Chasing Positive Bodies

We study the problem of chasing positive bodies in l(1): given a sequence of bodies K-t = {x(t) is an element of R-+(n) vertical bar C(t)x(t) >= 1, P(t)x(t) <= 1} revealed online, where C-t and P-t are nonnegative matrices, the goal is to (approximately) maintain a point x(t) is an element of K-t such that Sigma(t) parallel to x(t) - x(t-1)parallel to(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 d)-competitive algorithm for this problem, where d is the maximum row sparsity of any matrix C-t. This bvypasses and improves exponentially over the lower bound of root 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 memoryless. 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 polylogarithmic factors. This is a significantly stronger notion than the notion of absolute recourse in the dynamic algorithms literature.