We study a customer choice model that captures purchasing behavior when there is a limit on the number of times that a customer will substitute among the offered products. Under this model, we assume each customer is characterized by a ranked preference list of products and, upon arrival, will purchase the highest ranking offered product. Since we restrict ourselves to settings in which customers consider a limited number of products, we assume that these rankings contain at most k products. We call this model the k-product nonparametric choice model. We focus on the assortment optimization problem under this choice model. In this problem, the retailer wants to find the revenue maximizing set of products to offer when the buying process of each customer is governed by the k-product nonparametric choice model. First, we show that the assortment problem is strongly NP-hard even for k = 2. Motivated by this result, we develop a linear programming-based randomized rounding algorithm that gives the best known approximation guarantee. We tighten the approximation guarantee further when each preference list contains at most two products and consider the case where there is a limit on the number of products that can be offered to the customers.

We develop a general framework for designing polynomial-time approximation schemes (PTASs) for various vehicle routing problems in trees. In these problems, the goal is to optimally route a fleet of vehicles, originating at a depot, to serve a set of clients, subject to various constraints. For example, in Minimum Makespan Vehicle Routing, the number of vehicles is fixed, and the objective is to minimize the longest distance traveled by a single vehicle. Our main insight is that we can often greatly restrict the set of potential solutions without adding too much to the optimal solution cost. This simplification relies on partitioning the tree into clusters such that there exists a near-optimal solution in which every vehicle that visits a given cluster takes on one of a few forms. In particular, only a small number of vehicles serve clients in any given cluster. By using these coarser building blocks, a dynamic programming algorithm can find a nearoptimal solution in polynomial time. We show that the framework is flexible enough to give PTASs for many problems, including Minimum Makespan Vehicle Routing, Distance-Constrained Vehicle Routing, Capacitated Vehicle Routing, and School Bus Routing, and can be extended to the multiple depot setting.

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