1 Introduction
A research department has to choose members for a recruiting committee. A selected committee should be gender balanced, ideally containing 50% of male and 50% of female. Additionally, a committee should represent different research areas in certain proportions: ideally it should contain 55% of researchers specializing in area , 25% of experts in area , and 20% in area . Another requirement is that the committee should contain 30% junior and 70% senior researchers, and finally, the repartition between local and external members should be kept in proportions 30% to 70 %. The pool of possible members is the following:
Name  Sex  Group  Age  Affiliation 

Ann  
Bob  
Charlie  
Donna  
Ernest  
George  
Helena  
John  
Kevin  
Laura 
In the given example, if the department wants to select members, then it is easy to see that there exists no such committee that would ideally satisfy all the criteria. Nevertheless, some committees are better than others: intuitively we feel the sex ratio should be either equal to 2:1 or to 1:2, the area ratio should be equal to 2:1:0, the age ratio to 1:2, and the affiliation ratio to 1:2. Such relaxed criteria can be achieved by selecting Ann, Donna, and George. Now, let us consider the above example for the case when . In such case, the ideal sex ratio should be equal to 2:2, the research area ratio to 2:1:1, the age ratio to 1:3, and the affiliation ratio to 1:3. It can be proved, however, that for there exists no committee satisfying such relaxed criteria. Intuitively, in such case the best committee is either {Ann, Charlie, Donna, George}, with two externals instead of three, or {Charles, Donna, George, Kevin}, with males being overrepresented.
In this paper we formalize the intuition given in the above example and define what it means for a committee to be optimal. When looking for an appropriate definition we follow an axiomatic approach. First, we notice that our model generalizes the apportionment problem for proportional representation [2]. The central question of the apportionment problem is how to distribute parliament seats between political parties, given the numbers of votes casted for each party. Indeed, we can consider our multiattribute problem, with the single attribute being a political affiliation of a candidate, and the desired distributions being the proportions of votes casted for different parties. In such case we can see that selecting a committee in our multiattribute proportional representation system boils down to selecting a parliament according to some apportionment method.
There is a variety of apportionment methods studied in the literature [1]. In this paper we do not review these methods in detail (we refer the reader to the survey of Balinski and Young [2]), but we rather focus on a specific set of their properties that have been analyzed, namely nonreversal, exactness and respect of quota, population monotonicity, and house monotonicity. We define the analogs of these properties for the multiattribute domain, and analyze our definition of an optimal committee for a multiattribute domain with respect to these properties.
To emphasize the analogy between our model and the apportionment methods, we should provide some discussion on where the desired proportions for attributes come from. Typically, but not always, they come from votes. For instance, each voter might give her preferred value for each attribute, and the ideal proportions coincide with the observed frequencies. For instance, out of 20 voters, 10 would have voted for a male and 10 for a female, 13 for a young person and 7 for a senior one, etc. It is worth mentioning that the voters might cast approval ballots, that is for each attribute they might define a set of approved values rather than pointing out the single most preferred one. On the other hand, sometimes, instead of votes, there are “global” preferences on the composition of the committee, expressed directly by the group, imposed by law, or by other constraints that should be respected as much as possible independently of voter preferences.
The multiattribute case, however, is also substantially different from the singleattribute one. In particular, multiattribute proportional representation systems exhibit computational problems that do not appear in the singleattribute setting. Indeed, in the second part of our paper we show that finding an optimal committee is often NPhard. However, we show that this challenge can be addressed by designing efficient approximation and fixedparameter tractable algorithms.
After positioning our work with respect to related areas in Section 2, we present our model in Section 3. In Sections 4 and 5 we discuss relevant properties of methods for multiattribute fair representation. In Section 6 we show that, although the computational of optimal committees is generally NPhard, there exist good approximation and fixedparameter tractable algorithms for finding them. In Section 7 we point to further research issues.
2 Related work
Our model is related to three distinct research areas:
Voting on multiattribute domains (see the work of Lang and Xia [13] for a survey). There, the aim is to output a single winning combination of attributes (e.g., in multiple referenda, a combination of binary values). Our model in case when can be viewed as a voting problem on a constrained multiattribute domain (constrained because not all combinations are feasible).
Multiwinner (or committee) elections. In particular, our model is related to the problem of finding a fully proportional representation [6, 18]. There, the voters vote directly for candidates and do not consider attributes that characterize them. Thus, in this literature, the term “proportional representation” has a different meaning: these methods are ‘representative’ because each voter feels represented by some member of the elected committee. The computational aspects of full proportional and its extensions have raised a lot of attention lately [21, 3, 7, 24, 17]. Our study of the properties of multiattribute proportional representation is close in spirit to the work of Elkind et al. [10], who gives a normative study of multiwinner election rules. Budgeted social choice [16] is technically close to committee elections, but it has a different motivation: the aim is to make a collective choice about a set of objects to be consumed by the group (perhaps, subject to some constraints) rather than about the set of candidates to represent voters.
Apportionment for partylist representation systems (see the work of Balinski and Young [2] for a survey). As we already pointed out, the apportionment methods correspond to the restriction of our model to a single attribute (albeit with a different motivation). While voting on multiattribute domains and multiwinner elections have lead to significant research effort in computational social choice, this is less the case for partylist representation systems. Ding and Lin [8] studied a gametheoretic model for a partylist proportional representation system under specific assumptions, and show that computing the Nash equilibria of the game is hard. Also related is the computation of biapportionment (assignment of seats to parties within regions), investigated in a few recent papers [22, 23, 14].
Constrained approval voting (CAP) [4, 20]
is probably the closest work to our setting (MAPR). In CAP there are also multiple attributes, candidates are represented by tuples of attribute values, there is a target composition of the committee and we try to find a committee close to this target. However, there are also substantial differences between MAPR and CAP. First, in CAP, the target composition of the committee, exogenously defined, consists of a target number of seats
for each combination of attributes (called a cell), that is, for each , we have a value ; while in MAPR we have a smaller input consisting of a target number for each value of each attribute. Note that the input in CAP is exponentially large in the number of attributes, which makes it infeasible in practice as soon as this number exceeds a few units (probably CAP was designed only for very small numbers of attributes, such as 2 or 3). Second, in CAP, the selection criterion of an optimal committee is made in two consecutive steps: first a set of admissible committeesis defined, and the choice between these admissible committees is made by using approval ballots, and the chosen committee is the admissible committee maximizing the sum, over all voters, of the number of candidates approved (there is no loss function to minimize as in MAPR). A simple translation of CAP into an integer linear programming problem is given in
[20, 25].3 The model
Let be a set of attributes, each with a finite domain . We say that is binary if . We let . Let be a set of candidates, also referred to as the candidate database. Each candidate is represented as a vector of attribute values .^{1}^{1}1By writing , we slightly abuse notation, that is, we consider both as an attribute name and as a function that maps any candidate to an attribute value; this will not lead to any ambiguity.
For each , by we denote a target distribution with . We set . Typically, voters have casted a ballot expressing their preferred value on every attribute , and is the fraction of voters who have as their preferred value for , but the results presented in the paper are independent from where the values come from (see the discussion in the Introduction).
The goal is to select a committee^{2}^{2}2We will stick to the terminology “committee” although the meaning of subsets of candidates has sometimes nothing to do with the election of a committee. of candidates (or items) such that the distribution of attribute values is as close as possible to . Formally, let denote the set of all subsets of of cardinality . Given , the representation vector for is defined as , where for each , and .
Definition 1
A committee is perfect for if for all .
Thus, a perfect committee matches exactly the target distribution. Clearly, there is no perfect committee if for some , is not an integer multiplicity of . In some of our results we will focus on target distributions such that for each the value is an integer. We will refer to such target distributions as to natural distributions.
We define metrics measuring how well a committee fits a target distribution, called loss functions.
Definition 2
A loss function maps and to , and satisfies if and only if .
There are a number of loss functions that can be considered. As often, the most classical loss functions use norms, with the most classical examples of , , and . We focus on two representative norms, , and , but we believe that other choices are also justified and may lead to interesting variants of our model. Consequently, we consider the following loss functions:

.

.

.
Now, we are ready to formally define the central problem addressed in the paper.
Definition 3 (OptimalRepresentation)
Given , , , , and a loss function , find a committee minimizing .
Example 1
For the example of the Introduction, we have = {sex, group, age, affiliation}, , and , etc. is optimal for , with , and for , with , but not for . is optimal for , with , but not for the other criteria.
4 The singleattribute case
In this section we focus on the singleattribute case (
). Without loss of generality, let us assume that the single attribute be party affiliation. Further, let us for a moment assume that for each value
there are at least candidates with value (this is typically the case in partylist elections). Then finding the optimal committee comes down to apportionment problem for partylist elections, where a fractional distribution has to be “rounded up” to an integervalued distribution such that .There are two main families of apportionment methods: largest remainders and highest average methods [2]. We shall not discuss highest average methods here, because they are weakly relevant to our model. For largest remainders methods, a quota is computed as a function of the number of seats and the number of voters . The number of votes for party is . The most common choice of a quota is the Hare quota, defined as ; the method based on the Hare quota is called the Hamilton method.^{3}^{3}3Other common choices are the Droop quota , the HagenbachBischoff quota and the Imperiali quota . Our aim is to generalize the Hamilton method to multiattribute domains.
Definition 4 (The largest remainder method.)
The largest remainder method with quota is defined as follows:

for all , is the ideal number of seats for party .

each party receives seats; let (called the remainder).

the remaining seats are given to the parties with the highest remainders .
Below we show that the largest remainder methods select a distribution minimizing , which in the case of Hamilton comes down to minimizing . After defining for all , we obtain the result that explains that our problem, with any of the three variants of loss functions, generalizes the Hamilton apportionment method.
Proposition 1
When and assuming there are at least items for each attribute, optimal subsets for , and coincide, and correspond to the subsets given by the Hamilton apportionment method.
Proof. Note that and are equivalent for . Recall that denotes the target number of seats for party . Let be a committee of size and let be the number of members of that belong to party . Since , we need to show that the following three assertions are equivalent:

minimizes .

minimizes .

is a Hamilton committee.
We first show . Assume is not a Hamilton committee: then there exists an attribute value (party) that receives strictly more or strictly less seats than it would receive according to the Hamilton method. Naturally, there must also exist an attribute that receives strictly less or strictly more seats, respectively. Formally, this means that there are two attribute values (parties), say and , such that the target number of seats for parties 1 and 2 are and , with integers and , and such that either and . We have . Consider the committee obtained from by giving one less seat to and one more to .

If then .

If then similarly, .

If and then we have and , hence .
In all three cases, does not minimize and is therefore not an optimal committee for .
We now show . Call a party lucky if and unlucky if . Then we have . Let, without loss of generality, be the lucky party with the highest value (if there are several such parties, we take arbitrary one of them) and be the unlucky party with the highest value . Assume is not a Hamilton committee: then 2 had a higher remainder than 1 before 1 got her last seat, that is, . Let be the committee obtained from by giving one less seat to and one more to : then either is a Hamilton committee, or it is not, and in this case we repeat the operation until we get a Hamilton committee . Because , is not an optimal committee for .
It remains to be shown that if is a Hamilton committee then if is both optimal for and . If there is a unique Hamiltonoptimal committee then this follows immediately from and . Assume there are several Hamiltonoptimal committees . Then there are parties, w.l.o.g., , with equal remainders , that is, , …, , and the Hamiltonoptimal committees differ only in the choice if those of these parties to give they give an extra seat. We easily check that for any two of these committees we have and .
Therefore, our model can be seen as a generalization of the Hamilton apportionment method to more than attribute. Note that our model can easily extend other largest remainder methods, and our results would be easily adapted. Interestingly, when , our three criteria no longer coincide. However, for binary domains, and coincide, since .
Proposition 2

For each and binary domains, optimal subsets for and may be disjoint, even for .

For each , optimal subsets for and can be disjoint.

For each , if at least one attribute has 4 values, then optimal subsets for and can be disjoint.

For and binary domains, optimal subsets for and may differ.
Proof. We prove point 1 for (the proof extends easily to by adding attributes on which all items, and the target, agree). We have four candidates: two ( and ) with attribute vectors , and two ( and ) with . The target distribution is and for . The optimal committees are and . The optimal committee is .
For Point 2: because optimal subsets for and coincide for binary domains, Point 1 implies that optimal subsets for and can be disjoint. The counterexample extends easily to nonbinary domains.
For Point 3: Let there be two attributes with values and with values ; four candidates: with value vector , with value vector , with value vector , and with value vector ; ; and for and for . The optimal committees for are all pairs except (with loss 1.8) while the optimal committee for is (with loss 0.6). Next, we show that and can be disjoint. The counterexample extends easily to more attributes and more values.
For Point 4, let , three candidates , and with value vectors , and ; and , , , and . , and are all optimal, but only and are optimal.
These negative results come from the constraints imposed by the candidate database, which prevent the selection on the different attributes to be done independently. In the example of the proof of point 1, for instance, since all items with the value for have value for , selecting items with implies selecting items with . However, if the database is sufficiently diverse so that no such constraints exist, the optimization can be done separately on each attribute. This is captured by the following notion.
Definition 5
A candidate database satisfy the Full Supply (FS) property with respect to if for any there are at least candidates in associated with value vector .
The candidate database of Example 1 does not satisfy FS, even for , because there is not a single candidate with group and age . If we ignore attributes group and affiliation, then we are left with 2 (resp., 3, 2, 3) candidates with value vector (resp. , , ): the reduced database satisfies FS for .
Proposition 3
Let be an optimal committee selection problem. If satisfies FS w.r.t. , then the following statements are equivalent:

is an optimal committee for

is an optimal committee for

for any attribute , is a Hamilton committee for the singleattribute problem , where is the projection of on .
Moreover, any (and ) optimal committee is optimal for . (The converse does not always hold.)
Proof. For each attribute and value , let be the number of seats with value given by the Hamilton method for the singleattribute problem . For all , let . Then take as item any item in the database with value vector , and remove it from the database; the full supply assumption guarantees that it will always be possible to find such an item. Let ; it is easy to check that is an optimal committee for and for .
To illustrate the constructive proof, consider 2 attributes with 3 values , and with 2 values ; ; and , . Then , , , , which leads to choose with value vector , with vector , with vector , and with vector .
5 Properties of multiattribute proportional representation
Several properties of apportionment methods have been studied, starting with Balinski and Young [1]. We omit their definition in the singleattribute case and directly give their generalizations to our more general model. Let be any optimal committee for some criterion given , and . We recall that denotes the number of elements of with the attribute equal to .

Nonreversal: for any attribute , and attribute values , , if then .

Exactness and respect of quota: for all , either or .

Population monotonicity (with respect to ): consider and such that (a) , (b) for all , , and (c) for all and all , . Then there is an optimal committee for such that .

House monotonicity: let be an optimal committee for , and . Then for all , . ^{4}^{4}4Some other properties, such as consistency, seem more difficult to generalize to the multiattribute case. Also, properties that deal with strategy proofness issues, such as resistance to party merging or party splitting, are less relevant in our setting than for political elections and we omit them.
In the singleattribute case, it is known for long that the Hamilton method satisfies all these properties except house monotonicity (this failure of house monotonicity is better known under the name Alabama paradox).
We start by noticing that if a property fails to be satisfied in the singleattribute case, a fortiori it is not satisfied in the multiattribute case. As a consequence, house monotonicity is not satisfied, even under the FS assumption. We now consider the other properties.
Proposition 4
Under the full supply assumption, nonreversal, exactness and respect of quota, and population monotonicity are all satisfied, for any of our loss functions. In the general case, nonreversal, exactness and respect of quota are not satisfied. If
is a binary variable, and for
, population monotonicity with respect to is satisfied; however it is not satisfied in the general case.Proof. Under FS, the result easily comes from Proposition 3 and the fact that the property holds in the singleattribute case.
In the general case, we give counterexamples. For exactness and respect of quota, we have two binary attributes, and two items , with value vectors and , , defined as , , , . The optimal committee is either or , and does not respect quota even though all values are integers.
For nonreversal we have two binary attributes and six items: , each with vector and , each with vector . We have a target distribution defined as follows: , , , . We set . The optimal committees for and are and all triples made up from two items out of and one out of . The optimal committees for are all triples made up from two items out of and one out of . In all cases, for all optimal committees we have although .
Now, we prove that population monotonicity holds for binary domains and for . Consider a binary attribute , with .
Assume that (and ), and that for all we have . Let be an optimal committee for and, for the sake of contradiction, assume that for all optimal committees for we have . Let be such a committee. The proof is a case by case study, with six cases to be considered: (C1) ; (C2) ; (C3) ; (C4) ; (C5) ; and (C6) .

Case 1: . In this case we have and the following holds:
(4) comes from the fact that is not optimal for . Since, there is one strong inequality in the sequence, we imply that is not optimal for , a contradiction.

Case 2: .
Again we obtain a contradiction.

Case 3: .
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