Supplemental Material: Finding Group Interactions in Social Clutter

To apply branch-and-bound algorithm to minimize the quality function (6) in the main paper, we first specify the spaces where Ts and Te may take a value. We denote the length of the shortest exemplar activity as Tmin, then we assume 1 ≤ Ts ≤ T − Tmin + 1 and Tmin + 1 ≤ Te ≤ T . Additional constraint may be imposed, such as Tmin ≤ Te − Ts. Given these information, the temporal branch-and-bound algorithm, as a companion to the 2-D case studied in [2], can be derived as in Algorithm 1. In this algorithm, f̂(Ts,low, Ts,upp, Te,low, Te,upp) is a lower bound of the values of the quality function evaluated on all intervals enclosed in [Ts,low, Ts,upp]× [Te,low, Te,upp]. To calculate this lower bound, we define