1 Introduction
In the Santa Claus problem (sometimes referred to as MaxMin Fair Allocation) we are given a set of players and a set of indivisible resources . In its full generality, each player has a utility function , where measures the happiness of player if he is assigned the resource set . The goal is to find a partition of the resources that maximizes the happiness of the least happy player. Formally, we want to find a partition of the resources that maximizes
Most of the recent literature on this problem focuses on cases where is a linear function for all players . If we assume all valuation functions are linear, the best approximation algorithm known for this problem, designed by Chakrabarty, Chuzhoy, and Khanna [4], has an approximation rate of and runs in time for . On the negative side, it is only known that computing a approximation is NPhard [12]. Apart from this there has been significant attention on the socalled restricted assignment case. Here the utility functions are defined by one linear function and a set of resources for each player . Intuitively, player is interested in the resources , whereas the other resources are worthless for him. The individual utility functions are then implicitly defined by . In a seminal work Bansal and Srividenko [3] provide a approximation algorithm for this case. This was improved by Feige [8] to an approximation. Further progress on the constant or the running time was made since then, see e.g. [1, 7, 6, 5, 10, 2, 15].
Let us now move to the nonlinear case. Indeed, the problem becomes hopelessly difficult without any restrictions on the utility functions. Consider the following reduction from set packing. There are sets of resources and all utility functions are equal and defined by if for some and otherwise. Deciding whether there are disjoint sets in (a classical NPhard problem) is equivalent to deciding whether the optimum of the Santa Claus problem is nonzero. In particular, obtaining any bounded approximation ratio for Santa Claus in this case is NPhard.
Two naturally arising properties of utility functions are monotonicity and submodularity, see for example the related submodular welfare problem [11, 17] where the goal is to maximize . A function is monotone, if for all . It is submodular, if for all and . The latter is also known as the diminishing returns property in economics. A standard assumption on monotone submodular functions (used throughout this work) is that the value on the empty set is zero, i.e., . Goemans, Harvey, Iwata, and Mirrokni [9] first considered the Santa Claus problem with monotone submodular utility functions as an application of their fundamental result on submodular functions. Together with the algorithm of [4] it implies an approximation in time .
In this paper we investigate the restricted assignment case with a monotone submodular utility function. That is, all utility functions are defined by , where is a monotone submodular function and is a subset of resources for each players . Before our work, the stateoftheart for this problem was the approximation algorithm mentioned above, since none of the previous results for the restricted assignment case with a linear utility function apply when the utility function becomes monotone submodular.
1.1 Overview of results and techniques
Our main result is an approximation algorithm for the submodular Santa Claus problem in the restricted assignment case.
Theorem 1.
There is a randomized polynomial time approximation algorithm for the restricted assignment case with a monotone submodular utility function.
Our way to this result is organised as follows. In Section 2, we first reduce our problem to a hypergraph matching problem (see next paragraph for a formal definition). We then solve this problem using Lovasz Local Lemma (LLL) in Section 3. In [3] the authors also reduce to a hypergraph matching problem which they then solve using LLL, although both parts are substantially simpler. The higher generality of our utility functions is reflected in the more general hypergraph matching problem. Namely, our problem is precisely the weighted variant of the (unweighted) problem in [3]. We will elaborate later in this section why the previous techniques do not easily extend to the weighted variant.
The hypergraph matching problem.
After the reduction in Section 2 we arrive at the following problem. There is a hypergraph with hyperedges over the vertices and . We write and . We will refer to hyperedges as configurations, the vertices in as players and as resources^{5}^{5}5We note that these do not have to be the same players and resources as in the Santa Claus problem we reduced from, but and do not increase.. Moreover, a hypergraph is said to be regular if all vertices in and have the same degree, that is, they are contained in the same number of configurations.
The hypergraph may contain multiple copies of the same configuration. Each configuration contains exactly one vertex in , that is, . Additionally, for each configuration the resources have weights . We emphasize that the same resource can be given different weights in two different configurations, that is, we may have for two different configurations .
We require to select for each player one configuration that contains . For each configuration that was selected we require to assign a subset of the resources in which has a total weight of at least to the player in . A resource can only be assigned to one player. We call such a solution an relaxed perfect matching. One seeks to minimize .
We show that every regular hypergraph has an relaxed perfect matching for some assuming that for all , that is, all weights are small compared to the total weight of the configuration. Moreover, we can find such a matching in randomized polynomial time. In the reduction we use this result to round a certain LP relaxation and essentially translates to the approximation rate. This result generalizes that of Bansal and Srividenko on hypergraph matching in the following way. They proved the same result for unit weights and uniform hyperedges, that is, for all and all hyperedges have the same number of resources^{6}^{6}6In fact they get a slightly better ratio of .. In the next paragraph we briefly go over the techniques to prove our result for the hypergraph matching problem.
Our techniques.
Already the extension from uniform to nonuniform hypergraphs (assuming unit weights) is highly nontrivial and captures the core difficulty of our result. Indeed, we show with a (perhaps surprising) reduction, that we can reduce our weighted hypergraph matching problem to the unweighted (but nonuniform) version by introducing some bounded dependencies between the choices of the different players. For sake of brevity we therefore focus in this section on the unweighted nonuniform variant, that is, we need to assign to each player a configuration and at least resources in . We show that for any regular hypergraph there exists such a matching for assuming that all configurations contain at least resources and we can find it in randomized polynomial time. Without the assumption of uniformity the problem becomes significantly more challenging. To see this, we lay out the techniques of Bansal and Srividenko that allowed them to solve the problem in the uniform case. We note that for the statement is easy to prove: We select for each player one of the configurations containing uniformly at random. Then by standard concentration bounds each resource is contained in at most
of the selected configurations with high probability. This implies that there is a fractional assignment of resources to configurations such that each of the selected configurations
receives of the resources in . By integrality of the bipartite matching polytope, there is also an integral assignment with this property.To improve to in the uniform case, Bansal and Srividenko proceed as follows. Let be the size of each configuration. First they reduce the degree of each player and resource to using the argument above, but taking configurations for each player. Then they sample uniformly at random resources and drop all others. This is sensible, because they manage to prove the (perhaps surprising) fact that an relaxed perfect matching with respect to the smaller set of resources is still an relaxed perfect matching with respect to all resources with high probability (when assigning the dropped resources to the selected configurations appropriately). Indeed, the smaller instance is easier to solve: With high probability all configurations have size and this greatly reduces the dependencies between the bad events of the random experiment above (the event that a resource is contained in too many selected configurations). This allows them to apply Lovász Local Lemma (LLL) in order to show that with positive probability the experiment succeeds for .
It is not obvious how to extend this approach to nonuniform hypergraphs: Sampling a fixed fraction of the resources will either make the small configurations empty—which makes it impossible to retain guarantees for the original instance—or it leaves the big configurations big—which fails to reduce the dependencies enough to apply LLL. Hence it requires new sophisticated ideas for nonuniform hypergraphs, which we describe next.
Suppose we are able to find a set of configurations (one for each player) such that for each the sum of intersections with smaller configurations is very small, say at most . Then it is easy to derive a relaxed perfect matching: We iterate over all from large to small and reassign all resources to (possibly stealing them from the configuration that previously had them). In this process every configuration gets stolen at most of its resources, in particular, it keeps the other half. However, it is nontrivial to obtain a property like the one mentioned above. If we take a random configuration for each player, the dependencies of the intersections are too complex. To avoid this we invoke an advanced variant of the sampling approach where we construct not only one set of resources, but a hierarchy of resource sets by repeatedly dropping a fraction of resources from the previous set. We then formulate bad events based on the intersections of a configuration with smaller configurations , but we write it only considering a resource set of convenient granularity (chosen based on the size of ). In this way we formulate a number of bad events using various sets . This succeeds in reducing the dependencies enough to apply LLL. Unfortunately, even with this new way of defining bad events, the guarantee that for each the sum of intersections with smaller configurations is at most is still too much to ask. We can only prove some weaker property which makes it more difficult to reconstruct a good solution from it. The reconstruction still starts from the biggest configurations and iterates to finish by including the smallest configurations but it requires a delicate induction where at each step, both the resource set expands and some new small configurations that were not considered before come into play.
Additional implications of nonuniform hypergraph matchings to the Santa Claus problem.
We believe this hypergraph matching problem is interesting in its own right. Our last contribution is to show that finding good matchings in unweighted hypergraphs with fewer assumptions than ours would have important applications for the Santa Claus problem with linear utility functions. We recall that here, each player has its own utility function that can be any linear function. In this case, the best approximation algorithm is due to Chakrabarty, Chuzhoy, and Khanna [4] who gave a approximation running in time . In particular, no subpolynomial approximation running in polynomial time is known. Consider as before a nonuniform hypergraph with unit weights ( for all such that ). Finding the smallest (or an approximation of it) such that there exists an relaxed perfect matching in is already a very nontrivial question to solve in polynomial time.
We show, via a reduction, that a approximation for this problem would yield a approximation for the Santa Claus problem with arbitrary linear utility functions. In particular, any subpolynomial approximation for this problem would significantly improve the stateoftheart^{7}^{7}7We mention that our result on relaxed matchings in Section 3 does not imply an approximation for this problem since we make additional assumptions on the regularity of the hypergraph or the size of hyperedges.. All the details of this last result can be found in Section 4.
A remark on local search techniques.
We focus here on an extension of the LLL technique of Bansal and Srividenko. However, another technique proved itself very successful for the Santa Claus problem in the restricted assignment case with a linear utility function. This is a local search technique discovered by Asadpour, Feige, and Saberi [2] who used it to give a nonconstructive proof that the integrality gap of the configuration LP of Bansal and Srividenko is at most . One can wonder if this technique could also be extended to the submodular case as we did with LLL. Unfortunately, this seems problematic as the local search arguments heavily rely on amortizing different volumes of configurations (i.e., the sum of their resources’ weights or the number of resources in the unweighted case). Amortizing the volumes of configurations works well, if each configuration has the same volume, which is the case for the problem derived from linear valuation functions, but not the one derived from submodular functions. If the volumes differ then these amortization arguments break and the authors of this paper believe this is a fundamental problem for generalizing those arguments.
2 Reduction to hypergraph matching problem
In this section we give a reduction of the restricted submodular Santa Claus problem to the hypergraph matching problem. As a starting point we solve the configuration LP, a linear programming relaxation of our problem. The LP is constructed using a parameter
which denotes the value of its solution. The goal is to find the maximal such that the LP is feasible. In the LP we have a variable for every player and every configuration . The configurations are defined as the sets of resources such that . We require every player to have at least one configuration and every resource to be contained in at most one configuration.Since this linear program has exponentially many variables, we cannot directly solve it in polynomial time. We will give a polynomial time constant approximation for it via its dual. This is similar to the linear variant in [3], but requires some more work. In their case they can reduce the problem to one where the separation problem of the dual can be solved in polynomial time. In our case even the separation problem can only be approximated. Nevertheless, this is sufficient to approximate the linear program in polynomial time.
Theorem 2.
The configuration LP of the restricted submodular Santa Claus problem can be approximated within a factor of in polynomial time.
We defer the proof of this theorem to Appendix B. Given a solution of the configuration LP we want to arrive at the hypergraph matching problem from the introduction such that an relaxed perfect matching of that problem corresponds to an approximate solution of the restricted submodular Santa Claus problem. Let denote the value of the solution . We will define a resource as fat if
Resources that are not fat are called thin. We call a configuration thin, if it contains only thin resources and denote by the set of thin configurations. Intuitively in order to obtain an approximate solution, it suffices to give each player either one fat resource or a thin configuration . For our next step towards the hypergraph problem we use a technique borrowed from Bansal and Srividenko [3]. This technique allows us to simplify the structure of the problem significantly using the solution of the configuration LP. Namely, one can find a partition of the players into clusters such that we only need to cover one player from each cluster with thin resources. All other players can then be covered by fat resources. Informally speaking, the following lemma is proved by sampling configurations randomly according to a distribution derived in a nontrivial way from the configuration LP.
Lemma 3.
Let . Given a solution of value for the configuration LP in randomized polynomial time we can find a partition of the players into clusters and multisets of configurations , , such that

for all and

Each small resource appears in at most configurations of .

given any there is a matching of fat resources to players such that each of these players gets a unique fat resource .
The role of the players in the lemma above is that each one of them gets a fat resource for certain. The proof follows closely that in [3]. For completeness we include it in Appendix B. We are now ready to define the hypergraph matching instance. The vertices of our hypergraph are the clusters and the thin resources. Let be the multisets of configurations as in Lemma 3. For each and there is a hyperedge containing and all resources in . Let ordered arbitrarily, but consistently. Then we define the weights as normalized marginal gains of resources if they are taken in this order, that is,
This implies that for each , .
Lemma 4.
Given an relaxed perfect matching to the instance as described by the reduction, one can find in polynomial time an approximation to the instance of restricted submodular Santa Claus.
Proof.
The relaxed perfect matching implies that cluster gets some small resources where for some and . By submodularity we have that . Therefore we can satisfy one player in each cluster using thin resources and by Lemma 23 all others using fat resources. ∎
The proof above is the most critical place in the paper where we make use of the submodularity of the valuation function . We note that since all resources considered are thin resources we have, by submodularity of , the assumption that
for all such that . This means that the weights are all small enough, as promised in introduction. From now on, we will assume that for all configurations . This is w.l.o.g. since we can just rescale the weights inside each configuration. This does not hurt the property that all weights are small enough.
2.1 Reduction to unweighted hypergraph matching
Before proceeding to the solution of this hypergraph matching problem, we first give a reduction to an unweighted variant of the problem. We will then solve this unweighted variant in the next section. First, we note that we can assume that all the weights are powers of by standard rounding arguments. This only loses a constant factor in the approximation rate. Second, we can assume that inside each configuration , each resource has a weight that is at least a . Formally, we can assume that
for all . If this is not the case for some , simply delete from all the resources that have a weight less than . By doing this, the total weight of is only decreased by a factor since it looses in total at most a weight of
(Recall that we rescaled the weights so that ).
Hence after these two operations, an relaxed perfect matching in the new hypergraph is still an relaxed perfect matching in the original hypergraph. From there we reduce to an unweighted variant of the matching problem. Note that each configuration contains resources of at most different possible weights (powers of from to ). We create the following new unweighted hypergraph . The resource set remains unchanged. For each player , we create players, which later correspond each to a distinct weight. We will say that the players obtained from duplicating the original player form a group. For every configuration containing player in the hypergraph , we add a set of configurations in . contains player and all resources that are given a weight in . In this new hypergraph, the resources are not weighted. Note that if the hypergraph is regular then is regular as well.
Additionally, for a group of player and a set of configurations (one for each player in the group), we say that this set of configurations is consistent if all the configurations selected are obtained from the same configuration in the original hypergraph (i.e. the selected configurations all belong to for some in ).
Formally, we focus of the following problem. Given the regular hypergraph , we want to select, for each group of players, a consistent set of configurations and assign to each player a subset of the resources in the corresponding configuration so that is assigned at least resources. No resource can be assigned to more than one player. We refer to this assignment as a consistent relaxed perfect matching. Note that in the case where is small (e.g. of constant size) we are not required to assign any resource to player .
Lemma 5.
A consistent relaxed matching in induces a relaxed matching in .
Proof.
Let us consider a group of players in corresponding to a player in . These players are assigned a consistent set of configurations that correspond to a partition of a configuration in . Moreover, each player is assigned resources from . We have two cases. If then we have that is assigned at least
resources from . On the other hand, if then the player might not be assigned anything. However, we claim that that the configurations of cardinality less than can represent at most a fraction of the total weight of the configuration in the original weighted hypergraph. To see this note that the total weight they represent is upper bounded by
Hence, the consistent relaxed matching in induces in a straightforward way a matching in where every player gets at least a fraction of the total weight of the appropriate configuration. This means that the consistent relaxed perfect matching in is indeed a relaxed perfect matching in . ∎
3 Matchings in regular hypergraphs
In this section we solve the hypergraph matching problem we arrived to in the previous section. For convenience, we give a self contained definition of the problem before formulating and proving our result.
Input:
We are given a hypergraph with hyperedges over the vertices (players) and (resources) with and . As in previous sections, we will refer to hyperedges as configurations. Each configuration contains exactly one vertex in , that is, . The set of players is partitioned into groups of size at most , we will use to denote a group. These groups are disjoint and contain all players. Finally there exists an integer such that for each group there are consistent sets of configurations. A consistent set of configurations for a group is a set of configurations such that all players in the group appear in exactly one of these configurations. We will denote by such a set and for a player , we will denote by the unique configuration in containing . Finally, no resource appears in more than configurations. We say that the hypergraph is regular (although some resources may appear in less than configurations).
Output:
We wish to select a matching that covers all players in . More precisely, for each group we want to select a consistent set of configurations (denoted by ). Then for each player , we wish to assign a subset of the resources in to the player such that:

No resource is assigned to more than one player in total.

For any group and any player , player is assigned at least
resources from .
We call this a consistent relaxed perfect matching. Our goal in this section will be to prove the following theorem.
Theorem 6.
Let be a regular (nonuniform) hypergraph where the set of players is partitioned into groups of size at most . Then we can, in randomized polynomial time, compute a consistent relaxed perfect matching for .
3.1 Overview and notations
To prove Theorem 6, we introduce the following notations. Let be the regularity parameter as described in the problem input (i.e. each group has consistent sets and each resource appears in no more than configurations). As we proved in Lemma 3 we can assume with standard sampling arguments that at a constant loss. If this is not the case because we might want to solve the hypergraph matching problem by itself (i.e. not obtained by the reduction in Section 2), the proof of Lemma 3 can be repeated in a very similar way here.
For a configuration , its size will be defined as (i.e. its cardinality over the resource set). For each player , we denote by the set of configurations that contain . We now group the configurations in by size: We denote by the configurations of size in and for we write for the configurations of size in . Moreover, define and . Let be the smallest number such that is empty. Note that .
Now consider the following random process.
Random Experiment 7.
We construct a nested sequence of resource sets as follows. Each is obtained from by deleting every resource in independently with probability .
In expectation only a fraction of resources in survives in . Also notice that for we have that .
The proof of Theorem 6 is organized as follows. In Section 3.2, we give some properties of the resource sets constructed by Random Experiment 7 that hold with high probability. Then in Section 3.3, we show that we can find a single consistent set of configurations for each group of players such that for each configuration selected, its intersection with smaller selected configurations is bounded if we restrict the resource set to an appropriate . Restricting the resource set is important to bound the dependencies of bad events in order to apply Lovasz Local Lemma. Finally in Section 3.4, we demonstrate how these configurations allows us to reconstruct a consistent relaxed perfect matching for an appropriate assignment of resources to configurations.
3.2 Properties of resource sets
In this subsection, we give a precise statement of the key properties that we need from Random Experiment 7. The first two lemmas have a straightforward proof. The last one is a generalization of an argument used by Bansal and Srividenko [3]. Since the proof is more technical and tedious, we also defer it to Appendix C along with the proof of the first two statements.
We start with the first property which bounds the size of the configurations when restricted to some . This property is useful to reduce the dependencies while applying LLL later.
Lemma 8.
The next property expresses that for any configuration the sum of intersections with configurations of a particular size does not deviate much from its expectation. In particular, for any configuration , the sum of it’s intersections with other configurations is at most as each resource is in atmost configurations. By the lemma stated below, we recover this up to a multiplicative constant factor when we consider the appropriately weighted sum of the intersection of with other configurations of smaller sizes where each configuration is restricted to the resource set .
Lemma 9.
We now define the notion of good solutions which is helpful in stating our last property. Let be a set of configurations, , , and . We say that an assignment of to is good if every configuration receives at least resources of and if no resource in is assigned more than times in total.
Below we obtain that given a good solution with respect to resource set , one can construct an almost good solution with respect to the bigger resource set . Informally, starting from a good solution with respect to the final resource set and iteratively applying this lemma would give us a good solution with respect to our complete set of resources.
Lemma 10.
Consider Random Experiment 7 with . Fix . Conditioned on the event that the bounds in Lemma 8 hold for , then with probability at least the following holds for all , , and such that for all and : If there is a good assignment of to , then there is a good assignment of to where
for all . Moreover, this assignment can be found in polynomial time.
Given the lemmata above, by a simple union bound one gets that all the properties of resource sets hold.
3.3 Selection of configurations
In this subsection, we give a random process that selects one consistent set of configurations for each group of players such that the intersection of the selected configurations with smaller configurations is bounded when considered on appropriate sets . We will denote the selected consistent set for group and for ease of notation we will denote the selected configuration for player . For any integer , we write if and otherwise. As for the configuration set, we will also denote and . The following lemma describes what are the properties we want to have while selecting the configurations. For better clarity we also recall what the properties of the sets that we need are. These hold with high probability by the lemmata of the previous section.
Lemma 11.
Let be sets of fewer and fewer resources. Assume that for each and we have
for all . Then there exists a selection of one consistent set for each group such for all , and then we have
Moreover, this selection of consistent sets can be found in polynomial time.
Before we prove this lemma, we give an intuition of the statement. Consider the sets constructed as in Random Experiment 7. Then for we have . Hence
Similarly for the righthand side we have
Hence the lemma says that each resource in is roughly covered times by smaller configurations.
We now proceed to prove the lemma by performing the following random experiment and by Lovasz Local Lemma show that there is a positive probability of success.
Random Experiment 12.
For each group , select one consistent set uniformly at random. Then for each player set .
For all and
we define the random variable
Let . Then
We define a set of bad events. As we will show later, if none of them occur, the properties from the premise hold. For each , , and let be the event that
There is an intuitive reason as to why we define these two different bad events. In the case , we are counting how many times is intersected by configurations that are much smaller than . Hence the size of this intersection can be written as a sum of independent random variables of value at most which is much smaller than the total size of the configuration . Since the random variables are in a much smaller range, Chernoff bounds give much better concentration guarantees and we can afford a very small deviation from the expectation. In the other case, we do not have this property hence we need a bigger deviation to maintain a sufficiently low probability of failure. However, this does not hurt the statement of Lemma 11 since we sum this bigger deviation only a constant number of times. With this intuition in mind, we claim the following.
Claim 13.
For each , , and we have
Proof.
Consider first the case that . By a Chernoff bound (see Proposition 22) with
we get
Now consider . We apply again a Chernoff bound with
This implies
Proposition 14 (Lovasz Local Lemma (LLL)).
Let be bad events, and let be a dependency graph for them, in which for every , event is mutually independent of all events for which . Let for be such that and . Then with positive probability no event holds.
Let , and . For event we set
We now analyze the dependencies of . The event depends only on random variables for groups that contain at least one player that has a configuration in which overlaps with . The number of such configurations (in particular, of such groups) is at most since the hypergraph is regular.
In each of these groups, we count at most players, each having configurations hence in total at most configurations.
Each configuration can only influence those events where . Since and since each resource appears in at most configurations, we see that each configuration can influence at most events.
Putting everything together, we see that the bad event is independent of all but at most
other bad events.
We can now verify the condition for Proposition 14 by calculating
By LLL we have that with positive probability none of the bad events happen. Let and . Then for we have
Moreover, for it holds that
We conclude that, for any ,
This proves Lemma 11.
Remark 15.
Since there are at most bad events and each bad event has (because ), the constructive variant of LLL by Moser and Tardos [14] can be applied to find a selection of configurations such that no bad events occur in randomized polynomial time.
3.4 Assignment of resources to configurations
In this subsection, we show how all the previously established properties allow us to find, in polynomial time, a good assignment of resources to the configurations chosen as in the previous subsection. We will denote as in the previous subsection if and otherwise. We also define and . Finally we define the parameter
which will define how many times each resource can be assigned to configurations in an intermediate solution. Note that . By our choice of , we have that . Lemma 11 implies the following bound. For sake of brevity, the proof is deferred to Appendix D.
Claim 16.
For any , any , and any
The main technical part of this section is the following lemma that is proved by induction.
Lemma 17.
For any , there exists an assignment of resources of to configurations in such that no resource is taken more than times and each configuration () receives at least
resources from .
Before proceeding to the proof, we first give intuition of why this is what we want to prove. Note that the term is roughly equal to by the properties of the resource sets (precisely Lemma 8). The second term
can be shown to be
by Claim 16. Hence by choosing to be we get that the bound in Lemma 17 will be . At the end of the induction, we have which indeed implies that we have an assignment in which configurations receive
resources and such that each resource is assigned to at most configurations.
Proof.
We start from the biggest configurations and then iteratively reconstruct a good solution for smaller and smaller configurations. Recall is the smallest integer such that is empty. Our base case for these configurations in is vacuously satisfied.
Now assume that we have a solution at level , i.e. an assignment of resources to configurations in such that no resource is taken more than times and each configuration such that receives at least