Voting is almost never done in void, as usually there are some relations between the alternatives on which the voters vote on. These relations shall be taken into consideration when selecting a winning committee of some given multiwinner election. As taking into account all possible relations between the alternatives is generally computationally intractable, in this paper we consider classes of alternatives; intuitively, the number of classes is significantly smaller than the number of alternatives, and thus there is some hope in reaching computational tractability. We model both intraclass relations and interclass relations by functions, which we refer to as synergy functions, and study the computational complexity of identifying the best committee, taking into account those synergy functions. Our model accommodates both positive and negative relations between alternatives; further, our efficient algorithms can also deal with a rich class of diversity wishes, which we show how to model using synergy functions.

In the (classical) Secretary Problem, one has to hire the best among n candidates. The candidates are interviewed, one at a time, at a uniformly random order, and one has to decide on the spot, whether to hire a candidate or continue interviewing. It is well known that the best candidate can be hired with a probability of 1/e (Dynkin, 1963). Recent works extend this problem to settings in which multiple candidates can be hired, subject to some constraint. Here, one wishes to hire a set of candidates maximizing a given objective set function. Almost all extensions considered in the literature assume the objective set function is either linear or submod- ular. Unfortunately, real world functions might not have either of these properties. Consider, for example, a scenario where one hires researchers for a project. Indeed, it can be that some researchers can substitute others for that matter. However, it can also be that some combinations of researchers result in synergy (see, e.g., Woolley et al., Science 2010, for a study on collective intelligence). The first phenomenon can be modeled by a submoudlar set function, while the latter cannot. In this work, we study the secretary problem with an arbitrary non-negative monotone valuation function, subject to a general matroid constraint. One can prove that, generally, only very poor results can be obtained for this class of objective functions. We tackle this hardness by combining the following: (1) Parametrizing our algorithms by the supermodular degree of the objective function (defined by Feige and Izsak, ITCS 2013), which, roughly speaking, measures the distance of a function from being submodular. (2) Suggesting an (arguably) natural model that permits approximation guarantees that are polynomial in the supermodular degree (as opposed to the standard model which allows only exponential guarantees). Our algorithms learn the input by running a non-trivial estimation algorithm on a portion of it whose size depends on the supermodular degree. We also provide better approximation guarantees for the special case of a uniform matroid constraint. To the best of our knowledge, our results represent the first algorithms for a secretary problem handling arbitrary non-negative monotone valuation functions.

We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m (where m is the total number of items) captures all monotone functions. The lowest level, MPH-1, captures all monotone submodular functions, and more generally, the class of functions known as XOS. Every monotone function that has a positive hypergraph representation of rank k (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-k. Every monotone function that has supermodular degree k (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH-(k+1). In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH-k.One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of k+1 if all players hold valuation functions in MPH-k. The other is an upper bound of 2k on the price of anarchy of simultaneous first price auctions.

The problem of maximizing a constrained monotone set function has many practical applications and generalizes many combinatorial problems such as k-Coverage, Max-SAT, Set Packing, Maximum Independent Set and Welfare Maximization. Unfortunately, it is generally not possible to maximize a monotone set function up to an acceptable approximation ratio, even subject to simple constraints. One highly studied approach to cope with this hardness is to restrict the set function, for example, by requiring it to be submodular. An outstanding disadvantage of imposing such a restriction on the set function is that no result is implied for set functions deviating from the restriction, even slightly. A more flexible approach, studied by Feige and Izsak [ITCS 2013], is to design an approximation algorithm whose approximation ratio depends on the complexity of the instance, as measured by some complexity measure. Specifically, they introduced a complexity measure called supermodular degree, measuring deviation from submodularity, and designed an algorithm for the welfare maximization problem with an approximation ratio that depends on this measure. In this work, we give the first (to the best of our knowledge) algorithm for maximizing an arbitrary monotone set function, subject to a k-extendible system. This class of constraints captures, for example, the intersection of k-matroids (note that a single matroid constraint is sufficient to capture the welfare maximization problem). Our approximation ratio deteriorates gracefully with the complexity of the set function and k. Our work can be seen as generalizing both the classic result of Fisher, Nemhauser and Wolsey [Mathematical Programming Study 1978], for maximizing a submodular set function subject to a k-extendible system, and the result of Feige and Izsak for the welfare maximization problem. Moreover, when our algorithm is applied to each one of these simpler cases, it obtains the same approximation ratio as of the respective original work. That is, the generalization does not incur any penalty. Finally, we also consider the less general problem of maximizing a monotone set function subject to a uniform matroid constraint, and give a somewhat better approximation ratio for it.

Given a set of items and a collection of players, each with a nonnegative monotone valuation set function over the items, the welfare maximization problem requires that every item be allocated to exactly one player, and one wishes to maximize the sum of values obtained by the players, as computed by applying the respective valuation function to the bundle of items allocated to the player. This problem in its full generality is NP-hard, and moreover, at least as hard to approximate as set-packing. Better approximation guarantees are known for restricted classes of valuation functions.