CAM Colloquium: David P. Williamson (ORIE/IS, Cornell) - Semidefinite programming relaxations of the traveling salesman problem
Finding a polynomial-time relaxation of the traveling salesman problem whose integrality gap better matches what is seen in practice has been an outstanding open problem in combinatorial optimization for some time. We study several semidefinite programming relaxations of the traveling salesman problem proposed in the literature and show that all known relaxations have an unbounded integrality gap. To obtain our results, we search for feasible solutions within a highly structured class of matrices; the problem of finding such solutions reduces to finding feasible solutions for a related linear program, which we do analytically. The solutions we find imply the unbounded integrality gap. Further, they imply several corollaries that help us better understand the semidefinite program and its relationship to other TSP relaxations. These results are joint work with Sam Gutekunst.
David P. Williamson started his career in IBM Research, then joined Cornell University in 2004 as a professor with a joint position in the School of Operations Research and Information Engineering and the Department of Information Science. David is known for his work on the topic of approximation algorithms, and is a coauthor of the book "The Design of Approximation Algorithms", published by Cambridge University Press. His work with Michel Goemans on the uses of semidefinite programming in approximation algorithms was awarded the 1999 SIAM Activity Group on Optimization prize, and the 2000 Fulkerson Prize from the Mathematical Programming Society and the American Mathematical Society. He was the editor-in-chief for the SIAM Journal on Discrete Mathematics from 2012-2016. He is an ACM Fellow and a SIAM Fellow.
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