1 Introduction
A large body of recent work in algorithmic game theory, artificial intelligence, and computational social choice has been directed towards understanding the problem of allocating indivisible goods among agents in “fair” manner; see, e.g.,
[BCE16] and [End17] for excellent expositions. This recent focus on indivisible goods is motivated, in part, by applications (such as division of inheritance and partitioning computational resources in a cloud computing environment) which inherently entail allocation of goods/resources that cannot be fractionally allocated. In fact, algorithms developed for finding fair allocations of indivisible goods have been implemented in specific settings; for instance, Course Match [BCKO16] is employed for course allocation at the Wharton School in the University of Pennsylvania and the website Spliddit (www.spliddit.org) [GP14] provides free, online access to fair division algorithms.Note that, though the theory of fair division is extensive, classical notions of fairness—such as envy freeness^{1}^{1}1An allocation is said to be envyfree if each agent values her bundle at least as much as she values any other agent’s bundle[Fol67, Var74].—typically address allocation of divisible goods (i.e., goods which can be fractionally allocated) and are not representative in the indivisible setting. For instance, while an envyfree allocation of divisible goods is guaranteed to exist [Str80], such an existence result does not hold for indivisible goods.^{2}^{2}2If we have a single indivisible good and two agents, then in any allocation, the losing agent is bound to be envious.
Motivated by these considerations, recent results have formulated and studied solution concepts to address fair division of indivisible goods [Bud11, PW14, BL14]. Arguably, the two most prominent notions of fairness in this context are (i) envy freeness up to one good () and (ii) the maximin share guarantee (). These solution concepts were defined by Budish [Bud11] and they, respectively, provide a cogent analogue of envyfreeness and proportionality^{3}^{3}3An allocation is said to be proportionally fair among agents, if every agent gets a bundle of value at least times her value for the entire set of goods. in the context of indivisible goods:

[label=()]

An allocation is said to be if every agent values her bundle at least as much as any other agent’s bundle, up to the removal of the most valuable good from the other agent’s bundle. allocations are guaranteed to exist; by contrast, for indivisible goods envyfree allocations might not exist. Another attractive feature of is that it is computationally tractable: even under combinatorial valuations, allocations can be found efficiently [LMMS04]. Furthermore, under additive valuations, this notion of fairness is compatible with Pareto efficiency [CKM16].

An allocation is said to satisfy if each agent receives a bundle of value at least as much as her maximin share. These shares provide an agentspecific fairness threshold, and are defined as the maximum value that an agent can guarantee for herself if she were to partition the set of goods into bundles and then, from those bundles, receive the minimum valued one; here, is the total number of agents. That is, maximin share can be interpreted via an application of the standard cutandchoose protocol over indivisible goods: if agent is (hypothetically) asked to partition the set of goods into bundles and the remaining agents were to select their bundles before , then a riskaverse agent would find a partition which maximizes the least valued bundle. Overall, this value that the agent can guarantee for herself is called the maximin share of . Even though allocations are not guaranteed to exist [PW14, KPW16], this notion admits efficient approximation guarantees. Specifically, under additive valuations, there exist polynomialtime algorithms for finding allocations wherein each agent receives a bundle of value at least times her maximin share [PW14, AMNS15, BK17].^{4}^{4}4For additive valuations, Ghodsi et al. [GHS17] provide an improved approximation guarantee of . Note that these results establish an absolute guarantee, i.e., they show that an approximately maximin fair allocation always exists.
The existence and computational results developed for and provide a sound understanding of fair division of indivisible goods. However, it is relevant to note that the vast majority of work in this thread of research is solely focussed on the unconstrained version of the problem.^{5}^{5}5The work of Bouveret et al. [BCE17] along with [GM17] and [GMT14] are notable exceptions. These results are discussed in Section 1.1. To address this limitation, and motivated by the fact that in many realworld settings the allocations are required to satisfy certain criteria, we study a relevant, constrained version of the fair division problem.
In particular, we consider a setting wherein the indivisible goods are categorized (i.e., we are given a partition of the set of goods) and a limit is specified on the number of goods that can be allocated from each category to any agent. Here, the objective is to find a fair allocation in which the subset of goods assigned to any agent satisfies the given cardinality constraints. We shall see that this corresponds to a fair allocation problem under a partition matroid constraint.
The following stylized example—adapted from [GMT14]—demonstrates the applicability of such constraints: A museum decides to open new branches, and thereby needs to transfer some of the exhibits from the main museum to the newly opened ones. The exhibits are categorized into, say, statues, paintings, and pottery. In addition, there is an upper limit on the number of exhibits that every newly opened branch can accommodate from each category. The question now is to find a feasible division of the exhibits which is fair to the curators of each of the new branches.
Our Contributions: We show that the existence and algorithmic guarantees established for and allocations in the unconstrained setting can, in fact, be achieved under cardinality constraints as well. Specifically, we establish the following results under additive valuations and cardinality (partition matroid) constraints:

allocations are guaranteed to exist. In particular, we develop a combinatorial algorithm which, for a given fair division instance with additive valuations and cardinality constraints, finds an allocation in polynomial time (Theorem 1).

In this constrained setting, a constantfactor approximate allocation always exists and can be computed in polynomial time (Theorem 2). Note that, in this setting, the value of the maximin share of each of the agents is obtained by considering only feasible allocations (see Equation (1) in Section 2). That is, here, the maximin share of an agent is defined to be maximum value that she can guarantee for herself if she were to partition the set of goods into bundles, each of which must satisfy the cardinality constraints, and from them receive the minimum valued one.

Specific extensions of —in particular, envyfree up to the least valued good ()^{6}^{6}6An allocation is said to be envyfree up to the least valued good () if no agent envies any other agent after removing any positively valued good from the other agent’s bundle [CKM16]. and envyfree up to one leastpreferred good ()^{7}^{7}7See Section 5 for a definition.—are not guaranteed to exist under cardinality constraints. In Section 5 we provide a (constrained) fair division instance which does not admit an or an allocation.
1.1 Related Work
The fairness notions of and were defined by Budish [Bud11] (see also [Mou90]), and these solution concepts have been extensively studied since then. The existence of allocations can be established via the cycleelimination algorithm of Lipton et al. [LMMS04]. Caragiannis et al. [CKM16] have shown that these two solution concepts are incomparable, i.e., one does not imply the other.
Fair division instances wherein the agents’ valuations are binary and additive always admit allocations [BL14]. However, Procaccia and Wang [PW14] along with Kurokawa et al. [KPW16] provided intricate counterexamples to refute the universal existence of allocations, even under additive valuations. This lead to a study of approximate maximin share allocations, , where each agent receives a bundle whose value to her is at least times her maximin share. Procaccia and Wang [PW14] developed an efficient algorithm to obtain a  allocation when the number of agents is constant. Later, Amanatidis et al. [AMNS15] established that a  allocation can be computed in polynomial (even in the number of players) time.
More recently, constantfactor approximation guarantees for in settings wherein the valuations are not necessarily additive have been established in [BK17] and [GHS17]. Specifically, for submodular valuations Ghodsi et al. [GHS17] have developed an efficient algorithm for computing  allocations.
As mentioned previously, the work on fair division of indivisible goods is primarily confined to the unconstrained setting. Exceptions include the work of Bouveret et al. [BCE17] and Ferraioli et al. [FGM14]. Bouveret et al. [BCE17] consider fair division of goods which correspond to vertices of a given graph, and the problem is to fairly allocate a connected subgraph to each agent. It is shown in [BCE17] that allocations might not exist for general graphs; however, it can be efficiently computed when the underlying graph is a tree.
Ferraioli et al. [FGM14] consider fair division problems where each agent must receive exactly goods, for a given integer , and provide an algorithm to efficiently compute a approximate allocation. A different kind of matroid constraint is considered in [GM17] and [GMT14]. In particular, the problem formulation in [GM17] and [GMT14] requires that the union of all the allocated goods is an independent set of a given matroid. This setup is incomparable to the one considered in this paper. Specifically, while we ensure that in the computed fair allocation each agent’s bundle satisfies the partition matroid constraint and all the goods are allocated, these requirements are not imposed in [GM17] and [GMT14].
2 Notation and Preliminaries
An instance of the fair division problem comprises of a tuple , where denotes the set of indivisible goods, denotes the set of agents, and s specify the valuation (preferences) of the agents, , over the set of goods. Throughout, we will assume that, for each agent , the valuation is additive, i.e., for all agents and subsets , . Also, we have for all and . For ease of notation, we will use , instead of , to denote the valuation of agent for a good .
Write to denote the set of all partitions of a subset of goods . An allocation, , refers to an partition , where is the subset of goods (bundle) allocated to agent . In this work we focus on finding fair allocations which satisfy given cardinality constraints. Specifically, we are given a partition of the set of goods consisting of different “categories” , and associated with each category we have a (cardinality) threshold . In this setup, an allocation is said to be feasible iff, for every bundle and category , the cardinality constraint holds: . Throughout, we will use to denote the set of feasible allocations, . To ensure that is nonempty, we require that the threshold for all categories .
We will use to denote an instance of the fair division problem subject to cardinality constraints. Overall, our goal is to find fair allocations contained in .^{8}^{8}8Note that the set might be exponential in size, but it is specified in an efficient manner via the partition and thresholds . Setting =, =, =, we get =. Hence, this formulation is a strict generalization of the unconstrained fair division problem. In this work, we provide existential and algorithmic results for the following fairness notions:

Envyfree up to one good (): In a fair division instance , an allocation is said to be iff for every pair of agents there exists a good such that .

Maximin Share Guarantee (): Given an instance , the (constrained) maximin share of agent is defined as
(1) An allocation is said satisfy iff for all agents , we have . Since allocations are not guaranteed to exist, the objective is to find feasible allocations wherein each agent gets a bundle of value at least times ; with factor being as large as possible. We call such allocations .
3 Main Results
The key results established in this paper are:
Theorem 1.
Given any fair division instance with additive valuations and cardinality constraints (), there exists a polynomial time algorithm for finding a feasible allocation.
This theorem is established in Section 4 and it implies that as long as the set of feasible allocations is nonempty it admits an (i.e., a fair) allocation.
Theorem 2.
Given any fair division instance with cardinality constraints () and additive valuations, a  allocation can be computed in polynomial time.
Analogous to the case, this theorem provides an absolute, existence guarantee for approximate maximin fair allocations under cardinality constraints. A proof of this result appears in Section 6.
4 Allocations Under Cardinality Constraints: Proof of Theorem 1
In the unconstrained setting, there exist efficient algorithms for finding allocations; see, e.g., the cycleelimination algorithm of Lipton et al. [LMMS04] and the roundrobin method by Caragiannis et al. [CKM16]. However, the allocations found by these algorithms are not guaranteed to satisfy the given cardinality constraints. We bypass this issue by developing a polynomialtime algorithm, Alg 1, for finding an allocation which is not only , but also feasible, i.e., .
Alg 1 is based on an interesting modification of the roundrobin algorithm: initially, Alg 1 selects an arbitrary order (permutation) over the agents . It then picks an unallocated category and executes the GreedyRoundRobin algorithm (Alg 2) with the agents, goods (from category ), and the selected order .
Alg 2 follows the ordering in a roundrobin fashion (i.e., it selects agents, one after the other, from to ), and iteratively assigns to the selected agent an unallocated good from that she desires the most. Finally, it returns an allocation .
After allocating all the goods of a category , Step of Alg 1 creates an envy graph^{9}^{9}9An envy graph, for an allocation , is a directed graph that captures the envy between agents in . Specifically, the nodes in the envy graph represent the agents and it contains a directed edge from to iff envies , i.e., iff . . It was established in [LMMS04] that one can always efficiently update a given partial allocation such that the resulting envy graph is acyclic:
Lemma 1.
[LMMS04]
Given a partial allocation of a subset of goods , we can find another partial allocation = of in polynomial time such that
(i) The valuations of the agents for their bundles do not decrease: for all .
(ii) The envy graph is acyclic.
Finally, is updated to be a topological ordering of the acyclic directed graph . This new ordering is then used for the next category of goods.
The feasibility of the computed allocation, , directly follows from the fact that (for each ) Alg 2 distributes the goods evenly among agents. In particular, the roundrobin nature of Alg 2 ensures that, , the set of goods allocated to agent from category satisfies:^{10}^{10}10Recall that the underlying feasible set of allocations is nonempty iff the integer limit . .
In Lemma 2 we show that, for each , the partial allocation obtained after the allocating the first categories, , is . Since Algorithm 1 allocates all the goods we get that the final allocation, , is in fact as well.
This, along with the observation that Alg 1 runs in polynomial time completes the proof of Theorem 1.
Next, we establish a proposition which will be used in the proof of Lemma 2.
Proposition 1.
Given any fair division instance with additive valuations and an ordering of agents , the allocation obtained by Alg 2 satisfies the following properties:
For any two indices , the agent does not envy agent , i.e., .
is .
Proof.
Write to denote the total number of rounds of Alg 2. In each round, for , agent gets to choose her most desired good among the unallocated goods before agent . Hence, if and denote the good assigned to agent and , respectively, in the th round, then for all . Since the valuations are additive, the stated property holds: .
It is known that the roundrobin algorithm results in an allocation [CKM16], we repeat the argument for completeness: If index is less than , then agent does not envy agent . On the other hand, even if the good allocated to agent in the th round is of value (under ) no less than the good allocated to agent in the th round: for all . Summing we get, . Thus, the allocation is . ∎
Lemma 2.
Alg 1 returns an allocation.
Proof.
We will show inductively that, for each , the partial allocation obtained after allocating the first categories, , is . Hence, the returned allocation, is as well.
Here, the base case () follows from Proposition 1; since and the proposition ensures that is .
By the induction hypothesis we have that is and, by construction, the corresponding envy graph is acyclic. Next, we will show that this continues to hold for the next category . Note that the ordering of agents for (executing Alg 2 over) the category is obtained by topologically sorting . Let be that topological ordering and write as the index of agent according to the ordering .
Now, if agent envies in , then there is a directed edge from to in and, hence, . In this case, Proposition 1 ensures that does not envy in , i.e., we have . The fact that is gives us , for some . Summing the last two inequalities and noting that and we get that is with respect to agent and .
The complementary case wherein does not envy in is analogous, since is guaranteed to be in itself. Overall, this establishes the stated claim, that the final allocation is . ∎
Lemma 2 along with the feasibility argument mentioned above (and the direct observation that Alg 1 runs in polynomial time) proves Theorem 1, which we restate below.
See 1
5 NonExistence of and Allocations Under Cardinality Constraints
This section considers comparative notions of fairness which are stronger than . In particular, we show that allocations which satisfy envyfreeness up to the least valued good () are not guaranteed to exist under cardinality constraints.^{11}^{11}11It is relevant to note that the existence of allocations in unconstrained, fair division instances (with additive valuations) remains an interesting, open question [CKM16]. The nonexistence result also holds for envyfree up to one lesspreferred good () allocations; defined in [BBKN18], this solution concept is weaker than , but stronger than . Formally,

Envyfree up to the least valued good () [CKM16]: An allocation is said to be iff for every pair of agents and for all goods (i.e., for all goods in which are positively valued by agent ) we have .

Envyfree up to one lesspreferred good () [BBKN18]: An allocation is said to be iff for every pair of agents at least one of the following conditions hold:

contains at most one good which is positively valued by , i.e.,

There exists a good such that and .

Note that, by definition, an allocation is necessarily and, similarly, an allocation is always .
The universal existence of allocations in unconstrained fair division instances (with additive valuations) was established in [BBKN18]. By contrast, the following example shows that, under cardinality constrains, allocations do not always exist. This also implies the nonexistence of under cardinality constraints.
Example 1.
Consider an instance with agents and goods. Here, the bundle of each agent is constrained to contain at most goods. That is, we have a (uniform matroid) single category , which contains all the goods, and the threshold .
The valuations of both the agents are identical and as follows: and . In this example, no feasible allocation will ensure .
In this example any feasible allocation will allocate exactly two goods to each agent. Therefore, one of the agents, say , will receive a bundle of value (containing the highvalued good and a lowvalued good), while the other agent, , will receive a bundle of value (containing two lowvalued goods). Since agent envies , can be ensured only if there exists a good of value at most in ’s bundle which, when removed, eliminates ’s envy. However, such a good does not exist in ’s bundle and, hence, no feasible allocation is .
6 Under Cardinality Constraints: Proof of Theorem 2
To obtain an approximate maximin share allocation () under cardinality constraints we define a nonnegative, monotone, submodular function , for each agent , which satisfies
Hence, a fair division problem under additive valuations and cardinality constraints reduces to an unconstrained fair division problem under monotone, submodular valuations . Recall that a function is said to be monotone and submodular iff for all subsets and , we have and .
We define the function as follows where,
Here, denotes the set of the most valued (by agent ) goods contained in . The following lemma asserts that s are submodular.
Lemma 3.
For each agent the function (defined above) is monotone, nonnegative, and submodular.
Proof.
Since, for each agent , the valuation is nonnegative, the function is nonnegative as well. Furthermore, for any agent and category , the function is monotone; it either increases or remains constant after a good is included in any bundle . Hence, is monotone.
Next, we will show that for any agent the function is submodular, i.e., it satisfies the following inequality, for any two subsets and good .
(2) 
Write to denote the category of the good , i.e., . Since including in the subsets only effects the value of , we have and .
Note that if , then we have and . Therefore, if the cardinality of is strictly less that , then the inequality (2) is in fact tight. Otherwise, if , then we have the following two cases

Case 1: . Here, we have . Furthermore, the monotonicity of function ensures that . Therefore, inequality (2) holds.
∎
Recall that the set of feasible allocations is nonempty if and only if the cardinality thresholds for all categories . Assuming nonempty set of feasible allocations, we now establish that the maximin value over all feasible allocations using valuations is equivalent to the maximin value over all possible partitions of using the function .
Lemma 4.
If for all the categories , the cardinality threshold satisfies (i.e., if the set of feasible allocations is nonempty), then the following equality holds for each agent :
Proof.
We fix an agent and first show that the lefthand side of the stated equality is upper bounded by the righthand side. Write . Since for all and , we have for all . Now, the inequality establishes the upper bound.
We complete the proof by showing that an inequality holds in the other direction as well. Write . Say , then there exists an index and such that . Since , an averaging argument implies that there exists another index for which . Now, consider the lowest valued (by agent ) good . Note that and . Hence, we can iteratively perform such swaps till all the cardinality constraints are satisfied. That is, we can obtain an allocation, say , which is feasible and satisfies for all . The feasibility of ensures that the following equality holds for all : . Therefore, . Hence, the lefthand side of the equality stated in the lemma is at least as much as the righthand side. ∎
Under submodular valuations a  allocation can be computed efficiently in the unconstrained setting [GHS17]. Therefore, for an unconstrained fair division instance over the goods and with submodular valuations of the agents as , we can, in polynomial time, find an allocation which is . Employing a swap argument— similar to the one used in the proof of Lemma 4—we can efficiently convert into a feasible allocation which satisfies , for all . That is, for the unconstrained instance, is a  allocation as well. In addition, the feasibility of implies that , for all . Overall, via Lemma 4 (i.e., the fact that the maximin shares of each agent in the constructed unconstrained instance is equal to the underlying value), we get that is a feasible,  allocation for the constrained instance. This completes the proof of Theorem 2, which is restated below.
See 2
7 Identical Valuations and Matroid Constraint
This section shows that if the additive valuations of the agents are identical (i.e., for all ), then an allocation is guaranteed to exist even under a matroid constraint.
Matroids have been studied extensively in mathematics and computer science; see, e.g. [Oxl92]. These structures provide an encompassing framework for representing combinatorial constraints; in particular, the cardinality constraints considered in the previous sections correspond to a particular matroid, called the partition matroid. Formally, a matroid is defined as a pair where is the ground set of elements and —referred to as independent sets—is a nonempty collection of subsets of that satisfies: (i) Hereditary property: If and , then , and (ii) Independent Set Exchange: If and , then there exist an element such that .
We consider a fair division instance where denotes the set of all allocations which satisfy the underlying matroid constraint, i.e., allocations whose constituent bundles are independent, for all . The main result of this section is as follows
Theorem 3.
Every fair division instance under additive, identical valuations and matroid constraint, , admits an allocation.
Proof.
We establish the result by showing that an allocation which maximizes the Nash Social Welfare over the set is necessarily . For an allocation , the Nash Social Welfare (
) is defined as the geometric mean of the agents’ valuations,
.Let denote identical, additive valuations of the agents. We assume for all and consider an optimal allocation , which satisfies .^{12}^{12}12If the optimal over is zero, then it must be the case that we have less than goods. For such an instance, an allocation wherein each agent gets at most one good is both feasible and . We will prove that is . Since , the stated claim follows.
Say, for contradiction, that is not an allocation, then we will show that there exists another allocation along with agents and , such that , for all other agents , and . The last inequality and the fact that imply . Therefore, we get , which contradicts the optimality of .
By definition, if is not , then there exists a pair of agents such that
Note that if there exists a good in such that is independent (i.e., ), then swapping from to gives us the desired allocation (with a strictly higher than ). Hence, we analyze the case in which no such good exists. In particular, we have for all . This condition implies that ; otherwise, the Independent Set Exchange property of matroids would ensure the existence of a good such that .
Write . The independence of and along with the inequality imply that there exist componentwise distinct pairs of goods such that and are independent for all . This fact is established in [Goe09] for the case wherein . Lemma 5 (proved below) complements this result and guarantees the existence of the relevant componentwise distinct pairs of goods, , even when . Next, we will complete the proof of the theorem using the pairs .
Since the pairs in are distinct, and . The envy between agent and (i.e., ) implies that there exists index for which
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