Special Seminar, Talk 1 - Vijay Vazirani - 05/29/2024

Описание: Wednesday, May 29, 2024
Speaker: Vijay Vazirani
Affiliation: University of California, Irvine


Title: A Theory of Alternating Paths and Blossoms, from the Perspective of Minimum Length


It is well known that the proof of some prominent results in mathematics took a very long time -- decades and even centuries. The first proof of the Micali-Vazirani (MV) algorithm, for finding a maximum cardinality matching in general graphs, was recently completed -- over four decades after the publication of the algorithm (1980). MV is still the most efficient known algorithm for the problem. In contrast, spectacular progress in the field of combinatorial optimization has led to improved running times for most other fundamental problems in the last three decades, including bipartite matching and max-flow.

The new ideas contained in the MV algorithm and its proof remain largely unknown, and hence unexplored, for use elsewhere.

The purpose of this two-talk-sequence is to rectify that shortcoming.

TALK 1:
I will start by providing basic algorithmic background, including the bipartite case and Edmonds' algorithm. The MV algorithm resorts to finding minimum length augmenting paths. However, such paths fail to satisfy an elementary property, called breadth first search honesty. I will show why, in the absence of this property, an exponential time algorithm appears to be called for, even for finding one such path. On the other hand, the MV algorithm accomplishes this, and additional tasks, in linear time. The saving grace is the various ``footholds'' offered by the underlying structure.

I will end this talk by showing the powerful graph search procedure of Double Depth First Search (DDFS) in a simplified setting. This is a key idea underlying not only the algorithm but also its proof.

For people who are interested in the first talk only, this may be a good ``take home message''.

TALK 2:
I will attempt to provide an in-depth understanding of the footholds mentioned above. This is encapsulated in an elaborate, new theory of alternating paths and blossoms, from the perspective of minimum length paths. I will also use these insights to provide an in-depth understanding of the algorithm.

These talks are based on this paper https://www.ics.uci.edu/~vazirani/Mat....