## 1 Introduction

In discrete fair division, given a set of goods and agents, the problem is to integrally allocate the set of goods to the agents in a fair and (economically) efficient manner [BCE16; End17; Azi19]

. In this thread of work, the Nash social welfare—defined as the geometric mean of the agents’ valuations for their assigned bundles—has emerged as a fundamental and prominent measure of the quality of an allocation. It provides a balance between two central objectives: the social welfare (the sum of the agents’ valuations) and the egalitarian welfare (the minimum valuation across the agents). Note that social welfare is a standard measure of (economic) efficiency, whereas egalitarian welfare is a fairness objective.

A Nash optimal allocation (i.e., an allocation that maximizes Nash social welfare) satisfies other fairness and efficiency criteria as well.
Such an allocation is clearly Pareto optimal. Furthermore, if agents have additive valuations, then a Nash optimal allocation is known to be fair in the sense that it is guaranteed to be *envy-free up to one good* (Ef1) [CKM19] and *proportional up to one good* (Prop1) [CFS17].^{1}^{1}1An allocation is said to be Ef1 if for any pair of agents and , there exists a good in ’s bundle, such that prefers her bundle to the one obtained after removing from ’s bundle. An allocation is said to be Prop1 if for each agent there exists a good with the property that including into ’s bundle ensures that achieves a proportional share, i.e., her valuation ends up being at least times her value for all the goods.

As an objective, Nash social welfare is scale invariant: multiplicatively scaling any agent’s valuation function by a nonnegative factor does not change the Nash optimal allocation. Furthermore, interesting connections have been established between market models and this welfare function; see, e.g., [CDG17; BKV18]. As a practical application, the website spliddit.org uses the Nash social welfare as the optimization objective when partitioning indivisible goods [GP15; CKM19].

However, computing a Nash optimal allocation is APX-hard, even when the agents have additive valuations [Lee17]. In terms of approximation algorithms, the problem of maximizing Nash social welfare has received considerable attention in recent years [CG15; CDG17; AGSS17; BGHM17; AGMV18; BKV18; GHM18]. In particular, a polynomial-time -approximation algorithm is known for additive valuations [BKV18]. This algorithm preserves Ef1, up to a factor of , and Pareto optimality. The approximation guarantee of also holds for budget-additive valuations [CCG18]. The work of Garg et al. [GKK20] extends this line of work by considering Nash social welfare maximization under submodular valuations.

Submodular valuations capture the diminishing marginal returns property. They constitute a subclass of subadditive valuations, which, in turn, model complement-freeness. Formally, a set function (defined over a set of indivisible goods) is subadditive if it satisfies , for all subsets of goods and . Complement-freeness is a very common assumption on valuation functions. Hence, fair division with subadditive valuations is an encompassing and important problem.

For Nash social welfare maximization under submodular valuations, Garg et al. [GKK20] obtain an -approximation algorithm. Prior to their work, the best known approximation ratio for submodular valuations was , which also extends to subadditive valuations [NR14]; here, denotes the number of goods and the number of agents. For a constant number of agents with submodular valuations, Garg et al. [GKK20] provide an -approximation algorithm and show that, even in this setting, improving upon is NP-hard.

In the context of allocating indivisible goods, two other well-studied welfare objectives are the social welfare and the egalitarian welfare. These represent, respectively, for an allocation, the average valuation of the agents and the minimum valuation of any agent. For the social welfare objective, a tight approximation factor of is known under submodular valuations [Von08]. For subadditive valuations, Feige [Fei09] shows that social welfare maximization admits a polynomial-time -approximation, assuming oracle access to *demand queries*.^{2}^{2}2A demand-query oracle, when queried with prices associated with the goods, returns , for an underlying valuation function . The current paper works with more basic value oracle, which when queried with a subset of goods returns the value this subset. Any value query can be simulated via a polynomial number of demand queries. However, the converse is not true [NRTV07].

For maximizing egalitarian welfare under additive valuations, Chakrabarty et al. [CCK09] provide an -approximation algorithm that runs in time , for any . Under submodular valuations, egalitarian welfare maximization admits an -approximation algorithm [GHIM09]. Khot and Ponnuswami [KP07] provide a -approximation algorithm for maximizing egalitarian welfare under subadditive valuations. As a lower bound, with submodular valuations, egalitarian welfare cannot be approximated within a factor of , unless [BD05].

In this work we develop a unified treatment of fairness and efficiency objectives, including the welfare functions mentioned above. In particular, we develop an approximation algorithm for computing allocations that maximize the *generalized mean* of the agents’ valuations. Formally, for exponent parameter , the *generalized mean* of a set of positive reals is defined as . For an allocation (partition) of the indivisible goods among the agents, we define the *-mean welfare* of as the generalized mean of the values ; here is the value that agent has for the bundle assigned to it. Indeed, with different values of , the -mean welfare encompasses a range of objectives: it corresponds to the social welfare (arithmetic mean) for , the Nash social welfare (geometric mean) for , and the egalitarian welfare for . In fact, -mean welfare functions with exactly correspond to the collection of functions characterized by a set of natural axioms, including the Pigou-Dalton transfer principle [Mou03]. Hence, -mean welfare functions, with , constitute an important and axiomatically-supported family of objectives.

Our Contributions. We develop a polynomial-time algorithm that, given a fair division instance with subadditive valuations and parameter , finds an allocation with -mean welfare at least times the optimal -mean welfare (Theorem 9). Our algorithm uses the standard value oracle model which, when queried with any subset of goods and an agent , returns the value that has for the subset. For different values of , our algorithm changes minimally, differing only in the weights of edges for a computed matching. We thus present a unified analysis for this broad class of welfare functions, suggesting further connections between these objectives than the previously mentioned axiomatization. Our result matches the best known -approximation for egalitarian welfare [KP07] and improves upon the -approximation guarantee of Garg et al. [GKK20] for Nash social welfare with submodular valuations. Arguably, our algorithm (and the analysis) is simpler than the one developed in [GKK20] and simultaneously more robust, since it obtains an improved approximation ratio for subadditive valuations and a notably broader class of welfare objectives.

For clarity of exposition, we first present an -approximation algorithm for maximizing Nash social welfare under subadditive valuations (Theorem 2). We then generalize the algorithm to the class of -mean welfare objectives.

We complement these algorithmic results by adapting a result of Dobzinski et al. [DNS10] to show that for XOS valuations, any -approximation for -mean welfare requires an exponential number of value queries (Section 5). Hence, in the value oracle model, our approximation guarantee is essentially tight for XOS and, hence, for subadditive valuations. We note that these are the first polynomial lower bounds on approximating either the Nash social welfare or the egalitarian welfare.

Nguyen and Rothe [NR14] obtain an -approximation guarantee for maximizing Nash social welfare with subadditive valuations. We establish two extensions of this result. First, we show that, under subadditive valuations, an -approximation for the -mean welfare can be obtained for all . However, for , we establish that it is NP-hard to obtain an -approximation, even under additive valuations. An analogous hardness result holds for with submodular valuations.

Independent Work. In work independent of ours, Chaudhury et al. [CGM20] also obtain an -approximation algorithm for maximizing generalized -means under subadditive valuations. Their approach varies significantly from the current paper and, in particular, builds upon results on finding allocations that are approximately envy-free up to any good (EFX). Notably their algorithm computes allocations that satisfy additional fairness properties, including Ef1 and either of two approximate versions of EFX.

Section 3 presents our approximation algorithm for maximizing Nash social welfare. Then, Section 4 shows that we can extend the algorithm for Nash social welfare to obtain the stated approximation bound for -mean welfare. The tightness of these results is established in Section 5. Section 6 presents the results for the -approximation guarantees.

## 2 Notation and Preliminaries

An instance of a fair division problem is a tuple , where denotes the set of indivisible goods that have to be allocated (partitioned) among the set of agents, . Here, represents the valuation function of agent . Specifically, is the value that agent has for a subset of goods . For and , write to denote agent ’s value for the good , i.e., it denotes .

We will assume throughout that the valuation function for each agent is (i) nonnegative: for all , (ii) normalized: , (iii) monotone: for all , and (iv) subadditive: for all subsets .

Submodular and XOS (fractionally subadditive) valuations constitute subclasses of subadditive valuations. Formally, a set function is said to be submodular if it satisfies the *diminishing marginal returns* property: , for all subsets and . A set function, , is said to be XOS if it is obtained by evaluating the maximum over a collection of additive functions , i.e., , for each subset .^{3}^{3}3Here, can be exponentially large in .

We use to denote the collection of all partitions of the indivisible goods . An *allocation* is an -partition of the goods. Here, denotes the subset of goods allocated to agent and will be referred to as a *bundle*.

Given a fair division instance , the *Nash social welfare* of allocation is defined as the geometric mean of the agents’ valuations under : .

We will throughout use to denote an allocation that maximizes the Nash social welfare for a given fair division instance. We refer to as a Nash optimal allocation. An allocation is an -approximate solution (with ) of the Nash social welfare maximization problem if .

Besides the Nash social welfare, we address a family of objectives defined by considering the *generalized means* of agents’ valuations. In particular, for parameter , the the generalized (Hölder) mean of nonnegative numbers is defined as .

Parameterized by , this family of functions captures multiple fairness and efficiency measures. In particular, when , reduces to the arithmetic mean. In the limit, is equal to the geometric mean as tends to zero. In addition, .

We define the *-mean welfare*, , of an allocation as

With equal to one, zero, and , the -mean welfare corresponds to the (average) social welfare, Nash social welfare, and egalitarian welfare, respectively.

The following proposition implies that for any , if instead of the -mean welfare, we maximize the egalitarian welfare, then the resulting allocation loses a negligible factor in the approximation ratio. The proof of this proposition is deferred to Appendix A.1.

###### Proposition 1.

For any nonnegative numbers and , we have

## 3 An -Approximation for Nash Social Welfare

This section presents an efficient -approximation algorithm for the Nash social welfare maximization problem, under subadditive valuations. Our algorithm, Algorithm 1 (Alg), requires access to the valuation functions through basic value queries, i.e., it only requires an oracle which, when queried with a subset of goods and an agent , returns .

We first describe the ideas behind our algorithm. Write denote a Nash optimal allocation in the given instance and let us, for now, assume that the agents have additive valuations, i.e., for all agents and subset of goods , we have . In the following two cases, we can readily obtain an approximation. In the first case, each agent has a few “high-value” goods, i.e., each agent has a good with the property that . In such a setting, we can construct a complete bipartite graph with agents on one side and all the goods on the other. Here, the weight of edge is set to be . In this bipartite graph, the matching has Nash social welfare at least times the optimal and, hence, this also holds for a left-perfect maximum-weight matching in this graph.

In the second case, all goods are of “low-value”, i.e., for all and we have . Here again an approximation can be obtained via a simple round-robin algorithm, wherein the agents (in an arbitrary order) repeatedly pick their highest valued good from those remaining. At a high level, our algorithm stitches together these two extreme cases by first matching high-value goods and then allocating the low-value ones.

We connect the two cases by considering the following quantity for each agent

(1) |

That is, is the (near) proportional value that each agent is guaranteed to achieve even after the removal of any -size subset of goods. Our algorithm leverages the following existential guarantee (Lemma 3): there necessarily exists a good with the property that

(2) |

This result ensures that, a single high-value good (in particular, ) coupled with a -approximation to all the low-value goods (i.e., ), is sufficient to ensure a -approximation *for each agent*. At this point, if we could (i) explicitly compute for each agent and (ii) for any size- subset of goods , assign the remaining goods such that each agent gets a bundle of value at least , then we would be done. This follows from the observation that in the complete bipartite graph with weight of edge set to , the weight of the matching is a approximation to the optimal Nash social welfare by equation (2) and, hence, the same guarantee holds for a maximum-weight matching in the graph. Condition (ii) ensures that each agent also receives at least after the initial assignment of the matched goods.

For additive valuations, both conditions (i) and (ii) can be satisfied. This template was employed in the SMatch algorithm (for additive valuations) of Garg et al. [GKK20]. However, for submodular (and subadditive) valuations, the quantity is hard to approximate within a sub-linear factor [SF11].

Therefore, instead of satisfying condition (i) explicitly, we maintain an upper bound for each agent . Our algorithm first obtains a maximum weight matching in the bipartite graph between agents and goods with the weight of edge set to . It assigns all the matched goods to the respective agents, removes these goods from further consideration in this iteration, and then carries out a procedure (described below) to ensure condition (ii). If, for agent , the bundle obtained in this procedure (i.e., the bundle obtained for after removing the matched goods) has value less than

, then we multiplicatively reduce the (over) estimate

for and repeat the algorithm.The procedure towards satisfying condition (ii) consists of two steps. Let be the set of goods that remain once we remove the matched goods from . In the first step, if there exists an agent and a good such that , we assign to and remove both from further consideration. An agent thus removed has value from the assigned good; note that, by definition, . After this step, we observe that for each remaining agent and good . In the second step, we run a *moving knife* subroutine (Algorithm 2) on the goods that are still unassigned. In this subroutine, the goods are initially ordered in an arbitrary fashion. A hypothetical knife is then moved across the goods from one side until an agent (who has yet to receive a bundle) calls out that the goods covered so far have a collective value of at least for her. These covered goods are then allocated to said agent and both the agent as well as this bundle is removed from further consideration. We show that this allocation satisfies condition (ii), i.e., the bundle assigned to each agent in this procedure has value at least (but it may be lower than the overestimate ).

Since we can guarantee for each agent , irrespective of which goods are removed in the matching step, never goes below , for any agent. Hence, at some point, every agent receives a bundle of value at least in the above two steps. We show that these bundles, with the goods matched with each agent, provide an approximation to the optimal Nash social welfare.

It is relevant to note we use solely for the purposes of analysis. Our algorithm executes with the overestimate and keeps reducing this value till it is realized (in the two-step procedure) for all the agents.

As mentioned previously, the SMatch algorithm (developed for additive valuations) of Garg et al. [GKK20] relies of conditions (i) and (ii). However, for submodular valuations their work diverges considerably from the current approach. In particular, the RepReMatch algorithm (developed for submodular valuations) in [GKK20] first finds a set of goods with the property that in the bipartite graph between all the agents and , there is a matching wherein every agent is matched to a good with value at least as much as her highest valued good in . To ensure this property the cardinality of needs to be . Intuitively, this requirement leads to a lower bound of on the approximation ratio obtained in [GKK20]. Furthermore, the steps in their algorithm to ensure condition (ii) do not extend to subadditive valuations either. Specifically, Garg et al. [GKK19] note that their algorithm gives an approximation ratio of for the case of subadditive valuations. The -approximation of Khot and Ponnuswami for egalitarian welfare [KP07] first guesses the optimal egalitarian welfare , and uses this to partition the goods into “large” ones (those with value higher than ) and “small” ones, for each agent. It then tries to ensure every agent receives a bundle with valuation at least . For Nash social welfare, guessing just a single value does not appear to help, since the Nash social welfare depends on the valuation of each agent.

The following theorem constitutes our main result for Nash social welfare.

###### Theorem 2.

Let be a fair division instance in which the valuation function , of each agent , is nonnegative, monotone, and subadditive. Given value oracle access to s, the algorithm Alg computes an approximation to the Nash optimal allocation in polynomial time.

The following lemma proves inequality (2). We state and prove it for an arbitrary allocation , rather than just for the Nash optimal allocation.

###### Lemma 3.

Let be a fair division instance with monotone, subadditive valuations and let be any allocation in . Let be the most valued (by ) good in (i.e., ) and be as defined in (1). Then, for each agent

Proof Consider any agent and note that . We will establish the lemma by considering two complementary cases.

Case I: There exists a good with the property that . Since is the most valued good in , we have and the desired inequality follows.

Case II: For all goods , . Recall that . Let be the set that induces , i.e., . Monotonicity of ensures that and

(3) |

Furthermore, given that in the current case for all , we have

(4) |

Here, the first inequality follows from the fact that is subadditive and the last since .

Therefore, we obtain the desired bound in terms of :

Thus, the the stated inequality holds even in this case.

The next lemma establishes the key property of Algorithm 2 (MovingKnife): if all the goods have low value for every agent, then MovingKnife returns a near-proportional allocation.

###### Lemma 4.

Consider a fair division instance wherein the agents have monotone, subadditive valuations. In addition, suppose for each agent and good we have , where . Then the allocation returned by Algorithm 2 (MovingKnife) satisfies for all .

Proof Given instance , the MovingKnife algorithm (Algorithm 2) considers the goods in an arbitrary order and adds these goods one by one into a bundle until an agent calls out that its value for is at least . We assign these goods to agent and remove them—along with —from consideration. The algorithm iterates over the remaining set of agents and goods. We will show that the while loop in the MovingKnife algorithm terminates with and, hence, assigns to each agent a bundle of desired value.

Consider an integer (count) . Let and denote the set of goods and agents, respectively, that are left unassigned after agents are assigned bundles in MovingKnife; note that . The arguments below establish that for each remaining agent ,

(5) |

Therefore, for any , the set of unassigned goods is nonempty and even the last agent (i.e., with ) receives a bundle of sufficiently high value.

To prove (5), consider any agent . Indeed, agent has not received any goods yet, but the agents in have been assigned bundles. Let be a bundle assigned to some agent in (i.e., for some ) and be the last good included in . Step 4 of the algorithm ensures that ; otherwise, would have been assigned to agent . Furthermore, the assumption (in the Lemma statement) gives us . Hence, using these inequalities and the subadditivity of , we get .

This inequality provides an upper bound on , the total value of the set of goods assigned among the agents in . Specifically, by the subadditivity of , . Therefore, .

Overall, every agent is eventually assigned a bundle of value at least in the while loop.

Next we show that in each iteration of the while loop in Alg (Algorithm 1), the value of the assigned bundle is at least as large as .

###### Lemma 5.

Given a fair division instance with subadditive valuations, let be the bundle assigned to agent in the iteration (for ) of the outer while loop (Step 10) in Alg. Then, for all agents and each iteration count , we have .

Proof During any iteration of the outer while loop (Step 10) in Alg and for any agent , the bundle either consists of a single good of high value (Step 15), or of the set of goods assigned to agent obtained after executing the MovingKnife subroutine (Step 17). We will show that in both cases the stated inequality holds.

Recall that . Equivalently, . Therefore, we have

(6) |

The relevant observation here is that, in any iteration , the set of goods from which the bundles s are populated satisfies . Specifically, in the iteration, we start with (Step 13). Subsequently, the inner while loop (Step 14) assigns at most goods and, hence, the number of goods passed on to the MovingKnife subroutine satisfies .

First, we note that the lemma holds for any agent that receive a singleton bundle in Step 15: . Here, the first inequality follows from the selection criterion applied to and the second inequality from equation (6) and the fact that .

Finally, we note that the bound also holds for the remaining agents that receive a bundle through the MovingKnife subroutine. As mentioned previously, at least goods are passed on as input to the subroutine, i.e., if MovingKnife is executed on instance , then we have . Inequality (6) ensures that for all . Finally, using Lemma 4, we get that the bundle assigned to agent satisfies the stated inequality: .

Hence, the stated claim follows.

We now show that the estimates s used in Alg also satisfy a lower bound similar to that in Lemma 5.

###### Lemma 6.

Given a fair division instance with subadditive valuations, let be the estimate associated with agent in the iteration (for ) of the outer while loop (Step 10) in Alg. Then, for all agents and each iteration count , we have .

Proof Note that for any agent , the quantity iff has positive value for at most goods. This observation implies that the initial for loop in Alg correctly identifies agents that have , and sets . For such agents for all . Hence, the lemma holds for any agent with .

We now consider agents with . For such an agent , the algorithm initially sets . Hence, for we have . An inductive argument shows that this inequality continues to hold as the algorithm progresses. In particular, if in the iteration the algorithm does not decrement the estimate (i.e., if ), then .

Even otherwise, if the algorithm multiplicatively decrements the estimate (in particular, sets ), then it must be the case that (i.e., ). That is, after the decrement we have ; the last inequality follows from Lemma 5. This completes the proof.

### 3.1 Proof of Theorem 2

In this section we prove Theorem 2 by showing that Alg runs in polynomial time (Lemma 7) and the computed allocation achieves the stated approximation ratio of (Lemma 8).

###### Lemma 7 (Runtime Analysis).

Proof By design, Alg iterates as long as . We will bound the number of times (i.e., the distinct values of for which) any agent is contained in and, hence, establish the stated runtime bound.

Recall that for any agent , the quantity iff has positive value for at most goods. For such agents Alg sets . Therefore, these agents are contained in , for all iterations , and do not contribute to the repetitions of the outer while loop.

For the remaining agents, with , the algorithm initially sets and we have

(7) |

Using Lemma 6 and the fact that the algorithm decrements by a multiplicative factor of whenever , we get that the number of times agent can be in the is at most

(since is subadditive, ) | ||||

(via inequality (7)) | ||||

Summing over all agents, we get that the number of times is at most . Hence, the stated lemma follows.

We now show that the allocation computed by Alg achieves the required approximation guarantee.

###### Lemma 8 (Approximation Guarantee).

For any given fair division instance with subadditive valuations, let denote the allocation computed by Alg. Then, ; here, denotes the Nash optimal allocation in .

Proof For the given instance , say Alg terminates after iterations of the outer while loop. That is, we have and, for each agent , the returned bundle . Here, is the good assigned to agent under the maximum weight matching (considered in the last iteration) and is the bundle populated for (either in Step 15 or in Step 17).

The fact that (i.e., ) gives us

(8) |

Lemma 3 (instantiated with ) implies that there exists a matching—, for all —with the property that . Using this inequality and Lemma 6 we get, for all :

(9) |

Recall that is a maximum weight matching in the bipartite graph (considered in Step 11 of Alg) with edge weights . Given that is some matching in the graph and is a maximum weight matching, we get . That is,

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