Fast approximations for combinatorial optimization via multiplicative weight updates
Kent Quanrud · Illinois Digital Environment for Access to Learning and Scholarship (University of Illinois at Urbana-Champaign) · 2019
We develop fast approximations for several LP relaxations that arise in discrete and combinatorial optimization. New results include improved running times for explicit mixed packing and covering problems, nearly linear time approximations for tree packings, nearly linear time approximations for the Held Karp bound (leading to a significantly faster $(3/2 + \\epsilon)$-approximation for metric TSP), faster approximations for covering LPs with knapsack covering constraints (the bottleneck for covering integer programs), and nearly linear time $(2+\\epsilon)$-approximations for $k$-cut via the LP. Along the way we develop new techniques for the MWU framework and put forth two frameworks, "lazy MWU" for deterministic algorithms and "randomized MWU" for randomized algorithms, that algorithm designers can use to obtain nearly linear running times for their own problems of interest. This thesis has been organized as a user friendly guide, where we include basic background and analysis of the MWU framework, establish clean interfaces for the two frameworks, and use the applications as examples of the frameworks.