1 Introduction
The problem of fair division was formally introduced by Steinhaus [32], and has since been extensively studied in economics and computer science [10, 28]. Recent work has focused on the problem of fair and efficient allocation of indivisible private goods. We label this setting as the model. Here, goods have to be partitioned among agents, and a good provides utility only to the agent who owns it. However, goods are not always private, and may provide utility to multiple agents simultaneously, e.g., books in a public library. The fair and efficient allocation of such indivisible public goods is an important problem.
In this paper we study the setting of , where a set of agents have to select a set of at most goods from a set of given goods. This simple cardinality constraint models several real world scenarios. While previous work has largely focused on the case, e.g., for voting and committee selection [2, 13], there is much less work available for the case of . This setting is important in its own right. We present a few compelling examples.
Example 1.
A public library wants to buy books that adhere to preferences of people who might use the library. Clearly, the number of books has to be much greater than the number of people using the library, hence .
Example 2.
A family (or a group of friends) wants to decide on a list of movies to watch together for a few months. Here too, . Another example of the same flavor is a committee tasked with inviting speakers at a yearlong weekly seminar.
Example 3.
Another important example is that of diverse search results for a query. Given a query (say of “computer scientist images”) on a database, we would like to output search results which reflect diversity in terms of specified features (like “gender, race and nationality”). Once again, .
A related setting of public decision making [15] models the scenario in which agents are faced with issues with multiple alternatives per issue, and they must arrive at a decision on each issue. Conitzer et al. [15] showed that this model subsumes the setting.
Connections between the models.
A central question motivating this work is:
Question 1.
Can we establish fundamental connections between the three models , , and ?
To answer this question, we first describe two wellstudied solution concepts for allocating goods in the and models, namely the maximum Nash welfare (MNW) and leximin mechanisms. These mechanisms have been shown to produce allocations that are fair and efficient in the models of and
. The MNW mechanism returns an allocation that maximizes the geometric mean of agents’ utilities, and the leximin mechanism returns an allocation that maximizes the minimum utility, and subject to this, maximizes the second minimum utility, and so on. We label the problems of computing the Nash welfare maximizing (resp. leximin optimal) allocation in the three models as
(resp. ).We answer Question 1 positively by presenting novel polynomialtime reductions from the model of to , and from to for the problem of computing a Nash welfare maximizing allocation.
(1) 
More notably, these reductions also work for the MNW problem when restricted to binary valuations. Apart from establishing fundamental connections between these models, our reductions also determine the complexity of the MNW problem, as we detail below. We also develop similar reductions between the models for the leximin mechanism, showing:
(2) 
Fairness and efficiency considerations.
We next describe the fairness and efficiency properties that the MNW and leximin mechanisms have been shown to satisfy in the and models.
The standard notion of economic efficiency is Paretooptimality (PO). An allocation is said to be PO if no other allocation makes an agent better off without making anyone worse off. The classical fairness notion of proportionality requires that every agent gets her proportional value, i.e., fraction of the maximum value she can obtain in any allocation. However, proportional allocations are not guaranteed to exist.^{4}^{4}4Consider for example, two agents and and six public goods . Agent has value for and has value for . All other valuations are . Suppose we want to select three of these goods. The proportional share of both agents is . However, in any allocation, the value of at least one agent is at most , implying that proportional allocations need not exist. Hence, we study the notion of Proportionality up to one good (Prop1) for . We say an allocation is Prop1 if for every agent who does not get her proportional value, gets her proportional value after swapping some unselected good with a selected one. For and , Prop1 is defined similarly – in the former, an agent is given an additional good [8, 26]; and in the latter, an agent is allowed to change the decision on a single issue [15]. While Prop1 is an individual fairness notion, it is still important for allocating public goods. For instance, in Example 1, we want allocations in which every agent has some books that cater to her taste, even if her taste differs from the rest of the agents. Likewise, in Example 2, a fair selection of movies must ensure that there are some movies every member can enjoy. We also consider the fairness notion of RoundRobin Share (RRS) [15], which demands that each agent receives at least the utility which she would get if agents were allowed to pick goods in a roundrobin fashion, with picking last.
In the and models, an MNW allocation satisfies Prop1 in conjunction with PO [11, 15]. Similarly in both these models, the leximinoptimal allocation satisfies RRS and PO [15]. It is therefore natural to ask:
Question 2.
What guarantee of fairness and efficiency do the MNW and leximin mechanisms provide in the model?
Answering this question, we show that an MNW allocation satisfies Prop1, approximation to RRS, and is PO. Further, a leximinoptimal allocation satisfies RRS, Prop1 and PO.
Complexity of computing MNW and leximinoptimal allocations.
Given the desirable fairness and efficiency properties of these mechanisms, we investigate the complexity of computing MNW and leximinoptimal allocations in the model. It is known that is hard [25] (hard to approximate) and [15] is hard. Likewise, too is hard [9]. Therefore, we ask:
Question 3.
What is the complexity of and ?
Since and are known to be hard, our reductions (1) and (2) immediately show that and are hard. However, we show stronger results that and remain hard even when the valuations are binary. These results are in stark contrast to the case, which admits polynomialtime algorithms for binary valuations [16, 20]. Further, our reductions between and also directly enable us to show NPhardness of and . Moreover, a feature of our reductions (Observation 6) enables us to shows that is hard even for binary valuations, highlighting the utility of our reductions. We also show that and remain hard even when there are only two agents. We note that for the case of two agents, the hardness of and does not imply hardness of and because our reductions between the models do not preserve the number of agents. We summarize our results in Table 1.
Problem  

MNW valuations  [7, 16]  hard (Theorem 16)  hard (Corollary 25) 
Leximin valuations  [7, 16]  hard (Theorem 22)  ? 
MNW two agents  hard  hard (Theorem 21)  ? 
Leximin two agents  hard  hard (Theorem 24)  ? 
In light of the above computational hardness, we turn to approximation algorithms and exact algorithms for special cases. We design a polynomialtime algorithm that returns an allocation which approximates the MNW to a factor when , and is also Prop1 and satisfies RRS. Finally, we obtain pseudopolynomial time algorithms for computing MNW and leximinoptimal allocations for constant . These are essentially tight in light of the hardness for constant .
1.1 Other related work
Maximum Nash welfare. The problem of approximating maximum Nash welfare for private goods is wellstudied, see e.g., [14, 6, 12, 21]. [18] showed that the MNW problem is NPhard for allocating public goods subject to matroid or packing constraints. It has also been studied in the context of voting, or multiwinner elections [1]. Fluschnik et al. [19] studied the fair multiagent knapsack problem, wherein each good has an associated budget, and a set of goods is to be selected subject to a budget constraint. In this context, they studied the objective of maximizing the geometric mean of where is the utility of the agent. They showed that maximizing this objective is hard, even for binary valuations or constantly many agents with equal budgets and presented a pseudopolynomial time algorithm for constant .
Leximin. Leximin was developed as a fairness notion in itself [30]. Plaut and Roughgarden [29] showed that for private goods, leximin can be used to construct allocations that are envyfree up to any good. Freeman et al. [20] showed that in the model the MNW and leximinoptimal allocations coincide when valuations are binary.
Core.
Core is a strong property that enforces both PO and proportionalitylike fairness guarantees for all subsets of agents. It is wellstudied in many settings, including game theory and computer science
[31, 24]. The core of indivisible public goods might be empty. Fain et al. [18] proved that under matroid constraints, a additive approximation to core exists. On an individual fairness level, 1additive core is weaker than Prop1 [18].Participatory Budgeting. The participatory budgeting problem [3, 4] consists of a set of agents (or voters), and a set of projects that require funds, a total available budget, and the preferences of the voters over the projects. The problem is to allocate the budget in a fair and efficient manner. Here typically . Fain et al. [17] showed that the fractional core outcome is polynomialtime computable. This could be modeled as a public goods problem with goods as the projects.
2 Notation and Preliminaries
Problem setting.
For , let denote . An instance of the allocation problem is given by a tuple of a set of agents, a set of public goods, an integer , and a set of valuation functions , one per agent, where each . Unless specified, we assume that . For a subset of goods , denotes the utility agent derives from the goods in . Unless specified, we assume the valuations are additive. In this case, each is specified by nonnegative integers , where denotes the value of agent for good . Then for , . We assume without loss of generality that for every agent , there is at least one good with . For brevity, we write in place of for a set . An allocation is a subset of goods which satisfies the cardinality constraint .
Nash welfare.
The Nash welfare (NW) of an allocation is given by An allocation with the maximum NW is called an MNW allocation or a Nash optimal allocation.^{5}^{5}5If the NW is 0 for all allocations, MNW allocations are defined as those which give nonzero utility to maximum number of agents, and then maximize the product of utilities for those agents. Note if , every agent positively values at least one good and thus MNW . We also refer to the product of the agents’ utilities as the Nash product. An allocation approximates MNW to a factor of if , where is an MNW allocation.
Leximin.
Given an allocation , let
denote the vector of agent’s utilities under
, sorted in nondecreasing order. For two allocations , we say leximindominates if there exists such that and . An allocation is leximinoptimal if no other allocation leximindominates it.Fairness notions.
We now discuss fairness notions for the setting. The proportional share of an agent , denoted by is a share of the maximum value she can obtain from any allocation. Formally:
The roundrobin share of agent , denoted by , is the minimum value an agent can be guaranteed if the agents pick goods in a roundrobin fashion, with picking last. Therefore, this value equals the maximum value of any sized subset. Formally:
For , an allocation is said to satisfy:

Proportionality (Prop) if , ;

Proportionality up to one good (Prop1) if , , such that

 if for all agents , .
Due to the cardinality constraints in the model, the notion of Prop1 requires that for every agent, there is a way to swap one preferred unpicked good with one picked good, after which the agent gets her proportional share. Since Prop1 in requires only giving an extra good, this makes the definition of Prop1 in slightly more demanding than that in .
Paretooptimality.
An allocation is said to Paretodominate an allocation if for all agents , , with at least one of the inequalities being strict. We say is Paretooptimal (PO) if there is no allocation that Paretodominates .
Related models.

. The classic problem of private goods allocation concerns partitioning a set of goods among the set of agents. Thus, a feasible allocation is an partition of , where agent is allotted , and derives utility only from .

. In this model, a set of agents are required to make decisions on a set of issues. Each issue has a set of alternatives, given by . A feasible allocation or outcome comprises of decisions, where is the decision made on issue . Assuming the valuations are additive, each agent has a value for the alternative of issue . The valuation of the agent for the outcome is then .
3 Relating the models
We first show rigorous mathematical connections between the and models w.r.t. computing optimal MNW and leximin allocations.
Theorem 4.
polynomialtime reduces to .
Proof.
Let be an instance of the model. For , the MNW problem is trivial, since we can select all the goods. For , we can construct an instance of from in polynomial time, such that given an MNW allocation of , we can compute an MNW allocation of in polynomial time. Let . We create public issues: corresponding to each good , we create an issue with two alternatives and . That is, , and for . We create , where . The first agents here correspond to the agents in . The last agents are of two types: agents of type , and agents of type . The valuations are as follows: each agent values alternative ‘’ of the issue at , the agents of type value only alternative ‘’, agents of type value only alternative ‘’. Formally, for , and an alternative of the issue , where :
Let be an allocation for the instance . For , let be the set of issues with decision in . That is, . Let . Then we have:
We now relate to the instance . The decision corresponds to selecting the public good . Let be the corresponding set of public goods. Then for any we have that , since for every . Thus:
(3) 
We now have to prove that satisfies . Let be the Nash product of any MNW allocation for the instance , . Clearly, . As , , since we assume every agent has at least one good that she values positively. Define , as . Then if is an MNW allocation for , (3) becomes:
(4) 
Let and denote the largest and secondlargest values that attains over its domain. We observe that increases in , and decreases in . Hence, implying:
We now show that that:
Claim 1.
.
Proof.
Recall that denotes the Nash product of any MNW allocation for the instance , for . We have , and we assume . Recall that function , was defined as .
Let and denote the largest and secondlargest values that attains over its domain. We observe that increases in , and decreases in . Hence:
Now observe that for :
and for :
using standard properties of logarithms. Thus:
Then we have by recalling that ,
which gives:
as required. Lastly, we consider the cases of and . In both cases, , which gives , as claimed. ∎
Using Claim 1 we have for all :
Hence, the quantity is maximized when . Recalling (4), we conclude that for the MNW allocation of , the corresponding set has cardinality exactly . Further also maximizes the NW among all allocations of the instance satisfying this cardinality constraint. Thus, in fact is an MNW allocation for . Finally, it is clear that this is a polynomial time reduction. ∎
We next relate the MNW problem in the model with the model.
Theorem 5.
polynomialtime reduces to .
Proof.
Let be a instance, using which we create a instance as follows. We create agents, i.e. . The first agents correspond to the agents in . The last are dummy agents. We create public goods: for each good , we create a set of copies , . We set . The valuations for , are:
i.e. each agent values exactly one copy, for each at , and for each good , there are exactly two dummy agents who value all copies of .
We now state and use the following claim, and prove it immediately after the proof of Theorem 8.
Claim 2.
Any MNW allocation of does not select two goods from same .
Consider any MNW allocation of . We construct a partition, of goods for from this in the following way. For , , define if , and 0 otherwise. Let . Thus, the value that agent gets in is
Thus, if , and the partition corresponding to as defined above gives an MNW solution for . On the other hand, if , then already gives nonzero value to all dummy agents by Claim 2. Thus, to maximize the total number of agents who get nonzero value, it maximizes the number of agents in who get nonzero value. Call this set . Thus partition has maximum number of agents getting a nonzero value. Finally, it maximizes the Nash product over . Claim 2 also implies that all dummy agents get value . Thus, . Thus even in this case the allocation corresponds to an MNW allocation in . ∎
Proof of Claim 2.
Consider first . Suppose for which two goods . Since exactly goods are picked in , there is some , for which no good is picked in for any . This implies that the agents get zero value in , making . However, choosing a good from each gives nonzero value to all dummy agents. At the same time, since , these goods can be chosen so that they give nonzero value to distinct agents in . This makes contradicting Nash optimality of .
Now, if Nash welfare of all allocations in is . Thus, the MNW allocation is the one that maximizes the number of agents who get non zero value and then maximizes the product of values for these agents. Consider any allocation , suppose for which two goods then again for some , agents and get value 0 making . At the same time, even if has goods from all different , since , and each one item from gives value only to one agent , the even in this case. Thus, if , all allocations have Nash welfare in also. Suppose the MNW allocation, had two goods from same for some . Then, there exists a such that no good is selected from . The two goods from give value to exactly four agents  the two dummy agents and two agents who receive their copy of good . Instead, if we exchange one of these goods to a good from , we give nonzero value to at least five agents  dummy agents and at least one of the agents in . We did not change the value of any other agents in this process. Thus, we increase the number of agents who get nonzero value, contradicting the maximality of . Thus, in both cases, all goods are picked from different . ∎
Observation 6.
A desirable feature of the above reductions for the MNW problem from instance to is that , i.e., the reduction only creates instances which have 0 and 1 as the only potentially additional values as compared to . We use this feature in establishing the computational complexity of computing an MNW allocation in the model with binary values, see Corollary 25.
We also show similar polynomialtime reductions between the three models for the problem of computing a leximinoptimal allocation.
Theorem 7.
polynomialtime reduces to .
Proof.
Let be an instance of the model. For , the leximin problem is trivial, since we can select all the goods. When , we can construct an instance of the model from in polynomial time, such that given a leximin allocation of , we can compute a leximin allocation of in polynomial time. To construct , we first create a set of agents. The first agents here correspond to the agents in . The last agents are used in the construction, and ensure that exactly goods are selected in .
We next create public issues: for each good , we create an issue with two alternatives and . That is, , and for .
The valuations are as follows: for an agent , and an alternative of the issue , where :
where is a sufficiently small constant. Essentially, each agent values the ‘’ decisions of the issue at , the agent values only the ‘’ decisions, and agent values only the ‘’ decisions.
Let be a leximin allocation for the instance . Clearly for all agents, since there is some allocation that gives positive utility to all agents, and the minimum utility only improves in the leximin solution. In particular for all . For , let be the set of issues with decision in . That is, . Let . we note that , and . Since , for each , we have . Suppose . Then any allocation with gives , which is a leximin improvement over , since . Hence .
We now explain how we can relate of to the public goods instance . Intuitively, the decision corresponds to selecting the public good , and corresponds to not selecting . Let be a set of public goods of cardinality . Then for any we have that , since for every . Further since , . Hence is a feasible solution for . Since for all , , is a leximin allocation for .
Since the number of agents and goods created in the reduction are polynomially many in the size of the instance , and all other computations can also be carried out in polynomial time, this is a polynomial time leximinpreserving reduction. ∎
Theorem 8.
polynomialtime reduces to .
Proof.
The proof follows from essentially the same reduction used to show Theorem 5. ∎
4 Properties of MNW and Leximin
We prove that MNW and leximinoptimal allocations satisfy desirable fairness and efficiency properties in the model as well. First, we show some interesting relations between our three fairness notions – , and in the model where .^{6}^{6}6Note that when , is 0. Any agent who gets value satisfies when trivially. Thus, and coincide when . On the other hand, the proportional value will be nonzero even when if the agent likes at least one good. Thus, there can be no multiplicative relation between and when . Our results are presented in the table below.
✓  (Lem. 10)  ✓(Lem. 9)  
(Lem. 11)  ✓  ✓  
✗(Ex. 13)  ✗(Ex. 13)  ✓ 
Lemma 9.
Any allocation that satisfies also satisfies .
Proof.
Fix any agent . Let be any allocation that satisfies . Let denote the top goods for agent . We assume that the goods both in and are ordered in decreasing order of valuations according to agent . Now, suppose that top goods of match with top goods of , i.e. and . Note that since is the top goods of agent , we cannot have that for any . We want to prove that implies . If was already satisfying proportionality, it is obvious that is . If , it is again easy to see that is . This is because, if then we already have top goods, giving a proportional allocation. If , then we can remove any good from and exchange it with to ensure proportionality, making the original allocation . Finally, if divides then we have proportionality implied by from Lemma 10.
Thus, we now assume that , with and that is not already a proportional allocation. We know that and . Thus,
(5) 
Now, . Thus,