Yuan. Beyond submodularity: a unified framework of randomized set selection with group fairness constraints. J. Comb. Optim., 45(4):102, 2023. URL https://doi. httpp::///o..rg/10..110 1007/
610878-023-01036-4
Alfredo Torrico, Mohit Singh, Sebastian Pokutta, Nika Haghtalab, Joseph (Seffi) Naor, and Nima Anari. Structured robust submodular maximization: Offline and online algorithms. INFORMS J. Comput., 33 (4):1590-1607, 2021. URL https://doi..... www/o...//...... 1tt.111..11111o..o 2020. 0998
Alan Tsang, Bryan Wilder, Eric Rice, Milind Tambe, and Yair Zick. Group-firrness in influence maxi- mization. In Proceedings of the Twenny-FFight International Joint Conference Artāficial Intelligence, IJCAI-19, pp. 5997 6005. International Joint Conferences on Artificial Intelligence Organization, 2019. URL https:///oi.or
24963/1jcai 2019/831
Rajan Udwani. Multi-objoctive maximization of monotone submodular functions with cardinality constraint. Advances in Newral Information Processing Systems, 31:9513 9524, 2018. URL https://pproeeddngs.
Abstraatttta-stttttttcaccccac... html
Jan Vondrak. Optimal approximation for the submodular welfare problem in the value oracle model. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing (STOC '08), pp. 67-74. Association for Computing Machinery, 2008. URL https://aoi.orr/// 0tt.://///0111111//3777777663330
Yanhao Wang, Jiping Zhong, and Fanxu Meng. Improved algorithm for regret ratio minimization in multi- objective submodular maximization. Proceedings of the AAAI Conference on Artificial Intelligence, 37 (10):12500-1556) 2023. URL lhttps://doll http://....or///....// 1109/660aaaaee /aaii. v
Yanhao Wang, Yuchen Li, Francesco Bonchi, and Ying Wang. Balancing utility and fairness in submodular maximization. In Proceedings of the 27th International Conference on Extending Database Technology, EDBT 2024, Paestum, Italy, March 25 March 28, pp. 14. OpenProceedingg
2024. URL https:
//doi.org/11 org/10. 48786/edbt. 2024.01
Laurence A. Wolsey, An analysis of the greedy algorithm for the submodular set covering problem. Combi- natorica, 2(4):385-38) 1982. CRL https:///ot:: 100.///00000//822559440
Grigory Yaroslavtsev, Samson Zhou, and Dmitrii Avdiukhin. "Bring your own greedy"+ Near- optimal 1/2-approximations for submodular knapsack. In Proceedings of the Twenty Third Interna- tional Conference on Artificial Intelligence and Statistics (AISTATS), pp. 3263-3274. PMLR, 2020. URL
http://proocedinge.lk. mlr. htt.:///........ee///ooo/oaaaaaa
16

---

Published in Transactions on Machine Learning Research (06/2024)
A Proofs of Lemmas and Theorems
A.1 Proof of Lemma 1
Lemma Any set S E with ( 2 d+1- must be a (óa - 8, aapprozzima solution to SMFR, where ò (0, 1 - 1/e] the approcimation factor of the algorithm used for SMK or SMM.
Proof. We first consider the two special cases of a = 0 and B = 0. When a = or B 0, i' ( d+
we will have g.1
> _ for every i E [d] or
f(S)
> 5- In the general case of a, > 0, if
DOPT,
xbPT
F (S) d we will have ((S) oepr,
> 1 - and (S)
> 1 for every i E [d] at the same time. Thus,
POPT
it holds that
S(S) 2 (1
E JaOPT, > 50((1 (1- 1OP, (ōa - JOPT,
E
E
2
and
OPT vi [d]. Therefore, S is (ōa - 2. 83 >approximate solution to SMFR.
(())) 2 (1 /
SOPT g.
2 - JOPT 2 (63
A.2 Proof of Lemma
Lemma 2. If there is no set S E I with F², (() = d + 1, no (a, -approiiiie solution to SMFR cxists.
Proof. If Fr (S) < <d+1, then we will have f(5) < aOPT, S aOPT or there is some E [d] with gi(S) < (S) < d + 1, S will not be an (a, app-ppppooo solution to SMFR. And if there is no set S E I with F (S) + 1, no (a, appprroxiie solution to SMFR exists.
SOPT 9, I 3.PT. g. Therefore,
A.3 Proof of Theorem 1
Thcorem 1. For SMFR. with a knapsack constraint Ix, EMFR-SATURATE runs in O(d((A) + n² time, where ((A) is the time complexily of the 8-approzimation algorithm for SMK, and provides set S of solutions with the following properties: (1) |8| = O (2) c(S) = O(k log )) for each S E SS, (3) for each (a*,3*)- -approzimmate Pareto optimal solution S* to SMFR, there must exist its corresponding solution S E S such that f(S) 2 (ōa" - E)OPT and gi( 2 (ō5* - e)OPT g., Vi E [d].
= log 2)
Proof. Let us first analyze the time complexity of SMFR-SATURATE for a knapsack constraint Ig. First, it runs the SMK algorithm + 1 times to compule OPT, and OPT 9; for every E [d]. Then, iterales over [el values of 8 in the for loop. For each value of 3, it attempts to use O(log 2) different values of a in the bisection search. Finally, the subroutine CostEffectiveCreedy takes O(n²) time for SC on each Fl.g- In summary, the time complexity of SMFR-SATURATE for a knapsack constraint In is O(d((A) + n² log !) time, where ((A) is the time complexity of the SMK algorithm.
For the solution S of SMFR-SATURATE, it is easy see that |S| S 1 and thus |S = O because SMFR- SATURATE adds at most one set to S for each value of B. Then, due to the condition in the while loop of the subroutine CostEffectiveGreste
it must hold that c(S) S k(1 + In 2d+2) and thus c(S) O log +) for Parcto optimal solution S" there must cxist a value of 3 in the for loop such that 0 < 3" - B < G. Let Samini be the solution of SMFR-SATURATES w.r.l. such B and its corresponding amin- Since Fx a.pen... (Samn.0) 2 d+1- 2. Samm. is a (8amin - 3. 83 - appprxxiiae e where Syr is the solution 2
cach S E S. Finally, given an (a*, "="""""ppooximmaa"
solution according to Lemma [ Furthermore, we have F amar. (Sgr) < d +
w.r.t. F' F' with relaxed knapsack constraint for a budget k(1 + In 2d+2) returned by the subroutine CostEffectiveGreedy in Algorithm [ and amaz amin < f. Suppose that S is the first intermediate subset. of Sgr with ((S'r) 2 knn 2d-7 constructed using the cost-effective greedy procedure. Let S
17

---

Published in Transactions on Machine Learning Research (06/2024)
arg m Fy (S) and OPT p of Fy
F ff
((S))
According to the monotonicity and submodularity
umazi
0meri:1 we have
c(u)
A c(v)
F amar (5%) S F
(
)
+
Σ
A
F
g (5 +
Σ
VES
WESt
for any Sgg C Sy after iterations and A ( item added to gr for any i = I..... S Based on the cost-effective greedy selection in Algorithm
F
Famar (S Utr))
Omar gr Let un be the i-th
or
A 1ui+nlS5p)
A(c d())
c(uf+1)
for any v E S% S and i 1S.....,1yy. - 1I] because d(u) S k for any v E St and thus no item from St is excluded from consideration due to budget violation when uH41 is added to s(i Therefore, we further obtain
A A(uit1b
1S
A A(u1+1 ((()) c(((11)
F a (5k) S F B (S +
Σ
du) S F
. (S
k,
c(uit1)
WEST VS
After rearranging the inequality above, we have
F ((p F" Famas. a ( (s(it s $(1 (1- c(uit2) (F) (Sp) F² g ( ($ )).
k
a(aits)
I exp( exp((="ujap) c(wit))) Therefore,
Morcover, since x S w for any 0, holds that
F' Fa a (5k) F Famonnt a (s((+1) S expp c(uitt)) ((5)) F Fomma a (S
(4)
h
By applying Eq. recursively to i 01....1./ 15r. 1, we have
dui-1)
F'. e (Sk) - F", m
(S ) S exp(
k (Fr (S)) - F" (
(Sk)
duf
exp( (uu))] )Fo (Sk) Fy
exp(
Pmars 1 (S(E)1)))
k
15 1- Σ
c(u(tz))
I expl
i=0
F ((5))
b
c(
clS gr)
exp(
F (5k) expl
OOPT p
k
Since (rS 2 l: In 2d+2, it holds that
S gr
F' F' Fa (S 2 (1 exp (
OPT
E 2d + 2 umaz D
OPT
umaxig
In addition, F (Sgr) 2 F Fa (S²r) since Syr S Sgr. Therefore, we have OPT
d + and, according to Lemma [ there does not exist any (amazz -appproooee solution of cost at most k. Since S* is an (x",, 3rh )approxximeee Pareto optimal solution and 3 < 3*, S* must be an (a*, -)-approximaaee solution of cost at most As such, we obtain amaz > a and amin > a' - Because we have shown that Saman. is a (8amin - 63 / -ppproxxiettee solution, Somcn, guarantcod to be a (ōa" - e, 5,8 approximate solution. If Saman..8 .8 is included in S, we will conclude the proof directly; otherwise, the solution in 8 dominating Samin. can confirm our conclusion.
e)- -
18

---

Published in Transactions on Machine Learning Research (06/2024)
A.4 Proof of Theorem 2
Theorem 2. For SMFR with a matroid constraint I(M), SMFR-SATURATE runs in ((dt(AA) or log" a) time, where ((A) is the time complexity of the 5-approcimation algorithm for SMM, and provides a set S of solutions with the following properties: (1) |S| = O (2) ||| = Or =) for each S E S, (3) for each (a", ,0²) "ap-)papppmmatt Pareto optimal solution S to SMFR, there must exist its corresponding solution S E S such that f(S) 2 (ôa" s and 9i(5) 2 (63 - g) Vi [d]
Proof. Let us analyze the time complexity of SMFR-SATURATE for a matroid constraint I(M). First, it runs the SMM algorithm d + 1 times to compute OPT', and OPT for every E [d]. Then, it iterates over [21 values of 3 in the for loop. For each value of 3, it attempts to use Ollog 5) different values a in the biscction search. Finally, the subroutine IterativeGreedy takes ((nr) time per round and runs in Ollog 2) rounds. In summary, the time complexity of SMFR-SATURATE for a matroid constraint I(M) is O(di(A) + or log & log () time, where ((A) is the time complexity of the SMM algorithm, and can be simplified as (() + nr log² 4).
For the solution S of SMFR-SATURATE, it is casy to see that |S] - [2] and thus |S = O()) because SMFR- SATURATE adds at most one set to S for each value of 3. Then, because the subroutine IteretiveGreddy runs in at most + floge d-11 rounds and the size of each S is bounded by the rank r of the matroid M, 5 it must hold that |S| I r (1+ + flog2 e 1+11) and thus |S] = -(r log ()) for each S E S. Finally, given an ""-""")apprxximaal
Pareto optimal solution S there must exist a value of B the for loop such that 0 -3 Let Samin.9 be the solution of SMFR-SATURATE w.r.t. such / and its corresponding Omin- Since F F F (S (S.minn))) Oppan
d Samin.3 is a (darmnn 5,0 5) -aaprooximaeee solution according to have F (mmez.0(grr) (S < d++1= - 2, where Sgr is the solution w.r.t. F" Fa returned O'mez
Lemma I Furthermore, by the subroutine Iterativees verredyy in Algorithm [ and Omaa Qmin < 2. Since IterativeGreedy runs 2 p-approximation greedy algorithm for submodular maximization with matroid constraints in cach round,
we have
F amaz Fdmaz,3(88 (S1) - F amaz (0) 2 (1 (1 max Famaa..(S') - F S'EI(M)
amax. (0)).
Since T(S)= = S(SU A) (SS) is nonnegative, monotone, and submodular if f() nonnegative, monotone, and submodular for any A V, we can extend the above result for each round >>> follows:
Fo Famas A - F (1 2 (1- )
max (Fx (F A (S) U (U F' F. (U))))))
2
S(EI(M)
2 (1
)
1
S S'EI(M) F F amaz. (1)+
max (Fr
2
By induction, we obtain the following:
F amar. S1)= (1-
max F SEI(M)
(S')=
OPT
27
llog =il
Since Sgr = U
Sj, we have
F
ammrn (S 2 (1- )
OPT
(1- 6
OPT Fr
2d +
Therefore, we have OPT 8 d + and, according to Lemma I there does not exist any (Amass 3)- approximate solution under matroid constraint I(M). Since 5* is an (a*, "app--ppoxii Pareto optimal solution and 3 < 3", S* must be an , -)pappooxee solution under matroid constraint I(M). As such, we obtain amaz > a and amin > a - 2. Because we have shown that Sam.n. a (8amin 5.63 9 approximate solution, Samin.3 3 is guarantced to be (ōa" c, 63 e>-approximate solution. If Samin..8 is included in S, we will conclude the proof directly; otherwise, the solution in S dominating Samin. @ can confirm our conclusion.
19

---

Published in Transactions on Machine Learning Research (06/2024)
B ILP Formulations
In this section, we present the inleger linear programming (ILP) formulations for the Maximum Coverage and Recommendation problems, specifically tailored to the SMFR problem, as defined in Section 5.1 and Section [.2 respectivcly. Any ILP solver can be employed to identify optimal solutions for small SMFR instances on Mazimum Coverage and Recommendation. For our experimental results in Section and Appendix we refer to this approach the OPT algorithm. Note that these formulations are specifically
designed for these settings and cannot be applied directly to general SMFR problems.
Problems and 6 are specialized versions of the standard ILP formulation of SMFR on Mazimum Coverage
in Section 5.1
with knapsack and partition matroid constraints, respectively.
max
Vj
(5)
max
Σ yj jE(ml
(6)
]E(m]
subject to
Σ [e[m]
C S k:
subject to
Σ 31 S k
11 E [p]
SIEV
Σ 21 2 yj*
Vj E [m]]
Σ 3 2 V/
Vj e [m]
EyES)
by E
Σ yj 2 3 OPT 91 esEC,
Vic E [d]
Σ Vj 2 OPT. g.
Vie [d]
eyEC;
Vj E {0,1]
Vj E [m]]
V1 E {0, 17,
Vj E [m] ee in]
2 E (0,1),
11 E [n]
2 E [0. 13:
These IT.Ps maximize the coverage (i.e., the utility function f in SMFR) on a universe U = fei....,,,.. 6m of m elements and a collection V [S..... Sn} of sets (5) S E [n]), subject to additional coverage constraints on each subset C1,..., Ca of U (w.r.t. cach roprosentativeness function 91...., Id in SMFR). In both formulations, x indicates whether St E V is included in the solution S, and yj indicates whether ej E U is covered by S. Problem is specific to the knapsack constraint defined on a budget & E Z and cost function c(-). Problem 6 is specific to the partition matroid constraint, where V is divided into disjoint partitions Vi...., Vip and at most & sets can be selected from each partition. Solving optimally Problems 5 and 0 with 8 =0 0 and U- = C. yields the value of OPT for each representativeness function 9: corresponding to the knapsack and the partition matroid constraints, respectively.
Problems and g are specialized versions of the ILP formulation for capacitated facility location with benefit matrix B = (bj = (v,, vz) : E [m], E [n]) E Rmxx (m = | and n IVI), specifically designed for SMFR on the Recommendation sctting in Section 5.2 with knapsack and partition matroid constraints, respectively.
max
Σ AE
max
Σ
(8)
[m(m)
5E(m]
[E|n)
subject to
Σ C/TT S k
subject to
Σ I7 2 k
Vt [p]
(E(m)
eVV
Σ Bil S 1,
Vj E [m]
Σ Y5 1,
Vj E [m]
(E(M)
Vşイ S Zt
Vje [m] E [n]]
Mjl S 3i,
Vj E [m]. ( E [m]]
Σ x 2 B OPT g2>
Wi E [d]
Σ I 2 3 3PT go
34 [d]
EIEC,
NEC;
U/ E (0,1), a E 10.
Vj E [m) E [m]
Vy E (0, 1].
Vj E [m], [ E [m]
vl [n]
2 E (0,
14 E [n]
htttpe://en http://wwwwkkppiii.a.rr////s//aa coverage_problert https://en. http://hww...kaaaaaaoo//////a/aa fac.ammllmmalll1tttoolooaio
20

---

Published in Transactions on Machine Learning Research (06/2024)
Given a set V= fei...., )n of n items, both ILPs maximize the total benefit (i.e., the utility function f in SMFR) provided by a set S C V for subset T S V of m items, subject to representativeness constraints on each Ci,..., Ca subset of V (i.e., the representativeness functions 91..... in SMFR). In both formulations, 21 indicates whether e E V is included in the solution S, and zyt indicates whether ej E T takes the benefit from item et E V. Problem is specific to the knapsack constraint defined on a budget & E 2+ and cost function c(-). Problem 8 is specific to the partition matroid constraint, where V is divided into p disjoint partitions Vi,..., V For the knapsack constraint, the value of OPT,, for each representativeness function 9, can be easily computed by sorting the items in C ascendingly according to their costs and finding the maximum number of items whose cumulative cost does not exceed k. For the partition matroid constraint, the value of OPTg. gt for each representativeness function is trivially the maximum between k and [Cal-
C Additional Experiments
In this section, we complement the experimental analysis described in Sections 5.1 and 5.2
C.1 Additional Experiments Maximum Coverage
In this section, we use the same data sets and settings as in Section 5.1 for the Marimum Coverage problem. For the Facebook data set, alternatively define the knapsack constraint in the same way as for the DBLP data set. For the DBLP data set, we alternatively define a partition matroid constraint based the geographic area of the researchers, with five groups: Asia, Europe, North America, Oceamia, and South America. We also set the same upper bound k E z for each geographic group, resulting in partition matroid of rank r = 5k. Figures 5 and 0 present the performance of cach algorithm for different instances of SMFR on Mazimun Coverage with knapsack and matroid constraints on the Facebook and DBLP data sets, respectively. Generally, we observe trends similar to those already presented Section 5.1 which further confirm the effectiveness of SMER-SATURATE.
C.2 Additional Experiments on Recommendation
In this section, we use the same data sets and settings as in Section 5.2 for the Recommmnndation problem. For the MovieLens data set, we alternatively define a knapsack constraint by assigning to each item (movic) a random integer cost in the range [1, 10). For the X-Wines data set, we alternatively define a partition raid constraint based on the continent of origin for wine production: Africa, Asia, Europe, North America, South America, and Oceania. We also set the same upper bound & E Z for each geographic group, resulting in a partition matroid of rank r = 6k. Figures and g present the performance of each algorithm for different instances of SMFR Recommendation with knapsack and matroid constraints the MovieLens and X- Wines data sets, respectively. Generally, observe trends similar to those already presented in Section 5.2 which further confirm the effectiveness of SAIFR-SATURATE.
C.3 Time Efficiency
Figure 9 reports the running time (in seconds) of IMFFRSATURATE. SATURATE, BSM-SATURATE, and SMSC for SMFR on both Maximum Coverage and Recommendation instances. We use the same settings as in Soctions 5.1 and 5.2 In cach plot, the x-axis represents the value of in the knapsack or matroid constraint and the y-axis represents the running time (in seconds) used by each algorithm to solve an SMFR instance. We present the results for d d= 1 and 4 in Figure g
All algorithms take less than minute to complete on each tested instance. AMFRRSATURATE is faster than SMSC in all cases. For the knapsack constraints, SMFR R-RASTRRAAAE, generally runs faster than or close to BSM-SATURATE. However, for the matroid constraints, SOMFR-SATURAN is slower than BSM-SATURATE. SATURATE is the fastest method in most configurations. This bocause SATURATE does not allow for any trade-off between utility (f) and representativeness (g) by design and thus is run only once for each instance. However, all other algorithms should be multiple times with different values of (3 or 7.
21

---

Published in Transactions on Machine Learning Research (06/2024)
C.4 Experiments Large Data Sets
Setup. To show the applicability of SMPR-SATURATE to data sels larger than those used in Section 5 we performed additional experiments on two larger real-world data sets: Pokec (Kosicky) for Mazimum Coverage and MovieLens-25M for Recommendation.
The Pokec (Kosicky) data set an undirected graph with 234, 320 nodes and 2,417, 175 edges, ext extracted from the Poked data set. Pokec itself is directed graph that represents the fallowr-ffollweer relationships of users in Slovakian social network. Pokec (Kosicky) is a subgraph of the Pokec graph induced by nodes representing Pokec users who reside in the Kosicky region while ignoring the directions of the edges. The
MovieLens-220 2etiieenns2252 data set is larger version of the MovieLens data set presented in Section 5 The only difference relies on the size: the MovieLens-25M. data set consists of 25, 000, 095 ratings from 162, 541 users on 62, 423 movies.
We evaluate the Mazimum Coverage problem under a knapsack constraint, and the MovieLens-25MM data set for the Rec- ommendation problem under partition matroid constraint. We apply the same pre-processing and settings used for the Mazimum Coverage problem the DBLP data set (Section 5.1 and the Recommendation problem on the MovieLens data set (Section 5.2 to the Pokec (Kosicky) and MowicLens-251 data sets, respectively.
algorithm and baseline methods on the Pokec (Kosicky) data set for the
Results. The top row of Figure 10 presents the performance of each algorithm for different instances of SMFR on Marimum Coverage with knapsack constraints on the Pokec (Kosicky) data set (scc Fig- 10b and 10ch; while the boltom row of Figure 10 presents the performance of each algorithm for different instances of SMFR Recommenddation with matroid constraints on the MovieLens-255. data set (Figures 10d
ures 10a
10e and 10ft. Generally, we observe trends similar to those already presented for the other data sets for the same problems under the same kind of constraints.
Figure 11 reports the running time (in seconds) of SMFR-SATURATE, SATURATE, BSM-SATURATE, and SMSC for SMFR on both Marimum Coverage instances with knapsack constraints on the Pokec (Kosicky) data set and Recommendation instances with matroid constraints on the MovieLens-255 data set. We use the same settings as in Appendix C.3 In each plot, the x-axis represents the value of l in the knapsack matroid constraint, and the y-axis represents the running time (in seconds) used by cach algorithm to solve an SMFR instance. We present. the results for d = 1 and 4 in Figure 11 Generally, we observe trends similar to those in Figures 9b 9i 9c and 9g (see Appendix C.3). Execution times generally grow linearly with the size of the data sets in both applications. The above results have confirmed the applicability of SMER-S RASURRTE to larger data sets.
22

---

Published in Transactions on Machine Learning Research (06/2024)
Greedy+Max
Saturate
BSM-Saturate
SMSC
OPT
SMFR-Saturate
1.00
1.00
1.00
0.95
0.95
0.90
0.90
0.83
0.85
0.85
0.75
0.75
0.75
0.70
0.65
0.65
0.05
+
0.60
0.60
0.60--
0.30
0.40
0.50
0.00
0.70
0.80
0.50 1:00
0.00
0.10
0.20
0.30
0.40
0.00
0.03 0.05
0.08
0.13 0.15 0.18 gar
0.10
gout
gos
(a) Facebook (d= 1, 20)
(b) Facebook (d 2, 2, 20)
(c) Facebook (d= = 1, 20)
120
120
120
100
100
80
A5
60
60
60-
40
40
20
60
100
20
80
40
100
(d) Facebook (d = 1, 0.8)
(e) Facebook (d 2, 3 2 0.4)
(f) Facebook (d = 4, 0.2)
Figure 5: Results for Marimum Coverage the Facebook data set, with knapsack constraints.
Greedy
Saturate
+
BSM-Saturate
SMSC
OPT
SMFR-Saturate
1.0
1.0
0.9
0.9
0.9
0.8
0.8
0.7
0.5
0.5
0.5
0.4
0.4
V
0.4
0.60
0.70
0.50
1:00
0.50
0.70
0.30
0.90
0.60
0.70
0.90
gaie
8aut
Baur
(a) DBLP (d = k = 35)
(b) DBLP (d = 2, = 35)
(c) DBLP (d = 4, k: = 35)
1600
1600
1600
1400
1400
1200
1200
1000
1000
800
800
600
600
400
200
200
10
20
30
40
50
10
20
30
40
50
10
30
40
50
k
(d) DBLP (d = 1, 2 0.8)
(e) DBLP (d = 2,8
3 2 0.4)
(f) DBLP (d = 4, 2 0.2)
Figure 6: Results for Marimum Coverage on the DBLP data set, with matroid constraints.
23

---

Published in Transactions on Machine Learning Research (06/2024)
Greedy+ Max
Saturate
+
SSM--aturate
SMSC
OPT
SMFR-Saturade
1.00
1.00
1.00 -3
X0
0.95
0.90
0.85
0.85
0.80
0.80
0.00
0.75
0.75
0.20
0.40
0.50
0.80
1.00
0.20
0.40
0.60
0.80
1.00
0.10
0.20
0.30
0.40
0.50
0.60
Butt
guut
(a) MowieLens
1, 50)
(b) MovieLens (d k 50)
(c) MovieLens (d 4, 50)
185
x
x
x
x
X
1.0
0.8
0.95
0.90
0.6
0.5
0.3
.7.
0.2
0.2
20
60
80
100
20
60
80
100
20
60
100
k
K
(d) MovieLens (d = 1, 2 0.8)
(e) MovieLens (d 2, 0.4)
(f) MowieLens (d 4,9 2 0.2)
Figure 7: Results for Recommendation on the MovieLens data set, with knapsack constraints.
Greedy
Saturate
BSM-Saturate
SMSC
OPT
SMFR-Saturate
S-S-O
1.0
1.0
p +
+4
0.9
0.9
0.0
0.9
0.7
x
0.7
0.7
0.5
0.4
0.6
0.3
0.92
0.94
0.96
0.98
1.00
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0.20
0.30
0.40
0.30
0.60
0.70
Bour
gout
B
(a) X- Wines (d 1 25)
(b) X- Wines (d 2, 25)
(c) X- Wines
25)
2600
2500
2400
*
2200
2000
2000
1750
1800
1500
1500
1400
1000
....
1200
1000
10
20
40
50
20
30
750
40
50
(d) X- Wines (d = 1,8 0.8)
(e) X- Wines (d 2, 3 > 0.4)
(f) X- Wines (d 4, 0.2)
Figure 8: Results for Recommendation the X- Wines data set, with matroid constraints.
24

---

Published in Transactions on Machine Learning Research (06/2024)
Saturate
BSM-Saturate
SMSC
Satu-Sat
20.0
Time (Ww4)
Time Exec
Time (Gec) 12
Time (aNc)
0.0
20
50
20
40
60
100
20
50
60
100
(a) Facebook; = 1, Mat.
(b) DBLP, d = 1, Kna.
(c) MovieLens, d 1, Mat.
(d) X- Wines, d = 1, Kna.
Time (sed
12
Time Ceec
Tilns Ceec
Time (aac)
77
10
10
40
30
20
40
60
100
20
50
100
(e) Facebook, = 4, Mat.
(1) DBLP, d 4, Kna.
(g) MovieLens, d 4, Mat. (h) X- Wines, d 4, Kna.
Figure 9: Running times (in seconds) of SMFR-SATURATE SATURATE, BSM-SATURATE, and SMSC for SMFR when d= = 1. Here, the Pacebook and DBLP data sets are used for Maximum Coveruge (MC); the Wines and MovieLens data sets are used for Recommendation (RE). In addition, the matroid constraints (Mat.) are imposed on the Facebook and MovieLens data sets; the knapsack constraints (Kna.) are imposed the DBLP and X- Wines data sets.
Saturate
+
Saturatee
SMSC
SMFR-Saturate
200
200
200
175
125
100
75-
50
20
40
60
80
100
20
40
60
80
100
20
40
60
80
100
k
k
(a) Pokec (Kosicky) (d = 1,8 0.8)
(b) Pokec (Kosicky) (d 2, 2 0.4)
(c) Pokec (Kosicky) (d 4, B 0.2)
1.6
1.4
1.4
1.2
1.2
1.0
1.0
V
0.8
0.8
0.8
0.6
V...........
0.6
0.4
0.4
10
30
(d) MovicLens-225 25M (d= 1,3 0.8)
(e) MovicLens-25M (d 2, 0.4)
(I) MovicLens-S2 (d= 4,3 0.2)
Figure 10: Results for Marimum Coverage on the Pokec (Kosicky) data set with knapsack constraints, and for Recommendation the MovicLens-25M data set with matroid constraints.
25

---

Published in Transactions on Machine Learning Research (06/2024)
Saturate
BSM-Sature Saturate
SMSC
SaturSaturate
500
2000
Time (sec)
1500
Time (sec)
400-
1000
500
200
77.
40
60
80
100
10
30
40
k
(a) Pokec (Kosicky), d = 1, Kna.
(b) MovicLens- 25M, d = 1, Mal.
3500
3000
Time (sec)
2500
Time (sec)
400
2000
1500
900
1000
200-
40
60
80
100
10
30
40
k
k
(c) Pokec (Kosicky): d = 4, Kna.
(d) MovieLens-P25 25M,,25,5, = 4, Mat.
Figure 11: Running times (in seconds) of SMFR-SATURATE, SATURATE, BSM-SATURATE, and SMSC for SMFR when d = 1,4. Here, the Pokec (Kosicky) data set used for Masimuum Coverage (MC) with knapsack constraints (Kna.); the MovieLens-255 data set used for Recommendation (RE) with matroid constraints (Mat.).
26


Reviews:
[{"review_id": "fPtTBF7p7V", "reviewer": "Reviewer_vmk1", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n- This work proposes an interesting a novel way to maximize monotone submodular\n  functions in the presence of structured constraints and $d > 1$\n  \"representative\" functions. In particular, this differs from past works that\n  simply aim to maximize the *minimum* of the \"fairness\" objectives $g_i$.\n- The experiments are quite strong for an applied paper that studies submodular\n  maximization.\n\n**Weaknesses**\n- The only concrete theoretical contribution is Theorem 1 and Theorem 2, which\n  formalize how to recast the SMFR to a sequence of submodular cover problems.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work introduces the problem of submodular maximization with fair representation (SMFR).\nIn particular, it aims to maximize a (normalized) monotone submodular function\nsubject to a knapsack or matroid constraint in a Pareto-optimal way with\nrespect to $d$ representative objective functions $g_i$.\nThe main idea is to recast an SMFR instance into mutiple instances of the\nsubmodular cover problem with different weights between the $f$ and $g_i$\nobjectives.\n\nThe authors show that if we have a $\\delta$-approximation algorithm for\nsubmodular maximization with a knapsack (or matroid) constraint that returns a\n$(\\alpha,\\beta)$-approximate Pareto optimal solution to SMFR, then there exists\na $(\\delta \\alpha - \\varepsilon, \\delta \\beta - \\varepsilon)$-approximate\nsolution with cost $O(k \\log (d \\ \\varepsilon))$ where $k$ is the knapsack\nbudget (or matroid rank) constraint. They also provide extensive experiments on\nreal-world data that compares their algorithm on max coverage and\nrecommendation tasks to true optimal solutions computed via integer\nprogramming.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "**Questions**\n- [page 02] In Eq. (1), how is the max operator defined for a tuple of real\n  values? If we use a lexicographic comparator, then the first representative\n  function $g_1$ carries more weight than $g_d$.\n- [page 02] In the definition of Pareto optimal for $(\\alpha,\n  \\beta)$-approximate solutions, instead of saying \"there does not exist any\n  ...\", should you instead relax this to \"you have not computed any\" (i.e., a\n  solution is Pareto-optimal w.r.t. a set of computed solutions? Otherwise, this\n  quantifier includes optimal solution that you may not have computed. The same\n  applies to the top of page 05.\n\n**Typos / suggestions**\n- [page 02] Consider putting Example 1 in a \\paragraph so that the text is not\n  all italcized, as it is somewhat hard to read for more than a few lines.\n- [page 04] Typo: \"which ensures that the difference in the utilities of any two groups is As ...\"\n- [page 04] In the beginning of Section 4, you do not need to reintroduce what\n  SMFR means, as it was defined earlier.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ejahwB1rWA", "reviewer": "Reviewer_bDDZ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n\n- The writing quality is good\n\n- The result of only O(1 / eps) solutions can already capture the Pareto frontier looks interesting and strong\n\n- The citations and comparison to related works are comprehensive\n\n- The experiments are solid\n\n\nWeakness:\n\n- The technical novelty is limited, and it seems to be a composition of several existing results in a relatively straightforward way\n\n- The size of the data sets used in the experiments looks small", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper suggests the so-called “submodular maximization with fair representation (SMFR)” problem and obtains Pareto approximation algorithms for the problem subject to knapsack and matroid constraints. In this problem, one is given a main (monotone) submodular function f, and d other submodular functions g_1, …, g_d. A subset S is called (alpha, beta)-approx, if it is alpha-approx to the max of main function f, and beta-approx to the max of all the other d functions g_1, …, g_d simultaneously. Roughly speaking, the Pareto approximation is a set P of solutions, such that for every alpha, beta, if there is a (alpha, beta)-approx solution S’ then there is a (\\delta alpha - eps, \\delta beta - eps)-approx to it in the solution set P, where \\delta is the approx-ratio of a black-box of an algorithm for submodular maximization subject to the matroid or knapsack constraints. The main result is that |P| = O(1 / eps) is sufficient for this goal, and the every solution set in P is of small size (linear in the size of the knapsack or the rank of the matroid). This can be found efficiently in poly time. Experiments are conducted to validate the effectiveness of the algorithm on real datasets.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- \\mathcal{I} is not quantified in (1). Maybe you want to change to “subject to a knapsack or matroid constraint \\mathcal{I}”, in the sentence immediately before (1)?\n\n- The meaning of max in (1) is not discussed in the paragraph immediately below it. While I understand that the exact definition is discussed in ”Our Contributions” part, it is still suggested to at least comment that the intuition is to maximize them simultaneously even though this goal can be tricky and you would discuss this next.\n\n- Page 2, I think “and at least one is strictly larger” should be put out of the parenthesis since this is an important part of the definition. And similarly for the one on page 5.\n\n- In the statement of lemma 1, the quantification of S sounds confusing. I suppose you meant “exists S \\in \\mathcal{I}”, instead of “for every S”. However, currently it reads “for any set S \\in \\mathcal{I}”, and it may confuse with “for every S”. Maybe replace “any” with “some”? Or, rephrase as e.g. “suppose S \\in \\mathcal{I} is any set that satisfies F’_{\\alpha, \\beta}(S) \\geq d + 1 - eps / 2”", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "vd0rOx3jhT", "reviewer": "Reviewer_6VLx", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths: \n-I think the topic is important and interesting and it is nice to see that previous work can be used to obtain an approximate solution to this setting. \n\n-The experiments have been broad comparing to various baselines and datasets. \n\n### Weaknesses: \nW1-The major weakness in my view has to do with the problem formulation. Specifically, Pareto optimality is in general a very weak guarantee. Why haven’t the authors considered other possibilities such as imposing bounds on the functions g_1, …, g_d or maximizing a weighted sum of the functions? These possibilities and their disadvantages should be discussed in the paper.  \n\nW2-The paper would be more clear if the point behind functions g_1, …, g_d is clarified early on and separated from the knapsack/matroid constraints. Example 1 in the paper makes it seem as though the constraints (knapsack or matroid) are there to ensure fairness, but further ahead in the paper this is done by the representativeness objectives  g_1, …, g_d \n\nW3-In the related work, the paper states that the previous work “returns only a single solution”. But is this not the case in this paper as well? \n\n\nW4-This issue is also about the Pareto optimality, consider the knapsack constraint in section 5.1. The functions g_i are counting vertices from different communities, can’t we end up with trivial solutions where for example one group has full representation and the other has none? Would that not be Pareto-optimal? What forbids this? This would not happen on the other hand if we imposed bounds.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper solves the problem of submodular function maximization under a knapsack or matroid constraint. Unlike previous work, the paper is not maximizing a single objective function but rather a main function f in addition to a collection of d representation functions g_1,..., g_d. The authors suggest finding an (alpha,beta) Pareto optimal solution where the alpha and beta are the approximation factors for f and g1,..,g_d respectively. The main technical result is that an approximation algorithm for submodular maximization under knapsack or matroid constraints can be used to obtain a Pareto optimal approximate solution with close approximation factors. The paper further shows experimental results on several datasets and shows that the proposed algorithm has better performance in comparison to some baselines.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see the points above under weaknesses. The paper would be much stronger if it could convince the reader that other more traditional options such as maximizing a weighted sum or imposing bounds are not the right thing to do here.", "broader_impact_concerns": "I don't think that a statement on ethical implications is needed.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers' consensus was that the proposed formulation is likely to be of limited interest. However, the formulation still makes sense, and the results are accurate and correct. So I think regarding the "accuracy, credibility and clarity", the paper is still doing good enough. They also agree that regardless of the technical depth and potential impact, the second criteria (for audience interest) is met. 

As a result, I would like to ask the authors to include a more thorough discussions of the advantages and limitations of this approach, as done through the rebuttal.
Decision: Accept (poster)