SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Bo Li $^{1}$  Fangxiao Wang $^{1}$  Yu Zhou $^{1}$
$^{1}$ Department of Computing, The Hong Kong Polytechnic University, Hong Kong, China
comp-bo.li@polyu.edu.hk,
fangxiao.wang@connect.polyu.hk, cszhou@comp.polyu.edu.hk
We study the maximin share (MMS) fair allocation of  $m$  indivisible chores to  $n$  agents who have costs for completing the assigned chores. It is known that exact MMS fairness cannot be guaranteed, and so far the best-known approximation for additive cost functions is  $\frac{13}{11}$  by Huang and Segal-Halevi [EC, 2023]; however, beyond additivity, very little is known. In this work, we first prove that no algorithm can ensure better than  $\min\{n, \frac{\log m}{\log \log m}\}$ -approximation if the cost functions are submodular. This result also shows a sharp contrast with the allocation of goods where constant approximations exist as shown by Barman and Krishnamurthy [TEAC, 2020] and Ghodsi et al. [AII, 2022]. We then prove that for subadditive costs, there always exists an allocation that is  $\min\{n, \lceil \log m \rceil \}$ -approximation, and thus the approximation ratio is asymptotically tight. Besides multiplicative approximation, we also consider the ordinal relaxation, 1-out-of-  $d$  MMS, which was recently proposed by Hosseini et al. [JAIR and AAMAS, 2022]. Our impossibility result implies that for any  $d \geq 2$ , a 1-out-of-  $d$  MMS allocation may not exist. Due to these hardness results for general subadditive costs, we turn to studying two specific subadditive costs, namely, bin packing and job scheduling. For both settings, we show that constant approximate allocations exist for both multiplicative and ordinal relaxations of MMS.
1.1 Background and related research
Although fair resource allocation has been widely studied in the past decade, the research is centered around additive functions [3]. However, in many real-world scenarios, the functions are more complicated. Particularly, the functions appear in machine learning and artificial intelligence (e.g., clustering [37], Sketches [18], Coresets [36], data distillation [41]) are often submodular, and we refer the readers to a recent survey that reviews some major problems within machine learning that have been touched by submodularity [12]. Therefore, in this work, we study the fair allocation problem when the agents have beyond additive cost functions. The mainly studied solution concept is maximin share (MMS) fairness [14], which is traditionally defined for the allocation of goods as a relaxation of proportionality (PROP). A PROP allocation requires that the utility of each agent is no smaller than the average utility when all items are allocated to her. PROP is too demanding in the sense that such an allocation does not exist even when there is a single item and two agents. Due to this strong impossibility result, the maximin share (MMS) of an agent is proposed to relax the average utility by the maximum utility the agent can guarantee herself if she is to partition the items into  $n$  bundles but is to receive the least favorite bundle.
For the allocation of goods, although MMS significantly weakens the fairness requirement, it was first shown by Kurokawa et al. [31, 32] that there are instances where no allocation is MMS fair for all agents. Accordingly, designing (efficient) algorithms to compute approximately MMS fair allocations steps into the center of the field of algorithmic fair allocation. Kurokawa et al. [32] first proved that there exists a  $\frac{2}{3}$ -approximate MMS fair allocation for additive utilities, and then Amanatidis et al. [1] designed a polynomial time algorithm with the same approximation guarantee. Later, Ghodsi et al. [23] improved the approximation ratio to  $\frac{3}{4}$ , and Garg and Taki [22] further improved it to  $\frac{3}{4} + o(1)$ . On the negative side, Feige et al. [20] proved that no algorithm can ensure better than  $\frac{39}{40}$  approximation. Beyond additive utilities, Barman and Krishnamurthy [6] initiated the study of approximately MMS fair allocation with submodular utilities, and proved that a 0.21-approximate MMS fair allocation can be computed by the round-robin algorithm. Ghodsi et al. [24] improved the approximation ratio to  $\frac{1}{3}$ , and moreover, they gave constant and logarithmic approximation guarantees for XOS and subadditive utilities, respectively. The approximations for XOS and subadditive utilities are recently improved by Seddighin and Seddighin [40]. There are also several works that delve into concrete combinatorial problems and seek to improve approximation ratios compared to more general functions. Li et al. [33] and Hummel and Hetland [30] introduced interval scheduling and independent set structures, respectively, into MMS fair allocation problems. In both cases, the induced utility functions correspond to special cases of XoS functions. Both of these two works enhanced the approximation ratios for their specific utility functions. Some other combinatorial problems that have been studied for goods include the knapsack problem [21, 9, 8] and matroid constraints [19].
For the parallel problem of chores, where agents need to spend costs on completing the assigned chores, less effort has been devoted. Aziz et al. [4] first proved that the round-robin algorithm ensures 2-approximation for additive costs. Barman and Krishnamurthy [6], Huang and Lu [28] and Huang and Segal-Halevi [29] respectively improved the approximation ratio to  $\frac{4}{3}$ ,  $\frac{11}{9}$  and  $\frac{13}{11}$ . Recently, Feige et al. [20] proved that with additive costs, no algorithm can be better than  $\frac{44}{43}$ -approximate. However, except a recent work that studies binary supermodular costs [10], very little is known beyond additivity.
Besides multiplicative approximations, we also consider the ordinal approximation of MMS, namely, 1-out-of-  $d$  MMS fairness which is recently studied in [5, 26, 27], and proportionality up to any item (PROPX) which is an alternative way to relax proportionality [38, 34]. For indivisible chores, Hosseini et al. [26] and Li et al. [34] respectively proved the existence of 1-out-of- $\lceil \frac{3n}{4} \rceil$  MMS fair and exact PROPX allocations. All the above works also assume additive costs. More works related to this paper in the literature can be seen in Appendix A.
1.2 Main results
In this work, we aim at understanding the extent to which MMS fairness can be satisfied when the cost functions are beyond additive. In a sharp contrast to the allocation of goods, we first show that no algorithm can ensure better than  $\min \{n, \frac{\log m}{\log \log m}\}$ -approximation when the cost functions are submodular. Further, we show that for general subadditive cost functions, there always exists an allocation that is  $\min \{n, \log m\}$ -approximate MMS, and thus the approximation ratio is asymptotically tight. Next, we consider the ordinal relaxation, 1-out-of-  $d$  MMS. It is trivial that 1-out-of-1 MMS is satisfied no matter how the items are allocated, and somewhat surprisingly, our impossibility result implies that for any  $d \geq 2$ , there is an instance for which no allocation is 1-out-of-  $d$  MMS.
Result 1 For general subadditive cost functions, the asymptotically tight multiplicative approximation ratio of MMS is  $\min \{n,\log m\}$ . Further, for any  $d\geq 2$ , a 1-out-of-  $d$  MMS allocation may not exist.
Result 1 combines Theorems 1, 2 and Corollary 1. The strong impossibility in Result 1 does not rule out the possibility of constant multiplicative or ordinal approximation of MMS fair allocations for specific subadditive costs. For this reason, we turn to studying two concrete settings with subadditive costs. The first setting encodes a bin packing problem, which has applications in many areas (e.g., semiconductor chip design, loading vehicles with weight capacity limits, and filling containers [17]). In the first setting, the items have sizes which can be different to different agents. The agents have bins that can be used to pack the items allocated to them with the goal of using as few bins as possible. The second setting encodes a job scheduling problem, which appears in many research
In this paper we use  $\log (\cdot)$  to denote  $\log_2(\cdot)$ .
areas, including data science, big data, high-performance computing, and cloud computing [25]. In the second setting, the items are jobs that need to be processed by the agents. Each job may be of different lengths to different agents and each agent controls a set of machines with possibly different speeds. Upon receiving a set of jobs, an agent's cost is determined by the corresponding minimum completion time when processing the jobs using her own machines (i.e., makespan). As will be clear, job scheduling setting is more general than the additive cost setting. Besides, it uncovers new research directions for group-wise fairness.
Result 2 For the bin packing and job scheduling settings, a 1-out-of-  $\lfloor \frac{n}{2} \rfloor$  MMS allocation and a 2-approximate MMS allocation always exist.
Result 2 combines Theorems 3, 4 and Corollaries 2, 3. Besides studying MMS fairness, in Appendix F, we also prove hardness results for two other relaxations of proportionality, i.e., PROP1 and PROPX.
For any integer  $k \geq 1$ , let  $[k] = \{1, \dots, k\}$ . In a fair allocation instance  $I = (N, M, \{v_i\}_{i \in N})$ , there are  $n$  agents denoted by  $N = [n]$  and  $m$  items denoted by  $M = \{e_1, \dots, e_m\}$ . Each agent  $i \in N$  has a cost function over the items,  $v_i: 2^M \to \mathbb{R}^+ \cup \{0\}$ . Note that for simplicity, we abuse  $v_i(\cdot)$  to denote a cost function. The items are chores, and particularly, upon receiving a set of items  $S \subseteq M$ ,  $v_i(S)$  represents the effort or cost agent  $i$  needs to spend on completing the chores in  $S$ . The cost functions are normalized and monotone, i.e.,  $v_i(\emptyset) = 0$  and  $v_i(S_1) \leq v_i(S_2)$  for any  $S_1 \subseteq S_2 \subseteq M$ . Note that no bounded approximation can be achieved for general cost functions, and we provide one such example in Appendix B.1. Thus we restrict our attention to the following three classes. A cost function  $v_i$  is subadditive if for any  $S_1, S_2 \subseteq M$ ,  $v_i(S_1 \cup S_2) \leq v_i(S_1) + v_i(S_2)$ . It is submodular if for any  $S_1 \subseteq S_2 \subseteq M$  and any  $e \in M \setminus S_2$ ,  $v_i(S_2 \cup \{e\}) - v_i(S_2) \leq v_i(S_1 \cup \{e\}) - v_i(S_1)$ . It is additive if for any  $S \subseteq M$ ,  $v_i(S) = \sum_{e \in S} v_i(e)$ . It is widely known that any additive function is also submodular, and any submodular function is also subadditive.
An allocation  $\mathbf{A} = (A_{1},\dots ,A_{n})$  is an  $n$  -partition of the items where  $A_{i}$  contains the items allocated to agent  $i$  such that  $A_{i}\cap A_{j} = \emptyset$  for any  $i\neq j$  and  $\bigcup_{i\in N}A_i = M$  . For any set  $S$  and integer  $d$  , let  $\Pi_d(S)$  be the set of all  $d$  -partitions of  $S$  . The maximin share (MMS) of agent  $i$  is
$$
\mathsf {M M S} _ {i} ^ {n} (I) = \min  _ {(X _ {1}, \dots , X _ {n}) \in \Pi_ {n} (M)} \max  _ {j \in [ n ]} v _ {i} (X _ {j}).
$$
Note that we may neglect  $n$  and  $I$  in  $\mathsf{MMS}_i^n (I)$  when there is no ambiguity. Note that the computation of MMS is NP-hard even when the costs are additive, which can be verified by a reduction from the Partition problem. Given an  $n$ -partition of  $M$ ,  $\mathbf{X} = (X_1,\dots ,X_n)$ , if  $v_{i}(X_{j})\leq \mathsf{MMS}_{i}$  for any  $j\in [n]$ , then  $\mathbf{X}$  is called an MMS defining partition for agent  $i$ . Note that the original definition of  $\mathsf{MMS}_i$  for chores is defined with non-positive values, where the minimum value of the bundles is maximized. In this work, to simplify the notions, we choose to use non-negative numbers (representing costs), and thus the definition is equivalently changed to be the maximum cost of the bundles being minimized. To be consistent with the literature, we still call it maximin share.
Definition 1 ( $\alpha$ -MMS) An allocation  $\mathbf{A} = (A_1, \ldots, A_n)$  is  $\alpha$ -approximate maximin share ( $\alpha$ -MMS) fair if  $v_i(A_i) \leq \alpha \cdot \mathrm{MMS}_i$  for all  $i \in N$ . The allocation is MMS fair if  $\alpha = 1$ .
Given the definition of MMS, for any agent  $i$  with subadditive cost  $v_{i}(\cdot)$ , we have the following bounds for  $\mathsf{MMS}_i$ ,
$$
\mathrm {M M S} _ {i} \geq \max  \left\{\max  _ {e \in M} v _ {i} (\{e \}), \frac {1}{n} \cdot v _ {i} (M) \right\}. \tag {1}
$$
Following recent works [5, 27, 26], we also consider the ordinal approximation of MMS, namely, 1-out-of-  $d$  MMS fairness. Intuitively, MMS fairness can be regarded as 1-out-of-  $n$  MMS (i.e., partitioning the items into  $n$  bundles but receiving the largest bundle). Since 1-out-of-  $n$  MMS allocations may not exist, we can instead find a maximum integer  $d \leq n$  such that a 1-out-of-  $d$  MMS allocation is guaranteed to exist. Given a  $d$ -partition of  $M$ ,  $\mathbf{X} = (X_1, \ldots, X_d)$ , if  $v_i(X_j) \leq \mathsf{MMS}_i^d$  for any  $j \in [d]$ , then  $\mathbf{X}$  is called a 1-out-of-  $d$  MMS defining partition for agent  $i$ . An allocation  $\mathbf{A}$  is 1-out-of-  $d$  MMS fair if for every agent  $i \in N$ ,  $v_i(A_i) \leq \mathsf{MMS}_i^d$ . More generally, given any  $\alpha \geq 1$ , we have the bi-factor approximation,  $\alpha$ -approximate 1-out-of-  $d$  MMS, if  $v_i(A_i) \leq \alpha \cdot \mathsf{MMS}_i^d$  for
every  $i \in N$ . By the definition, we have the following simple observation, whose proof is deferred to Appendix B.2.
Observation 1 For any instance with subadditive costs, given any integer  $1 \leq d \leq n$ , a 1-out-of-d MMS allocation is  $\lceil \frac{n}{d} \rceil$ -MMS fair.
3 General subadditive cost setting
By Inequality 1, if the costs are subadditive, allocating all items to a single agent ensures an approximation of  $n$ , which is the most unfair algorithm. Surprisingly, such an unfair algorithm achieves the optimal approximation ratio of MMS even if the costs are submodular.
Theorem 1 For any  $n \geq 2$ , there is an instance with submodular costs for which no allocation is better than  $n$ -MMS or  $\frac{\log m}{\log \log m}$ -MMS.
Proof. For any fixed  $n \geq 2$ , we construct an instance that contains  $n$  agents and  $m = n^n$  items. By taking logarithm of  $m = n^n$  twice, it is easy to obtain
$$
n = \frac {\log m}{\log \log m - \log \log n} \geq \frac {\log m}{\log \log m}.
$$
Thus, in the following, it suffices to show that no allocation can be better than  $n$ -MMS. Let each item correspond to a point in an  $n$ -dimensional coordinate system, i.e.,
$$
M = \left\{\left(x _ {1}, x _ {2}, \dots , x _ {n}\right) \mid x _ {i} \in [ n ] \text {f o r a l l} i \in [ n ] \right\}.
$$
For each agent  $i \in N$ , we define  $n$  covering planes  $\{C_{il}\}_{l \in [n]}$  and for each  $l \in [n]$ ,
$$
C _ {i l} = \left\{\left(x _ {1}, x _ {2}, \dots , x _ {n}\right) \mid x _ {i} = l \text {a n d} x _ {j} \in [ n ] \text {f o r a l l} j \in [ n ] \backslash \{i \} \right\}. \tag {2}
$$
Note that  $\{C_{il}\}_{l\in [n]}$  forms an exact cover of the points in  $M$ , i.e.,  $\bigcup_{l}C_{il} = M$  and  $C_{il}\cap C_{iz} = \emptyset$  for all  $l\neq z$ . For any set of items  $S\subseteq M$ ,  $v_{i}(S)$  equals the minimum number of planes in  $\{C_{il}\}_{l\in [n]}$  that can cover  $S$ . Therefore,  $v_{i}(S)\in [n]$  for all  $S$ . We first show  $v_{i}(\cdot)$  is submodular for every  $i$ . For any  $S\subseteq T\subseteq M$  and any  $e\in M\setminus T$ , if  $e$  is not in the same covering plane as any point in  $T$ ,  $e$  is not in the same covering plane as any point in  $S$ , either. Thus,  $v_{i}(T\cup \{e\}) - v_{i}(T) = 1$  implies  $v_{i}(S\cup \{e\}) - v_{i}(S) = 1$ , and accordingly,
$$
v _ {i} (T \cup \{e \}) - v _ {i} (T) \leq v _ {i} (S \cup \{e \}) - v _ {i} (S).
$$
Since  $\{C_{il}\}_{l\in [n]}$  is an exact cover of  $M$ ,  $\mathrm{MMS}_i = 1$  for every  $i$ , where the MMS defining partition is simply  $\{C_{il}\}_{l\in [n]}$ . Then to prove the theorem, it suffices to show that for any allocation of  $M$ , there is at least one agent whose cost is  $n$ . For the sake of contradiction, we assume there is an allocation  $\mathbf{A} = (A_1,\dots ,A_n)$  where every agent has cost at most  $n - 1$ . This means that for every  $i\in N$ , there exists a plane  $C_{il_i}$  such that  $A_{i}\cap C_{il_{i}} = \emptyset$ . Consider the point  $\mathbf{b} = (l_1,\ldots ,l_n)$ , it is clear that  $\mathbf{b}\in C_{il_i}$  and thus  $\mathbf{b}\notin A_{i}$  for all  $i$ . This means that  $\mathbf{b}$  is not allocated to any agent, a contradiction. Therefore, such an allocation  $\mathbf{A}$  does not exist which completes the proof of the theorem.
To facilitate the understanding of Theorem 1, in Appendix C.1, we visualize an instance with 3 agents and 27 items where no allocation is better than 3-MMS. The hard instance in Theorem 1 also implies the following lower bound for 1-out-of-  $d$  MMS, whose proof is in Appendix C.2.
Corollary 1 For any  $2 \leq d \leq n$ , there is an instance with submodular cost functions for which no allocation is 1-out-of-d MMS.
Theorem 2 For any instance with subadditive cost functions, there always exists a  $\min \{n, \lceil \log m \rceil\}$ -MMS allocation.
Proof. We describe the algorithm that computes a  $\min \{n, \lceil \log m \rceil\}$ -MMS allocation in Algorithm 1. First, if  $\log m \geq n$ , we are safe to arbitrarily allocate the items to the agents, which ensures  $n$ -approximation.
The tricky case is when  $\log m < n$ , where we cannot allocate too many items to a single agent. For this case, we first look at agent 1's MMS defining partition  $\mathbf{D}^{1} = (D_{1}^{1},\dots ,D_{n}^{1})$ , where  $v_{1}(D_{j}^{1})\leq \mathrm{MMS}_{1}$
Algorithm 1 Computing a  $\min \{n, \lceil \log m \rceil\}$ -MMS allocation for subadditive costs
Input: An instance  $(N,M,\{v_{i}\}_{i\in N})$  with general subadditive costs.
Output: An allocation  $\mathbf{A} = (A_{1},\dots ,A_{n})$  such that  $v_{i}(A_{i})\leq \lceil \log m\rceil \cdot \mathrm{MMS}_{i}$  for all  $i\in N$
1: Initialize  $A_{i}\gets \emptyset$  for every  $i\in N$
2: if  $\log m\geq n$  then
3:  $A_{1}\gets M$
4: else
5:  $i\gets 1$  and  $M_0\gets M$
6: end if
7: while  $M_{i-1} \neq \emptyset$  and  $i \leq n$  do
8: Let  $(D_1^i,\ldots ,D_n^i)$  be one of  $i$  's MMS defining partitions over  $M$
9: Let  $R_{j}^{i} = D_{j}^{i}\cap M_{i - 1}$  for all  $j\in [n]$ . Re-index the bundles such that  $|R_1^i |\geq \dots \geq |R_n^i |$
0:  $A_{i}\gets \bigcup_{j\in [\log m]}R_{j}^{i}$  and  $M_{i}\leftarrow M_{i - 1}\setminus A_{i}$
1:  $i\gets i + 1$
2: end while
for all  $j \in [n]$  and we assume that they are ordered by the number of items, i.e.,  $|D_1^1| \geq \dots \geq |D_n^1|$ . In order to ensure that agent 1's cost is no more than  $\lceil \log m \rceil$  times her MMS, we ask her to take away  $\lceil \log m \rceil$  largest bundles (in terms of number of items) in  $\mathbf{D}^1$ , i.e.,  $A_1 = \bigcup_{j \in [\lceil \log m \rceil]} D_j^1$ . Since the cost function is subadditive,
$$
v _ {1} (A _ {1}) \leq \sum_ {j \in [ \lceil \log m \rceil ]} v _ {1} \left(D _ {j} ^ {1}\right) \leq \lceil \log m \rceil \cdot \mathsf {M M S} _ {1}.
$$
Moreover, since on average each bundle in  $\mathbf{D}^1$  contains  $\frac{m}{n}$  items and  $A_{1}$  contains the bundles with largest number of items,  $|A_1|\geq \lceil \log m\rceil \cdot \frac{m}{n}\geq \frac{\log m}{n}\cdot m$ . That is, at least  $\frac{\log m}{n}$  fraction of the items are taken away by agent 1. Let  $M_{1} = M\setminus A_{1}$  be the set of remaining items, and we have
$$
\left| M _ {1} \right| \leq \left(1 - \frac {\log m}{n}\right) \cdot m.
$$
We next ask agent 2 to take away items in a similar way to agent 1. Let  $\mathbf{D}^2 = (D_1^2, \ldots, D_n^2)$  be one of agent 2's MMS defining partitions, and  $\mathbf{R}^2 = (R_1^2, \ldots, R_n^2)$  be the remaining items in these bundles, i.e.,  $R_j^2 = D_j^2 \cap M_1$ . Again, we assume  $|R_1^2| \geq \dots \geq |R_n^2|$ . Letting  $A_2 = \bigcup_{j \in [\log m]} R_j^2$  and  $M_2 = M_1 \setminus A_2$ , we have  $v_2(A_2) \leq \lceil \log m \rceil \cdot \mathrm{MMS}_2$ . Moreover, since on average each bundle in  $\mathbf{R}^2$  contains  $\frac{|M_1|}{n}$  items and  $A_2$  contains the bundles with largest number of items,
$$
| A _ {2} | \geq \lceil \log m \rceil \cdot \frac {| M _ {1} |}{n} \geq \frac {\log m}{n} \cdot | M _ {1} |,
$$
$$
\left| M _ {2} \right| \leq \left(1 - \frac {\log m}{n}\right) \cdot \left| M _ {1} \right| \leq \left(1 - \frac {\log m}{n}\right) ^ {2} \cdot m. \tag {3}
$$
We continue with the above procedure for agents  $i = 3, \ldots, n$  with the formal description shown in Algorithm 1. It is straightforward that every agent  $i$  who gets a bundle  $A_i$  has cost at most  $\lceil \log m \rceil \cdot \mathrm{MMS}_i$ . Further, by induction, Equation 3 holds for all agents  $i \leq n$ , i.e.,
$$
\left| M _ {i} \right| \leq \left(1 - \frac {\log m}{n}\right) \cdot \left| M _ {i - 1} \right| \leq \left(1 - \frac {\log m}{n}\right) ^ {i} \cdot m.
$$
To show the validity of the Algorithm, it remains to show that the algorithm can allocate all items, i.e.,  $M_{n} = \emptyset$ . This can be seen from the following inequalities,
$$
\left| M _ {n} \right| \leq \left(1 - \frac {\log m}{n}\right) ^ {n} \cdot m = \left(1 - \frac {\log m}{n}\right) ^ {\frac {n}{\log m} \cdot \log m} \cdot m <   \left(\frac {1}{e}\right) ^ {\log m} \cdot m <   \frac {1}{m} \cdot m = 1.
$$
Since  $|M_n| < 1$ ,  $M_n$  must be empty, which completes the proof of the theorem.
Note that Theorem 1 does not rule out the possibility of beating the approximation ratio for specific subadditive costs. In the next two sections, we turn to studying two specific settings, where we are able to beat the lower bounds in Theorem 1 and Corollary 1 by designing algorithms that can guarantee constant ordinal and multiplicative approximations of MMS. We will mostly consider the ordinal approximation of MMS. By Observation 1, the ordinal approximation gives a result of the multiplicative one, which can be improved by slightly modifying the designed algorithms.
4 Bin packing setting
4.1 Model
The first setting encodes a bin packing problem where the items have sizes and need to be packed into bins by the agents. The items may be of different sizes to different agents. Specifically, each item  $e_j \in M$  has size  $s_{i,j} \geq 0$  to each agent  $i \in N$ . For a set of items  $S$ ,  $s_i(S) = \sum_{e_j \in S} s_{i,j}$ . Each agent  $i \in N$  has unlimited number of bins with the same capacity  $c_i$ . Without loss of generality, we assume that  $c_1 \geq \dots \geq c_n$  and  $c_i \geq \max_{e_j \in M} s_{i,j}$  for all  $i \in N$ .
Upon receiving a set of items  $S \subseteq M$ , agent  $i$ 's cost  $v_{i}(S)^{2}$  is determined by the minimum number of bins (with capacity  $c_{i}$ ) that can pack all items in  $S$ . Note that the calculation of  $v_{i}(S)$  involves solving a classic bin packing problem which is NP-hard. For any two sets  $S_{1}$  and  $S_{2}$ ,  $v_{i}(S_{1} \cup S_{2}) \leq v_{i}(S_{1}) + v_{i}(S_{2})$  since the optimal packing of  $S_{1} \cup S_{2}$  is no worse than packing  $S_{1}$  and  $S_{2}$  separately and thus  $v_{i}(\cdot)$  is subadditive. Accordingly,  $\mathsf{MMS}_i^d$  is essentially the minimum number  $k_{i}$  such that the items can be partitioned into  $d$  bundles and the items in each bundle can be packed into no more than  $k_{i}$  bins. The definition of  $\mathsf{MMS}_i^d$  gives  $\mathsf{MMS}_i^d \cdot c_i \geq \frac{s_i(M)}{d}$  for all  $i \in N$ .