In this paper, we consider the classic fair division problem of allocating m divisible items to n agents with linear valuations over the items. We define novel notions of fair shares from the perspective of individual agents via the cake-cutting process. These shares generalize the notion of proportionality by taking into account the valuations of other agents via constraints capturing envy. We study what fraction (approximation) of these shares are achievable in the worst case, and present tight and non-trivial approximation bounds as a function of n and m. In particular, we show a tight approximation bound of Θ(√n) for various notions of such shares. We show this bound via a novel application of dual fitting, which may be of independent interest. We also present a bound of O(m^(2/3)) for a strict notion of share, with an almost matching lower bound. We further develop weaker notions of shares whose approximation bounds interpolate smoothly between proportionality and the shares described above. We finally present empirical results showing that our definitions lead to more reasonable shares than the standard fair share notion of proportionality.

We present a novel framework for finding complex activities matching user-described queries in cluttered surveillance videos. The wide diversity of queries coupled with the unavailability of annotated activity data limits our ability to train activity models. To bridge the semantic gap, we propose letting users describe an activity as a semantic graph with object attributes and inter-object relationships associated with nodes and edges, respectively. We learn node/edge-level visual predictors during training and, at test-time, propose retrieving activity by identifying likely locations that match the semantic graph. We formulate a novel conditional random field-based probabilistic activity localization objective that accounts for misdetections, misclassifications and track losses, and outputs a likelihood score for a candidate grounded location of the query in the video. We seek groundings that maximize overall precision and recall. To handle the combinatorial search over all high-probability groundings, we propose a highest precision subgraph matching algorithm. Our method outperforms existing retrieval methods on benchmarked datasets.