Honestly, the parent is pretty accurate. No one is claiming that P = NP. However, the technology to solved mixed integer programs has improved dramatically over the last 30 years and that improvement in performance has outpaced computational speed by multiple orders of magnitude. It's the algorithms.
I just went to pull up some numbers. The following comes from a talk that Bob Bixby gave at ISMP in 2012. He is the original author of CPLEX and one of the current owners of Gurobi. Between 1991 and 2007, CPLEX achieved a 29530x speedup in their solvers. Their 1997/98 breakthrough year attributes the following speedups, Cutting planes: 33.3x, presolve: 7.7x, variable selection: 2.7x, node presolve 1.3x, heuristics: 1.1x, dive probing 1.1x. I don't have a paper reference for these numbers and I don't think he has published them, but I was at the talk.
The point is that integer programming solvers perform unreasonably well. There is theory as to why. Yes, there is still a lot of searching. However, search in-and-of-itself is not sufficient to solve the problems that we regularly solve now. Further, that increase in performance is not just heuristics.
Laugh. Probably! I gave a talk at that conference titled, "Software Abstractions for Matrix-Free PDE Optimization with Cone Constraints." I still work in the field, so you want to talk algorithms sometime, feel free to send me an email. I keep my email off of HN to limit spam, but if you search for the lead author on that presentation, it should list my website.
I just went to pull up some numbers. The following comes from a talk that Bob Bixby gave at ISMP in 2012. He is the original author of CPLEX and one of the current owners of Gurobi. Between 1991 and 2007, CPLEX achieved a 29530x speedup in their solvers. Their 1997/98 breakthrough year attributes the following speedups, Cutting planes: 33.3x, presolve: 7.7x, variable selection: 2.7x, node presolve 1.3x, heuristics: 1.1x, dive probing 1.1x. I don't have a paper reference for these numbers and I don't think he has published them, but I was at the talk.
The point is that integer programming solvers perform unreasonably well. There is theory as to why. Yes, there is still a lot of searching. However, search in-and-of-itself is not sufficient to solve the problems that we regularly solve now. Further, that increase in performance is not just heuristics.