Combinatorial Assortment Optimization

Assortment optimization refers to the problem of designing a slate of products to offer potential customers, such as stocking the shelves in a convenience store. The price of each product is fixed in advance, and a probabilistic choice function describes which product a customer will choose from any given subset. We introduce the combinatorial assortment problem, where each customer may select a bundle of products. We consider a model of consumer choice where the relative value of different bundles is described by a valuation function, while individual customers may differ in their absolute willingness to pay, and study the complexity of the resulting optimization problem. We show that any sub-polynomial approximation to the problem requires exponentially many demand queries when the valuation function is XOS, and that no FPTAS exists even for succinctly-representable submodular valuations. On the positive side, we show how to obtain constant approximations under a “well-priced” condition, where each product’s price is sufficiently high. We also provide an exact algorithm for -additive valuations, and show how to extend our results to a learning setting where the seller must infer the customers’ preferences from their purchasing behavior.

1 Introduction

Imagine that you are an inventory manager, tasked with selecting which products to display on the shelves in a retail store. These products are acquired from different producers, who control the suggested retail prices. Your goal is to find a profitable assortment of items to offer, given a model of how customers choose which item(s) to ultimately purchase from the subset you display. This assortment problem captures a natural tradeoff. If you offer only the most expensive items, then many customers might simply leave the store without purchasing anything. On the other hand, a variety of inexpensive items might cannibalize sales from pricier goods and dilute the overall revenue. Given a collection of possible items, and a model of customer preferences, which subset of items should you display to maximize revenue?

The assortment problem is of practical importance for brick and mortar stores, but is also relevant to online shopping platforms that must choose which products to display in response to a search query and whose price is exogenous, set by a third party. Customers have limited patience and are more likely to select products from the first page of results, so the platform is incentivized to display a well-chosen slate of products. Since an online platform may need to choose from a vast array of potential products, it is important to find computationally feasible solutions.

There is a growing literature on assortment in the field of revenue management, typically focusing on cases where each customer wants at most a single item. In such unit-demand settings, the problem is captured by a choice function that maps an assortment to a probability distribution describing which good in a customer will ultimately purchase. Commonly-studied choice functions include multinomial logit functions [17], exponential choice functions [2], and mixture models [3], among others. On the other hand, the computer science literature has mostly focused on combinatorial versions of revenue or welfare maximization when the designer controls the prices of items (see e.g. multi-dimensional revenue maximization [4, 6, 10]) or the mode of interaction with the consumer (see e.g. combinatorial auctions [9, 13]). The important case of assortment optimization, where the platform designer is constrained to only design the set of available items, has been largely left untouched by the combinatorial optimization community. The goal of our work is to bridge this gap and explore the intersection of assortment and combinatorial optimization.

We introduce the combinatorial assortment problem, where consumers may choose to purchase bundles of goods. For example, a customer may want to buy a camera, possibly in combination with accessories, which may be either of the same brand as the camera or a cheaper off-brand variety. These items may be complementary (a camera plus an accessory), or substitutes for each other (a brand-name accessory or a generic version of the same accessory). We ask: given the relationship between the items for sale, and possibly a cardinality constraint on the number of items that can be shown, what is a revenue-maximizing selection to offer?

We consider a model of consumer choice motivated by vertical customer differentiation. In this model, the relationship between the items is fixed and common to all potential buyers, but customers vary in their willingness-to-pay. Formally, the value that a buyer has for a certain bundle of goods is taken to be , where is a valuation function common to all buyers and is a buyer-specific multiplier that represent’s the buyer’s type. This captures settings where the relative quality and relationship between the items is unambiguous, but customers vary in their ability to extract value from the items. For example, if the items are cameras and accessories, a professional photographer might derive a value equal to of the reference value for any bundle, whereas an amateur might only derive of the reference value. Our market exhibits vertical differentiation in that all customers agree on the relative comparisons between bundles, so that if one bundle is more valuable and cheaper than another, everyone will buy the former. In comparison, horizontally-differentiated choice functions like multinomial logit perturb the common component of the valuation by an additive constant; this causes customers to disagree on which bundles are more valuable, so that if one item is cheaper and has a higher common value than another, a positive fraction of customers would still prefer the latter.

1.1 Our Results and Techniques

communication lower bound-hard (succinct)-hard (2-demand)Exact algorithm
(a) Arbitrary prices.
- (w/o constraints)- (w/ constraints)
(b) Well-priced items.
Figure 1: Computational landscape of Combinatorial Assortment Optimization. For arbitrary prices, the negative results for XOS and SM (submodular) valuations hold even without cardinality constraints, and the exact algorithm for K-ADD (-additive) valuations applies even with cardinality constraints. For the case of well-priced items (see Section 4), we give a constant-approximate algorithm for general valuations without cardinality constraints, or for GS (gross substitutes) valuations with cardinality constraints.

Our goal is to explore the computational complexity of combinatorial assortment. We will characterize the limits of polynomial time computation or approximability and provide conditions under which simple heuristics such as a greedy algorithm, or exhaustively searching over small assortments, are optimal or approximately so. Interestingly, we will see a stark difference in the computational landscape, depending on how well the items are priced (with respect to the distribution over consumer types). It turns out that assuming the item prices are not too low can make otherwise computationally hard assortment problems easy to solve or approximate (see Figure 1).

The bulk of our results apply in the case where the valuation function is known to the assortment planner, and the types are unknown but drawn from a known prior distribution . We then investigate the difficulty of the assortment problem as a function of the structural assumptions imposed on the valuation . At the end of the paper we extend many of our algorithmic results to a setting where the planner must learn these parameters from samples.

Negative Results

Our main results are summarized in Figure 1. We begin by showing that, in general, the combinatorial assortment problem is inherently difficult. Even in the deterministic case, where all buyers have the exact same preferences and these are known to the optimizer (i.e., the type distribution is a point mass at ), it is hard to approximate the revenue of the optimal assortment to a factor of for any constant , where is the number of items to choose from. This is true even if there is no constraint on the number of items to be shown, and even if the valuation function is an XOS function, a subclass of subadditive functions.111A valuation is subadditive if, for any sets of items and , . A valuation is XOS if it is the maximum of a collection of additive functions. Notably, this is a class of valuations where the welfare maximization problem can be well-approximated [13, 9].

This hardness result takes the form of a communication complexity bound, independent of any computational hardness assumptions. We show that an approximation algorithm requires an exponential amount of communication with an oracle that can answer demand queries about the valuation function . Note that it is too much to hope for a lower bound in a fully general model of communication with a valuation oracle, since in particular the oracle could simply communicate the optimal assortment, which can be described in polynomially many bits. Instead, our proof considers a communication model in which information about the valuation is split between two oracles, and show that exponential communication between the oracles is necessary to obtain any reasonable approximation. We then show how the pair of oracles can simulate a demand query oracle. One implication of this result is that any assortment algorithm with a sub-polynomial approximation factor requires exponentially many demand queries about the valuation function .

We next show that even for valuation functions that can be described succinctly,222Formally: an XOS valuation that is the maximum of only additive functions. it is still NP-hard to compute the optimal assortment. Like the communication complexity result, this holds even if all buyers have type . If we move beyond this deterministic case and allow buyer types to be drawn from an arbitrary distribution, then we show that there is no FPTAS for the combinatorial assortment problem with XOS (or even submodular) valuations even if each customer wants at most two items.333When customers demand at most 2 items, the XOS condition is equivalent to submodularity. A valuation is submodular if, for any sets of items and , . This is equivalent to each item having diminishing marginal value, and is more restrictive than subadditivity. Furthermore, the natural greedy heuristics that adds items to the assortment one by one, maximizing the marginal revenue increase on each step, fails to obtain a constant approximation for submodular valuations, even in the deterministic case where is a point mass.

Algorithmic Results

Motivated by these lower bounds, we characterize settings in which natural methods achieve good approximations, and where exact solutions can be computed in polynomial time. We first characterize settings where displaying all items is a good approximation to the optimal revenue. As mentioned earlier, offering all items might be highly suboptimal in the presence of “cheap” items that might cannabilize sales from more profitable items. We show that such an issue is inherently due to items being sold at too low a price. We say that the goods are “well-priced” if, roughly speaking, the price of each bundle is at least its optimal (i.e., Myerson) reserve price, in a world where only that bundle is for sale. When goods are substitutes, this is equivalent to each individual item’s price being at least its Myerson price. This may be the case if the individual product retailers are behaving like monopolists and not responding to the assortment planner, such as when the platform is driving only a small portion of the producer’s overall revenue. We show that if the goods are well-priced, and the type distribution satisfies the standard regularity property, then offering all items is a -approximation to the optimal revenue.

Theorem 1.1.

For combinatorial assortment with well-priced items and regular type distribution, the assortment that selects all items is a -approximation to the optimal expected revenue.

We also show that if there is a cardinality constraint on the number of items that can be shown, then greedily accepting items to maximize marginal revenue also yields a constant approximation when the valuations satisfy the gross substitutes condition, which is a stronger notion of substitutability than submodularity.

Theorem 1.2.

For cardinality-constrained combinatorial assortment with well-priced items, a gross substitutes valuation, and regular type distribution, the assortment that selects items greedily by revenue is a -approximation to the optimal expected revenue.

In addition to these approximation results, we present an exact algorithm for combinatorial assortment when the valuation function is -demand additive. That is, when each buyer desires at most items, and the value for such a bundle is the sum of the individual item values. This class extends unit-demand valuations to bundles of more than a single item. For this setting, we describe a dynamic programming solution that runs in time . Our solution builds an optimal assortment by first optimizing for high-type buyers and incrementally modifying the assortment to cater to lower types. This algorithm does not require any assumptions about items being well-priced, and applies whether or not there are cardinality constraints on the assortment.

Finally, for -demand valuations that may not be additive, we show that under a certain revenue-concavity assumption on the type distribution, the optimal assortment will have size at most .

Extension: Welfare Maximization

We conclude by considering two extensions. First, we note that most of our positive results apply also to the goal of maximizing welfare, rather than maximizing revenue. The welfare maximization problem is still non-trivial, since the presence of cheap goods can result in lower-valued items being purchased. However, we show that if items are well-priced then offering all items is, in fact, the welfare-optimal assortment. Note that this is a stronger result than for revenue-maximization, where we established a -approximation. Under a cardinality constraint, the greedy algorithm for assortment yields a approximation to the optimal welfare for well-priced items and gross substitutes valuations. Finally, our dynamic program for additive -demand valuations applies just as well to the welfare objective, and can be used to compute a welfare-optimal assortment. Kleinberg et al. [11] study the learnability of a class of comparison-based choice functions.

Extension: Learning

The second extension concerns a setting where and are not known to the seller. Rather, the seller must learn these through demand queries: repeatedly choosing a slate of items and observing a buyer’s choice. We show that the dynamic programming solution for -demand additive valuations can be implemented in this learning setting, with the loss of an additive error factor, using queries.

1.2 Related Work

There is a growing literature on (unit-demand) assortment optimization in the management science literature. Talluri and van Ryzin [17] provide a closed-form solution when buyer choices follow the multinomial logit model. Rusmevichientog et al. [16] extend this solution to the case of cardinality-restricted assortment, and Davis et al. [7] show how to solve for the optimal assortment under more general nested logit models. When the choice function is described by a mixture of multinomial logit models, the assortment problem is NP-hard but various integer programming methods and approximation algorithms are known [3, 8, 15].

There has also been work studying learning in assortment, where the product slate can be adjusted to learn customer preferences. Caro and Gallein [5] consider learning in a model of assortment without substitution effects, where the demand for each product is unaffected by the other products in the assortment. Ulu et al. [18] study the dynamic learning problem when products exhibit purely horizontally differentiation, as modeled by location on a line segment. Agrawal et al. [1] consider a multi-armed bandit model of dynamic assortment, and show how to achieve near-optimal regret for multinomial logit choice models. Kleinberg et al. [11] consider a general class of comparison-based choice models, and study the complexity of learning their model from samples.

The combinatorial assortment problem can be viewed as a restricted form of mechanism design, where the design space consists only of choosing which subset of items to display. This is more restrictive than sequential posted pricing, where the designer can also choose the price at which each item can be sold (e.g., [6]).

2 The Combinatorial Assortment Optimization Problem

There is a set of items. Each item has a fixed price . We assume items are indexed so that . There is an unbounded supply (i.e., number of copies) of each item.

There is a collection of buyers, each of whom wish to purchase a subset of the items. Each buyer has a value for each subset of goods. Here is a common valuation that determines the relationship between the goods, for all buyers, and is a buyer-specific scaling factor. We refer to as the type of buyer . We assume that each is sampled independently from a distribution , which we refer to as the type distribution. We sometimes also call the multiplicative noise of buyer . When is a point mass on 1 (i.e., for each buyer ), we call the problem noiseless. We call the general problem noisy.

Given a subset of items displayed to a buyer , the buyer will pick maximizing and pay . Our goal as a seller is to pick an optimal assortment, which is a subset of at most items that maximizes the expected revenue. Here is a parameter of the problem. We will focus first on the unconstrained case of , then consider general in Section 5. For most of the paper we will assume that and are known to the seller and given as inputs to the optimization problem. In Section 5 we relax this assumption and suppose and are fixed but unknown to the seller, who must learn about them by interacting with buyers.

Valuation classes.

We focus on variants of the combinatorial assortment problem where the valuation function lies in a given class. We assume that valuations are monotone non-decreasing and normalized so that . In this paper we will focus on the following valuation classes, which encode forms of substitutability between items.

  • additive: there exist such that .

  • XOS: there exist additive valuations (i.e., clauses) such that .

  • submodular: for all , .

  • gross substitutes: for all and , one of the following is true:444We use the M#-exchange characterization of gross substitutes, since it will be convenient for our proofs [14].

    1. .

    2. There exists , .

We will also be interested in valuations that encode a constraint that a buyer does not derive benefit from receiving more than a certain number of items.

Definition 2.1.

Valuation is -demand if, for all , . That is, the buyer derives no benefit from receiving more than items. We say that valuation is additive (resp. XOS, submodular) -demand if there is an additive (resp. XOS, submodular) valuation such that, for all , .

We note that these valuation classes can be ordered from most to least restrictive, as follows: Additive -demand gross substitutes submodular XOS.

3 Hardness of Combinatorial Assortment

In this section we explore the hardness of the Combinatorial Assortment problem. We give a general hardness of approximation result for XOS valuations, even in the noiseless setting. We then show that even when valuations can be succinctly represented, the problem remains NP-hard. We also demonstrate that even when valuations are submodular, the natural greedy heuristic fails to obtain a good approximation. All missing proofs can be found in Appendix B.

Hardness of approximation, even without noise.

We begin by considering the noiseless setting, where is a point mass at and hence the valuation of the buyer is known exactly. Our first result shows that for XOS valuations, the combinatorial nature of the problem leads to strong hardness of approximation. Indeed, it may take exponential many demand queries to achieve better than an -approximation to the combinatorial assortment problem.

Theorem 3.1.

For XOS valuations, any -approximate algorithm for the combinatorial assortment problem requires demand queries.

Note that Theorem 3.1 is a query complexity bound, and puts no limitations on the algorithm’s running time. Theorem 3.1 can be extended to a more general statement about communication complexity under a certain query model. See Remark B.1 for details. The general result will suggest that combinatorial assortment problem is hard to approximate with a sub-exponential number of a certain class of queries. Note that we cannot hope for Theorem 3.1 to extend to a fully general communication complexity bound with an arbitrary query model: if arbitrary queries are allowed, one could directly ask for the optimal assortment, which can be succinctly described.

The proof of Theorem 3.1 follows by reducing from the communication complexity of the equality function to the combinatorial assortment problem. Two players, Alice and Bob, play a communication game where they each hold an (exponentially-long) input string and want to determine if they hold the same string. They each use their input strings to construct XOS function clauses, and the input to the combinatorial assortment problem will be the XOS valuation function containing both Alice and Bob’s clauses. Each of Alice’s clauses corresponds to a large set of items, and assigns small values; each of Bob’s corresponds to a small set, and assigns large values. The buyer will only ever buy a set of items corresponding to one of these clauses. The optimizer would prefer that the buyer chooses one of Alice’s large sets. However, the clauses are constructed so that if Alice and Bob’s inputs are equal, then each of Alice’s clauses is “dominated” by one of Bob’s clauses, so there is no assortment where the buyer purchases many items. However, if the inputs are unequal, then at least one of Alice’s clauses is “uncovered,” and the corresponding items would be purchased if they were the only items available. By carefully designing the XOS clauses in this way, we can show that approximation of the combinatorial assortment problem will also solve the equality problem.

Hardness for succinct valuations.

Theorem 3.1’s hardness is a communication bound, and relies on the fact that an XOS function may require exponentially many bits to fully describe. As we now show, the combinatorial assortment problem remains hard even for XOS valuations with succint descriptions. In particular, the problem is NP-hard, again in the noiseless setting, even if we restrict to valuations with only two clauses (i.e., the maximum of two additive functions).

Theorem 3.2.

For any XOS valuations with only 2 clauses, finding the optimal revenue is NP-hard in the noiseless case and the offline setting.

The idea of the proof is to relate the optimal revenue of the combinatorial assortment problem to the solution to a knapsack problem, implementing the knapsack constraints by comparing values between the two clauses in the combinatorial assortment problem.

Hardness for -demand valuations in the noisy setting.

One might also wonder if the hardness results above are driven by the large sets of goods desired by the buyers. What if we restrict attention to -demand buyers, where is a small constant?

One observation is that in the noiseless setting, the optimal assortment for a -demand valuation will contain at most items, so the problem can be solved in time by evaluating the revenue for all subsets of size . So this question is interesting only in the more general noisy setting.

Theorem 3.3 shows that even for submodular 2-demand valuations, there can be no FPTAS for combinatorial assortment. Therefore, we can only hope to get an efficient algorithm for -demand valuations if we add add more restrictions, for example, to require the valuations to also be additive.

Theorem 3.3.

For submodular -demand valuations, it is NP-hard to approximate the optimal revenue within approximation factor , for some large enough constant . In particular, there is no FPTAS in this setting unless .

The proof is a reduction from the -clique problem. Given a graph, we construct a 2-demand valuation and a distribution over the types. We embed the edge information into the prices of pairs of items. is carefully chosen such that an assortment has large revenue if and only if it corresponds to a set of vertices which form a -clique in the original graph.

Greedy assortment fails for submodular valuations.

We’ve shown that there is no FPTAS for submodular valuations in the general noisy setting. One might wonder if it’s possible to obtain a constant approximation, however, by using a simple heuristic. One natural idea for submodular valuations is to use a greedy approach: repeatedly add the revenue-maximizing item to the assortment, until either no item remains or until adding any one item causes revenue to decrease. The following example shows that this heuristic can lead to approximation , even without noise.

Example 3.1.

There are items, which we’ll label . The valuation is:

One can verify that this valuation is indeed submodular. Suppose and for all . The greedy algorithm selects item first, as it generates revenue which is larger than , the revenue from any other single item. However, having selected item , the greedy algorithm would not add more items, since if the assortment is for any , the buyer would choose to buy only item leading to a loss of revenue. So greedy obtains revenue . The optimal assortment takes all items other than , for a revenue of .

4 Structural and Algorithmic Results

Approximate assortment for well-priced items.

As mentioned in the introduction, it can be highly suboptimal to select all items in the combinatorial assortment problem, since the presence of a cheap but valuable item might cannibalize revenue from more expensive items. One might wonder, then, if such a situation can be made less severe if the items are all priced “reasonably.” For example, suppose that each individual item is assigned the price that would maximize revenue when that item is sold by itself. Indeed, we would argue that such prices are very reasonable if the items are typically sold separately, and it is precisely the assortment platform that presents these items in combination with each other. We will show that under such an assumption, plus a regularity assumption on the type distribution, it is approximately revenue-optimal to show all of the items. Let us first define formally the assumptions needed for our result.

Definition 4.1 (Regularity).

We say that type distribution is regular if the virtual value function is non-decreasing, where denotes the density function of distribution .

Regularity is a common assumption in the revenue maximization literature. Many natural distributions are regular, including uniform, gaussian, and exponential distributions.

Definition 4.2 (Revenue Curve).

The revenue curve of a type distribution is .

We can think of as describing the revenue obtained if we were to offer a single item with value and price to a buyer whose type is drawn from distribution . As we show the total revenue of an assortment can be expressed as a function of .

The optimal reserve (or Myerson reserve) for is the value that maximizes (or the supremum over such values , if the maximum is not unique).

Definition 4.3 (Well-priced).

Suppose type distribution is regular with Myerson reserve and non-increasing density after .555In fact, our results for well-priced combinatorial assortment hold for distributions that satisfy a weaker condition than regularity. It is enough for the well-pricedness condition to hold for some value (not necessarily the Myerson reserve) such that the density function is non-increasing after , and the revenue curve is non-increasing after . Then the combinatorial assortment problem with type distribution is well-priced if, for each subset of items ,

We think of as a desired threshold on the type of buyers who purchase items. For example, if we focus on a single item , then reserve corresponds to a price of , as this is the price at which a buyer with type would choose to purchase. The well-priced condition requires that the price assigned to any set of items is at least the reserve , scaled appropriately to the value of the set. Note that if is subadditive, it is enough for each individual item to be well-priced as this implies the condition for any larger set of items as well.

We show that for well-priced instance of the combinatorial assortment problem, selecting all items yields a -approximation to the optimal revenue. The proof can be found at Appendix C.1.

Theorem 4.4.

Choosing is a -approximation to the optimal revenue for well-priced combinatorial assortment.

The idea behind Theorem 4.4 is to show that the revenue curve can be well-approximated by a modified revenue curve that is convex on the range . We show that for convex curves, maximizing revenue reduces to the problem of maximizing utility, and hence the (modified) revenue is maximized by the assortment that maximizes buyer utility, which is to display all items.

Exact assortment when revenue is concave.

This approximation result used intuition that when revenue curves are convex, it is preferable to show as many items as possible. As it turns out, the reverse intuition holds as well: if the revenue curve is concave, then it is preferable to show fewer items. In particular, if buyers are -demand, then the optimal assortment will consist of at most items. The proof of Theorem 4.5 can be found in Appendix D.1.

Theorem 4.5.

Suppose that buyers are -demand, and the revenue function is concave over the support of the type distribution . Then there exists an optimal assortment with .

Recall that in Section 3 we noted that, in general, the optimal assortment for -demand buyers may contain far more than items. In particular, a heuristic that simply enumerates all assortments of size at most will not find an optimal solution in general. Theorem 4.5 shows that such a heuristic does find an optimal solution in cases where the revenue curve is concave.

Example 4.1.

Suppose that buyers are -demand with uniform type distribution over . The revenue curve is concave for all and thus by Theorem 4.5 the optimal assortment consists of at most items.

Remark 4.1.

If in Example 4.1 items are well-priced, Theorem 4.4 implies that even though the optimal assortment is small, showing all items yields a -approximation to the optimal revenue.

Exact combinatorial assortment for -demand buyers.

We showed in Section 3 that the combinatorial assortment problem is hard even for submodular -demand buyers. We instead turn to additive -demand valuations, which include unit-demand valuations as a special case. We show that for constant , there is a polynomial-time algorithm that solves the combinatorial assignment problem. The proof of Theorem 4.6 can be found in Appendix E.1. Importantly, this result applies even in the general noisy cases where buyer values are not fully known in advance. Like the submodular -demand case considered in Section 3, the optimal assortment for additive -demand valuations may include many more than items.

Theorem 4.6.

For additive -demand valuations, there exists an algorithm that finds the revenue-optimal assortment in time666Our algorithms depend on the type distribution, which may be continuous. The runtime bound assumes that the CDF of this distribution can be queried in time. See Appendix A for a detailed discussion. .

The algorithm we propose is a dynamic program, which incrementally builds an optimal assortment by considering how the purchasing behavior of a buyer changes with . To build intuition for our dynamic program, consider first the unit-demand case of . In this case, each buyer chooses at most a single item to purchase. The utility derived by purchasing item is , which we can plot as a line mapping to utility. A choice of assortment then corresponds to a subset of possible lines; and for any given value of , the item with the highest utility would be chosen. We can think of this as tracing the maximum over this set of lines; see Figure 2. Given this pictorial representation our DP algorithm computes the optimal revenue “from left to right” adding lines/items to the assortment but only keeping track of the last line that was added.

In the case , we are interested in the revenue obtained by tracing the top lines given an assortment. Computing the optimal revenue is inherently more difficult in this case, as it doesn’t suffice to store only the top lines at any given point (see Figure 6 in Appendix E.1). However, we are able to extend our DP by showing that any lines that were among the top earlier but are not in the top at the current point won’t be among the top lines for any . This allows us to only keep track of the top lines, resulting in runtime.

Figure 2: The dark solid line represents a unit-demand buyer’s utility for the assortment . It is the upper envelope of the lines corresponding to items 2 and 4.

5 Extensions

Constrained assortment.

To this point we focused exclusively on the case of unconstrained assortment, where . For general , the lower bounds from Section 3 still apply. Also, the dynamic program for exact revenue-optimal assortment for additive -demand valuations solves the constrained case; one need only track the remaining budget for additional items as part of the program. See Corollary E.6 for a detailed proof.

Theorem 5.1.

For additive -demand valuations and any cardinality constraint , there exists an algorithm that finds the revenue-optimal assortment of at most items in time .

Theorem 4.4 specified conditions under which it is approximately optimal to select all items. Under a cardinality constraint, this solution may not be feasible. However, if the buyer valuations are gross substitutes, a greedy assortment algorithm is approximately optimal. The idea is to reduce from revenue maximization to utility maximization, as in Theorem 4.4, then note that the total utility derived from the buyers is a submodular function. See Appendix C.2 for details.

Theorem 5.2.

For gross substitutes valuations and well-priced items, a -approximation to the revenue-optimal assortment of size at most can be computed in time .

Welfare maximizing assortment.

We have focused on revenue-maximization, but assortment optimization for welfare maximization is also non-trivial. The presence of cheap items in the assortment can reduce the total welfare and should be excluded. We note that the algorithm we developed for revenue-maximization under additive -demand valuations can be easily adjusted for welfare maximization (see Remark E.1 for details). Also, if items are well-priced, our results for revenue maximization apply to welfare maximization with even better constants. In particular, for unconstrained assortment, selecting the slate of all items is welfare-optimal if items are well-priced. See Appendix C.3.

Learning assortments from demand samples.

Suppose and are not known to the seller. Instead, the algorithm can learn about and via samples, taken by choosing a slate of items to sell and observing a buyer’s choice. Details appear in Appendix F.

We show how to implement our dynamic program for -additive valuations in this learning setting, by characterizing the algorithm’s robustness to noise. We show that if the algorithm can make queries, then our dynamic programming solution will be within an additive factor to the optimal revenue.

We also show that a variant of Theorem 4.5 applies to the learning setting. This requires choosing the best of a polynomial number of assortments. Since the highest revenue is bounded, standard concentration arguments imply that we can evaluate the revenue of any given assortment to within a small additive error by making polynomially many queries.

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Appendix A A Note on Computation

The algorithm of Theorem 4.6, as well as all the algorithms presented in this work, depend on the distribution of types which may be continuous. The only assumption required for their runtime is that the CDF of this distribution can be queried in time. As the CDF may be a real number, our algorithms require a real RAM model where basic calculus can be performed as a single operation. This assumption can be easily dropped if the CDF oracle returns only the CDF within accuracy of , where is the number of bits required to represent valuations and prices. As the revenue of an assortment equals and all probabilities are accurate within , this results in an additive error in revenue calculation of at most .

Additionally, revenue calculations depend only on the CDF at points where consumers are indifferent about the set of items to purchase. These are points of the form for . Since we assume both valuations and prices can be represented using -bit numbers, the algorithms require that the CDF is only queried in rational numbers where both numerator and denominator are -bit integers.

The algorithm of Theorem 5.2 requires computing a modified revenue curve which is convex so that the resulting optimization problem is submodular. The optimal such curve corresponds to the lower convex envelope of the actual revenue curve . While this curve might be easily computable in closed form in some cases, it cannot be computed efficiently using only query access to the CDF. To overcome this issue, we first preprocess the revenue curve by rounding it into powers of . This results in a different revenue maximization problem whose solution is close to the original one. Moreover, this revenue curve consists of very few pieces and can be listed explicitly and thus computing the convex envelope and all revenue calculations can be done efficiently. See details in Appendix C.2.

Appendix B Missing Proofs of Section 3

b.1 Hardness results in the noiseless setting

Lemma B.1.

For any constant , there exist sets ’s which have sizes and are subsets of such that

Proof.

We are just going to pick random subset of size . Then we will show the probability that is positive.

Fix some pair . Define random variable to be one if and . Otherwise will be 0. We have

Although ’s are not independent, they are negative correlated. We can apply the multiplicative Chernoff bound:

Then by a Union bound over all pairs , we have

Theorem B.2 (Restatement of Theorem 3.1).

For XOS valuations, any algorithm (no restrictions on the running time) which approximates the optimal revenue within factor smaller than needs demand queries in the noiseless case.

Proof.

By Lemma B.1, we can find sets ’s which have sizes and are subsets of such that

Let , For each , define to be an arbitrary partition of , i.e.,

  1. for all .

  2. for all .

Let be a subset of . Define the XOS valuation as the following by specifying its clauses:

  1. For each in , has clause such that for all and for all . These clauses are called c-clauses.

  2. For each in and each , has clause such that for and for . These clauses are called d-clauses.

Suppose there’s an algorithm (can be randomized) which guarantees better than -approximation on the optimal revenue for XOS valuations. Let’s assume uses queries in expectation.

Now consider the following communication problem:

  1. Alice gets input which is a subset of and . And Bob gets input which is a subset of and .

  2. The goal is just to decide whether by communication between Alice and Bob.

This problem is just equality problem, which has zero-error randomized communication complexity (see Example 3.9 of [12]).

Now consider the following protocol for the communication problem based on algorithm :

  1. Alice and Bob run algorithm locally on valuation and item price 1 (i.e. ). They use public randomness if needs randomness. Notice that without any information of they can run most part of except demand queries. They are going to simulate the value queries by the following communication procedure.

  2. Whenever is making a demand query, Alice sends buyer’s favorite subset over all the clauses she knows (i.e. ’s for all ). Alice also sends the utility of that subset. Bob does the similar thing. Then they can figure out the result of the demand query by picking the subset with better utility.

  3. After running , if the output of is larger than , output “not equal”. Otherwise output “equal”.

Since Alice and Bob use bits of communication for each demand query, the expected communication complexity of is .

Now let’s prove correctly solves equality.

  1. When , we are going to show that the optimal revenue is at most . Let’s assume the optimal revenue is achieved by the seller showing and the buyer buying .

    1. If is evaluated on some c-clause, let’s assume the clause is . Then we know . Since , we know . Since is a partition of , we know there exists such that . The utility of buying is at least . On the other hand, the utility of buying is . We get a contradiction now and therefore it’s never the case that is evaluated on some c-clause.

    2. If is evaluated on some d-clause, then we know will be smaller than as any d-clause are non-zero on items. Therefore in this case the revenue is at most .

  2. When , as , there exists such that and . We are going to show that if the seller show subset , the buyer will buy the entire set. And therefore the optimal revenue is . Formally, we will show for any subset , the utility of the buying is at most the utility of buying for the buyer. It is clear that the utility of buying is . For any ,

    1. If is evaluated on some c-clause, then the utility of buying is at most . The equality is only achieved when .

    2. If is evaluated on some d-clause, let that d-clause be . Since , we know , therefore . Therefore the utility of buying is at most .

Since guarantees better than -approximation, when , it would output something larger than . And when , it will output something at most . Therefore protocol correctly solves equality. Then by the communication lower bound, we know that . Therefore . ∎

Remark B.1.

It’s easy to check that the proof of Theorem 3.1 still works if we switch demand queries with other queries which can be computed by Alice and Bob using polynomial many bits of communication. For example, a value query can be simulated by bits as Alice and Bob can just report the value of the set in their own parts and then take the maximum.

Lemma B.3.

Among all the subsets that achieve the optimal revenue for the seller, there exist a subset such that if the seller shows , the buyer would pick all the items in .

Proof.

Let be an arbitrary set that achieves the optimal revenue for the seller. Assume in this case, buyer chooses . Then if we just show the buyer , the buyer will still pick . ∎

Theorem B.4 (Restatement of Theorem 3.2).

For any XOS valuations with only 2 clauses, finding the optimal revenue is NP-hard in the noiseless case.

Proof.

We will reduce from the knapsack problem which is NP-hard, i.e.:

maximize
subject to

Suppose we have an algorithm to find the optimal revenue for XOS valuations with 2 clauses. We are going to solve the knapsack problem in the following way:

We set . We then construct XOS valuation with 2 clauses and . We set , and price ’s as:

  1. For , , and .

  2. For , , and .

Notice that the optimal revenue is at least because the seller can always show only item . And as , to achieve the optimal revenue, the seller needs to make sure the buyer purchases item , which means the value needs to be evaluated on .

Now we are going to show the optimal revenue is equal to the maximum objective of the knapsack problem plus .

  1. By Lemma B.3, let be the set that when the seller shows , the buyer would buy and the seller gets optimal revenue. As discussed before . Then we know that . This implies . Therefore, if we pick iff , we have and at least the optimal revenue minus . Therefore the optimal revenue is at most the maximum objective of the knapsack problem plus .

  2. Now consider the optimal solution for the knapsack problem ’s. Define . It’s easy to check that the buyer would pick if the seller shows because . Therefore the optimal revenue is at least the maximum objective of the knapsack problem plus .

We finish the proof by noticing the fact that the decision version of the knapsack problem is NP-hard. ∎

b.2 Hardness for -demand valuations in the noisy setting

Theorem B.5 (Restatement of Theorem 3.3).

For submodular -demand valuations, it’s NP-hard to approximate the optimal revenue within approximation factor in the noisy case for some large enough constant . Therefore there’s no FPTAS to compute the optimal revenue in this setting unless .

Proof.

The reduction is from -clique. Let the -clique instance be graph . Let . Now consider the following mathematical program:

maximize
subject to

The objective is maximized when , otherwise the objective will be negative. It’s easy to see maximized value is if and only if graph has a -clique. The objective can be rewritten as

Next we are going to show that the assortment algorithm can be used to solve the following mathematical program when and . This is more general than the above mathematical program. Therefore it will imply the assortment algorithm can be used to solve the -clique problem.

maximize
subject to

Let be different positive integers such that the pairwise sums of are all different and . We can find such sequence by using numbers that are . The existence can be proved by randomly picking integers and using probabilistic argument. Now let be the sorted list of and the square roots of ’s pairwise sums (i.e. ). Here . Let and for and . So now we have . Figure 3 shows what it looks with these options as lines.

Figure 3: Example of ’s and ’s viewed as lines

Now consider the following 2-demand valuation with items. Each , corresponds to an item with price and value . Each , also corresponds to an item with price and value . If two items are both corresponding to some and , the value of the bundle will be . Other bundles of 2 items will have values equal to the higher value of one of its items. Since and this valuation is 2-demand, its also submodular.

It’s also that no matter what the seller shows, the buyer would either buy nothing, or a single item, or 2 items that correspond to some and . Therefore the sets the buyer could buy correspond to ’s and ’s. The prices of these sets are or . The values of these sets are or .

Now consider a distribution on the multiplicative noise that is specified by the following CDF .

  1. For , , we have .

  2. For , , we have .

  3. For , we have .

  4. The probability of multiplicative noise smaller than is . It does not matter where they actually distribute at for our problem since these buyers won’t buy anything anyways.

Now define for . We have

  1. For , , we have .

  2. For , , we have . In other words, is linear between and .

  3. For , we have .

Notice that is a convex function. And is a piece-wise linear function based on points on curve . So is also convex.

Now we are going to specify the expected revenue for valuation and multiplicative noise distribution . Suppose the set shown by the seller is and is the set of ’s and ’s corresponding to available set options for the buyer when is shown. Let and elements in are sorted as . For notation convenience, let . It’s easy to check that is has the highest utility for the buyer if . The revenue can be written as

We are going to prove the following lemma to characterize the set ’s which achieve the optimal revenue.

Lemma B.6.

If achieves the optimal revenue if and only if it contains all the items correspond to .

Proof.

We prove the two directions separately:

  1. “only if”: We prove by contradiction. Suppose does not have some item corresponds . We will show by adding this item, the revenue will strictly increased. There are two cases:

    1. When , adding this item would increase the revenue by .

    2. When , let be the index such that , adding the item would increase the revenue by

      Since is a convex function, we know

      Moreover, since and , we know and locate on two different piece-wise linear functions of . Therefore we know the previous inequality is strict, i.e.

      This will imply that adding item corresponds to would increase the revenue by some positive value.

  2. “if”: Consider the option set for , we will first show that if and , then to won’t change the revenue. Similarly as the “only if” part, adding will increase the revenue by

    Since and , we know that , and are on the same piece-wise linear function, therefore

    This means the revenue does not change.

    Now we can apply this claim to several times and we know that the revenue of showing is the same the revenue of showing everything.

    Thus we know that if contains all the items correspond to , no matter what other items have, the revenue is the same. This together with the “only if” part will imply that the “if” part.

Notice that from Lemma B.6, we know that if does not contain all the items correspond to , the revenue of will be strictly worse than the optimal revenue. Let’s assume optimal revenue to be . Let’s also assume ’s revenue will be at least worse than . It’s easy to see that this is not too small. It should be for some constant .

Now consider another distribution on the multiplicative noise. will be used to embed the mathematical program. is discrete, we will describe it by its support and density. For each ,

  1. If equals to some , will have density at location slightly smaller than .

  2. IF equals to some , will have density at location slightly larger than .

Here is just some scaling factor that makes sure that the densities sum up to 1. Now suppose has all the items correspond to . Let indicate whether item corresponds to is included in . And let be the revenue of only showing items correspond to on distribution , then the revenue of on can be written as

It is very similar to the objective of the mathematical program.

Finally we consider a distribution where . Now we are going to understand the optimal revenue for . No matter what we show, the revenue from part will be at most . On the other hand, for , if we don’t show some set with all the items correspond to , we will at least lose revenue. Therefore, to achieve the optimal revenue for , we also need to include all the items correspond to . Then the optimal revenue can be written as

Since and can be computed in polynomial time, computing the optimal revenue will also solve the mathematical program. Notice that for the -clique problem, the optimized value of the mathematical program is differed by some constant between the case when the instance has a -clique and the case when the instance does not have a -clique. It is easy to check that is for some large enough constant . Therefore approximating the optimal revenue within approximation factor for some large enough constant will also solve the -clique problem which is NP-hard.

Finally one thing worth mentioning is that in the proof we define as a continuous distribution, it might be annoying to make it to be an input for algorithms. Actually we can discretize this distribution by picking a discrete distribution that has the same CDF as at all locations which are pairwise sums of . It’s easy to check that for our valuation , the preferences of options stay the same between any two adjacent locations of these locations. So discretizing in this way won’t affect the proof. ∎

Appendix C Missing Proofs for Well-Priced Items (Thm 4.4,