Automated tracking is key to many computer vision applications. However, many tracking systems struggle to perform in real-time due to the high computational cost of detecting people, especially in ultra high resolution images. We propose a new algorithm called \emph{PartiMax} that greatly reduces this cost by applying the person detector only to the relevant parts of the image. PartiMax exploits information in the particle filter to select $k$ of the $n$ candidate \emph{pixel boxes} in the image. We prove that PartiMax is guaranteed to make a near-optimal selection with error bounds that are independent of the problem size. Furthermore, empirical results on a real-life dataset show that our system runs in real-time by processing only 10\% of the pixel boxes in the image while still retaining 80\% of the original tracking performance achieved when processing all pixel boxes.