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In the weighted load balancing problem, the input is an n-vertex bipartite graph between a set of clients and a set of servers, and each client comes with some nonnegative real weight. The output is an assignment that maps each client to one of its adjacent servers, and the load of a server is then the sum of the weights of the clients assigned to it. The goal is to find an assignment that is well-balanced, typically captured by (approximately) minimizing either the ℓ∞-or ℓ2-norm of the server loads. Generalizing both of these objectives, the all-norm load balancing problem asks for an assignment that approximately minimizes all ℓp-norm objectives for p ≥ 1, including p = ∞, simultaneously.Our main result is a deterministic O(log n)-pass O(1)-approximation semi-streaming algorithm for the all-norm load balancing problem. Prior to our work, only an O(log n)pass O(log n)-approximation algorithm for the ℓ∞-norm objective was known in the semi-streaming setting.Our algorithm uses a novel application of the multiplicative weights update method to a mixed covering/packing convex program for the all-norm load balancing problem involving an infinite number of constraints.

Abstract. We present a fast algorithm for uniform sampling of contiguous minimum cuts separating a source vertex from a set of sink vertices in a weighted undirected planar graph with n vertices embedded in the plane. The algorithm takes O(n) time per sample, after an initial O(n 3 ) preprocessing time during which the algorithm computes the number of all such contiguous minimum cuts. Contiguous cuts (that is, cuts where a naturally defined boundary around the cut set forms a simply connected planar region) have applications in computer vision and medical imaging [6,14].