On the complexity of package recommendation problems

Recommendation systems aim to recommend items that are likely to be of interest to users. This paper investigates several issues fundamental to such systems. (1) We model recommendation systems for packages of items. We use queries to specify multi-criteria for item selections and express compatibility constraints on items in a package, and use functions to compute the cost and usefulness of items to a user. (2) We study recommendations of points of interest, to suggest top-k packages. We also investigate recommendations of top-k items, as a special case. In addition, when sensible suggestions cannot be found, we propose query relaxation recommendations to help users revise their selection criteria, or adjustment recommendations to guide vendors to modify their item collections. (3) We identify several problems, to decide whether a set of packages makes a top-k recommendation, whether a rating bound is maximum for selecting top-k packages, whether we can relax the selection query to find packages that users want, and whether we can update a bounded number of items such that the users' requirements can be satisfied. We also study function problems for computing top-k packages, and counting problems to find how many packages meet the user's criteria. (4) We establish the upper and lower bounds of these problems, all matching, for combined and data complexity. These results reveal the impact of variable sizes of packages, the presence of compatibility constraints, as well as a variety of query languages for specifying selection criteria and compatibility constraints, on the analyses of these problems.