摘要：Traditionally, we view graphs as purely combinatorial objects and tend to design our graph algorithms to be combinatorial as well. In fact, in the context of algorithms, "combinatorial" became a synonym of "fast". Recent work, however, shows that a number of such "inherently combinatorial" graph problems can be solved much faster using methods that are very "non-combinatorial". Specifically, by approaching these problems with tools and notions borrowed from linear algebra and, more broadly, from continuous optimization. A notable examples here are the recent lines of work on the maximum flow problem, the bipartite matching problem, and the shortest path problem in graphs with negative-length arcs. This raises an intriguing question: Is continuous optimization a more suitable and principled optics for fast graph algorithms than the classic combinatorial view? In this talk, I will discuss this question as well as the developments that motivated it.
关键词：maximum flow problem; bipartite matchings; interior-point methods; gradient decent method; Laplacian linear systems
