[4] He fled Czechoslovakia in 1968, three days after the Soviet invasion,[5] and completed his Ph.D. in Mathematics at the University of Waterloo, under the supervision of Crispin St. J.

[4] He quickly recognized the importance of cutting planes for attacking combinatorial optimization problems such as computing maximum independent sets and, in particular, introduced the notion of a cutting-plane proof.

Between 1988 and 2005, the team of David L. Applegate, Robert E. Bixby, Vašek Chvátal, and William J. Cook developed one such solver, Concorde.

[22][23] The team was awarded The Beale-Orchard-Hays Prize for Excellence in Computational Mathematical Programming in 2000 for their ten-page paper [24] enumerating some of Concorde's refinements of the branch and cut method that led to the solution of a 13,509-city instance and it was awarded the Frederick W. Lanchester Prize in 2007 for their book, The Traveling Salesman Problem: A Computational Study.

Chvátal is also known for proving the art gallery theorem,[25][26][27][28] for researching a self-describing digital sequence,[29][30] for his work with David Sankoff on the Chvátal–Sankoff constants controlling the behavior of the longest common subsequence problem on random inputs,[31] and for his work with Endre Szemerédi on hard instances for resolution theorem proving.