Rothblum’s description of the stable marriage polyhedron is TDI

Rothblum showed in that the convex hull of the stable matchings of a bipartite preference system can be described by an elegant system of linear inequalities. In this note we show that the description given by Rothblum is totaly dual integral. Our proof is based on the results of Gusfield and Irving on rotations

An extension of Lehman's theorem and ideal set functions

Lehman’s theorem on the structure of minimally nonideal clutters is a fundamental result in polyhedral combinatorics. One approach to extending it has been to give a common generalization with the characterization of minimally imperfect clutters (Sebő, 1998; Gasparyan et al., 2003). We give a new generalization of this kind, which combines two types of covering inequalities and works well with the natural definition of minors. We also show how to extend the notion of idealness to unit-increasing set functions, in a way that is compatible with minors and blocking operations

Total dual integrality of Rothblum´s description of the stable marriage polyhedron

Rothblum showed that the convex hull of the stable matchings of a bipartite preference system can be described by an elegant system of linear inequalities. In this paper we prove that the description given by Rothblum is totally dual integral. We give a constructive proof based on the results of Gusfield and Irving on rotations, which gives rise to a strongly polynomial algorithm for finding an integer optimal dual solution

Location-aware online learning for top-k recommendation

We address the problem of recommending highly volatile items for users, both with potentially ambiguous location that may change in time. The three main ingredients of our method include (1) using online machine learning for the highly volatile items; (2) learning the personalized importance of hierarchical geolocation (for example, town, region, country, continent); finally (3) modeling temporal relevance by counting recent items with an exponential decay in recency.For (1), we consider a time-aware setting, where evaluation is cumbersome by traditional measures since we have different top recommendations at different times. We describe a time-aware framework based on individual item discounted gain. For (2), we observe that trends and geolocation turns out to be more important than personalized user preferences: user-item and content-item matrix factorization improves in combination with our geo-trend learning methods, but in itself, they are greatly inferior to our location based models. In fact, since our best performing methods are based on spatiotemporal data, they are applicable in the user cold start setting as well and perform even better than content based cold start methods. Finally for (3), we estimate the probability that the item will be viewed by its previous views to obtain a powerful model that combines item popularity and recency.To generate realistic data for measuring our new methods, we rely on Twitter messages with known GPS location and consider hashtags as items that we recommend the users to be included in their next message. © 2016 Elsevier B.V