Okay, the question of P versus NP is important. Now keep in mind that I admitted this when read the rest below:<p>Contention: In this research question of P versus NP and in the paper, we are looking at:<p>(1) A Lot of Hype.<p>(2) A Search for a Very Long Term Academic Job.<p>(3) Significant Amounts of Nonsense.<p>Details:<p>(1) A Lot of Hype.<p>(1.A) Nowhere in the article are there any very good explanations that a polynomial algorithm that shows that P = NP would be fast in any practical sense.<p>Indeed, the article has:<p>"Technically we could have P = NP, but not have practical algorithms for most NP-complete problems. But suppose in fact we do have very quick algorithms for all these problems."<p>So, to make such a polynomial algorithm of practical interest, we have to just "suppose" that it will be fast in practical terms.<p>(1.B) With the "suppose" above, the article has:<p>"Since all the NP-complete optimization problems become easy, everything will be much more efficient. Transportation of all forms will be scheduled optimally to move people and goods around quicker and cheaper. Manufacturers can improve their production to increase speed and create less waste. And I'm just scratching the surface."<p>No, it's just "scratching" and not "just scratching the surface."<p>I can absolutely, positively assure all readers that there are plenty of reasonably efficient and powerful means to attack such problems in practice now. In fact, the people flying airplanes, running manufacturing plants, designing large telecommunications networks, etc. are not much interested in attacking these problems with optimization. The main reason is: They just don't want to be bothered. In particular, these problems have long been part of the field of 'operations research', that has long been a dead field, a "late parrot", a dead duck.<p>Just what is it about people don't want to be bothered that is so difficult for the author of the paper to understand?<p>(1.C) Solve It All.<p>The suggestion in the article is that the question of P versus NP is the grand question and, thus, the last big problem in computational complexity.<p>Let's see: For many of the optimization problems in, say, airline scheduling, manufacturing scheduling, telecommunications network design, given an optimal solution, over the coming few hours, days, or weeks, real world uncertainty commonly will make that solution out of date and far from 'optimal'. So, the real problem that needs to be attacked in practice is optimization over time under uncertainty, and there was no hint of such problems in the paper or that showing that P = NP would provide solutions. Net, it is not clear from the paper that the NP-complete problems cover all the challenges that remain.<p>A lot of hype.<p>(2) A Search for a Very Long Term Academic Job.<p>The paper ends with:<p>"Perhaps we will see a resolution of the P versus NP problem in the near future but I almost hope not."<p>Of course he hopes not: As long as the problem is not solved, a lot of researchers chipping away on apparently quite distant parts continue to have a very stable career!<p>(3) Significant Amounts of Nonsense.<p>The article has:<p>"everything will be much more efficient. Transportation of all forms will be scheduled optimally to move people and goods around quicker and cheaper. Manufacturers can improve their production to increase speed and create less waste."<p>Glad he's interested in "less waste". But, let's see on three points:<p>(3.A) Approximately Optimal<p>Commonly in such cost minimization optimization problems now, we report two numbers: First we report the cost of the feasible, but perhaps not optimal, solution we did find. Second we report a lower bound on the cost of an optimal solution. When these two numbers are close for our practical problem, we quit and accept the feasible and approximately optimal solution.<p>The last time I did this, I had a 0-1 integer linear program with 600,000 variables and 40,013 constraints and found a feasible solution with cost only 0.025% higher than the lower bound, in 905 seconds on a 90 mHz PC.<p>So, the practical problem is, can we find techniques that get a feasible solution and a lower bound close enough for practice nearly always on the practical problems we face?<p>Nowhere did the paper recognize this problem or indicate a close connection with the challenge of P versus NP.<p>Yes, we can ask, given the optimization problem and a cost c, is there a feasible solution with cost less than c? So, since we can check a proposed solution quickly, this is a problem in NP. Then on this problem we can do a binary search on c and converge to optimality. So if this NP problem is in P, then with the binary search our optimal algorithm is also in P.<p>But it is not clear if this is the same problem as, can we get a feasible solution (in reasonable time, nearly always, on our practical problems) with cost c only 1% higher than a lower bound u? Or only 1% above the cost of an optimal solution (we don't know the cost of an optimal solution).<p>So, the question P versus NP is much more difficult than demanded by practice.<p>(3.B) The Big Savings.<p>The paper has,<p>"everything will be much more efficient."<p>This conclusion is unsupported, wildly unjustified, and from experience nonsense. It is not the least bit clear that optimal solutions will on average cost significantly less than the approximately optimal solutions commonly available now.<p>(3.C) The Cartoon.<p>Early in the reference,<p>Michael R. Garey and David S. Johnson, 'Computers and Intractability: A Guide to the Theory of NP-Completeness', ISBN 0-7167-1045-5, W. H. Freeman, San Francisco, 1979.<p>and praised in the paper, is a cartoon with an executive sitting behind a desk, a researcher standing just in front of the desk and stretching behind him over the horizon a long line of researchers, and the researcher explaining to the executive that he, the researcher, can't solve the executive's problem but neither can any of the researchers in the long line because none of them could settle P versus NP.<p>Nonsense. Made up, junk-think, make-work, prof-scam, busy-work nonsense: The executive's problem was just to save nearly all the money nearly all the time on the real problems, or at least to save some significant money sometimes, and not to guarantee to save all the money, down to the last tiny fraction of one penny, with polynomial computer time, on worst case problems, the worst that can exist even in theory.<p>Instead the researcher deliberately bamboozled the executive by converting his problem into one the researcher could have an excuse to work on for the rest of his career without getting a solution.<p>There is one more curious point.<p>The paper mentioned:<p>"Consider the traveling salesperson problem again with distances between cities given as the crow flies (Euclidean distance). This problem remains NP-complete but Arora4 gives an efficient algorithm that gets very close to the best possible route."<p>where his reference is<p>Arora, S. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45, 5 (Sept. 1998), 753–782.<p>While I don't know this paper, there is the highly curious,<p>Richard M. Karp, "The Probabilistic Analysis of Some Combinatorial Search Algorithms," pages 1-19, 'Algorithms and Complexity: New Directions and Recent Results', edited by J. F. Traub, Academic Press, New York, 1976.<p>So, here's what to do: Given a traveling salesman problem in the plane (or any finite dimensional space) with just Euclidean distances, pick a city, from that city build a minimum spanning tree connecting all the cities (well-known to be polynomial and fast). Then for the traveling salesman tour, just do a depth-first traversal of that tree except do not 'backtrack' in the tree and revisit cities and, instead, just take the direct link to the next city to be visited in the traversal.<p>Then for cities selected randomly with meager and reasonable assumptions, and as the number of cities n grows, the solutions have distance as close as we please to optimality with probability as high as we please less than 1.<p>So, for big problems, as long as all we are trying to do is save some travel distance, no problem. For small problems, enumerate!<p>Broadly, the question of P versus NP does not connect very well with the real needs of optimization in practice.<p>Ah, never let the real facts get in the way of an exciting story!