Nonoverlapping Grid-aligned Rectangle Placement for High Value Areas

We consider heuristic and optimal solutions to a discrete geometric bin packing problem that arises in a resource allocation problem. An imaging sensor is assigned to collect data over a large area, but some subregions are more valuable than others. To capture these high-value regions with higher fidelity, we can assign some number of non-overlapping rectangular subsets, called “subfootprints.” The sensor image is partitioned into squares called “chips”, and each chip is further partitioned into pixels. Pixels may have different values. Subfootprints are restricted to rectangular collections of chips, but we are free to choose different rectangle heights, widths, and areas. We seek the optimal arrangement over the family of possible rectangle shapes and sizes. We provide a mixed-integer linear program optimization formulation, as well as a greedy heuristic, to solve this problem. For the meta-problem, we have some freedom to align the chip boundaries to different pixels. However, it is too expensive to solve the optimization formulation for each alignment. However, we show that the greedy heuristic can inform which pixel alignments are worth solving the optimization over. We use a variant of k-means clustering to group greedy solutions by their transport shape-similarity. For each cluster, we run the optimization problem over the greedy layout with the highest value. In practice this efficiently explores the geometric configuration space, and produces solutions close to the global optimum. We show a contrived example using surveillance of the Mississippi River. Our software is available as open-source in the Github repository “GeoPlace.”