Title: Chasing Positive Bodies
| Speaker: | Niv Buchbinder |
| Affiliation: | Tel Aviv University |
| Location: | MC 5417 |
Abstract: We study the problem of chasing positive bodies in \ell_1: given a sequence of bodies K_t\subset R^n revealed online, where each K_t is defined by a mixed packing-covering linear program, the goal is to (approximately) maintain a point x_t \in K_t such that \sum_t \|x_t - x_{t-1}\|_1 is minimized. This captures the fully-dynamic low-recourse variant of any problem that can be expressed as a mixed packing-covering linear program and thus also the fractional version of many central problems in dynamic algorithms such as set cover, load balancing, hyperedge orientation, minimum spanning tree, and matching.
We give an O(\log n)-competitive algorithm for this problem. This bypasses and improves exponentially over the lower bound of \sqrt{n} known for general convex bodies. Our algorithm is based on iterated information projections, and, in contrast to general convex body chasing algorithms, is entirely memory-less.
We also show how to round our solution dynamically to obtain the first fully dynamic algorithms with {competitive recourse} for all the stated problems above; i.e. their recourse is less than the recourse of every other algorithm on every update sequence, up to poly-logarithmic factors. This is a significantly stronger notion than the notion of {absolute recourse} in the dynamic algorithms literature.