We introduce and study a variation of the submodular maximization problem motivated by applications in online retail. A platform displays a list of products to a user in response to a search query. The user inspects the first k items in the list for a k chosen at random from a given distribution, and decides whether to purchase an item from that set based on a choice model. The goal of the platform is to maximize the engagement of the shopper defined as the probability of purchase. This problem gives rise to a less-studied variation of submodular maximization in which we are asked to choose an ordering of a set of elements to maximize a linear combination of different submodular functions.

We build a model to study the implications of utilization-based minimum earning regulations of the kind recently enacted by New York City for its ride-hailing providers. We identify the precise conditions under which a utilization-based minimum earnings rule causes marketplace instability, where stability is defined as the ability of platforms to keep wages bounded while maintaining the current flexible (free-entry) work model. We also calibrate our model using publicly available data, showing the limited power of the law to increase earnings within an open marketplace. We argue that affected ride-hailing companies might respond to the law by reducing driver flexibility.

We consider the restricted assignment version of the problem of max-min fair allocation of indivisible goods, also known as the Santa Claus problem . There are m items and n players. Every item has some nonnegative value, and every player is interested in only some of the items. The goal is to distribute the items to the players in a way that maximizes the minimum of the sum of the values of the items given to any player. It was previously shown via a nonconstructive proof that uses the Lovász local lemma that the integrality gap of a certain configuration LP for the problem is no worse than some (unspecified) constant. This gives a polynomial-time algorithm to estimate the optimum value of the problem within a constant factor, but does not provide a polynomial-time algorithm for finding a corresponding allocation. We use a different approach to analyze the integrality gap. Our approach is based upon local search techniques for finding perfect matchings in certain classes of hypergraphs. As a result, we prove that the integrality gap of the configuration LP is no worse than 1/4. Our proof provides a local search algorithm which finds the corresponding allocation, but is nonconstructive in the sense that this algorithm is not known to converge to a local optimum in a polynomial number of steps.

We consider the Asymmetric Traveling Salesman problem for costs satisfying the triangle inequality.We derive a randomized algorithm which delivers a solution within a factor O(log n/ log log n) of the optimum with high probability.

In this paper, we give the first approximation algorithm for the problem of max-min fair allocation of indivisible goods. An instance of this problem consists of a set of k people and m indivisible goods. Each person has a known linear utility function over the set of goods which might be different from the utility functions of other people. The goal is to distribute the goods among the people and maximize the minimum utility received by them. The approximation ratio of our algorithm is $\Omega(\frac{1}{\sqrt{k}\log^{3}k})$. As a crucial part of our algorithm, we design and analyze an iterative method for rounding a fractional matching on a tree which might be of independent interest. We also provide better bounds when we are allowed to exclude a small fraction of the people from the problem.

In this paper we give the first approximation algorithm for the problem of max-min fair allocation of indivisible goods. The approximation ratio of our algorithm is Ω1√k log3 k. As a part of our algorithm, we design an iterative method for rounding a fractional matching on a tree which might be of independent interest.