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These solvers really show that NP-hardness is no reason to give up. For example, they can solve surprisingly large Traveling Salesmen instances to proven optima

• dragontamer 3y ago
  NP-hard problems commonly come up as human-solvable puzzles. Like Sudoku... or perhaps a more applicable problem... layout and routing of electronic components on a PCB and/or chip. Or even assembly-language register allocation (coloring and packing problem). Trained Humans are surprisingly good at these problems, far better than expected given how much computational power we have today. So its clear we don't understand something fundamental with regards to NP-hardness. And that's why the research continues, to bring human-like intuition into these problems and provide more automation to these tasks.
  • imtringued 3y ago
    Computers have to solve hard exact problems. You don't seem to understand that heuristics can significantly speed up NP hard problems, especially if you are willing to give up exact solutions.
  • azornathogron 3y ago
    I do find this interesting... > Trained Humans are surprisingly good at these problems Surprising why, and by what metric? (speed? Or quality of solution?)
    • codeflo 3y ago
      A problem is NP hard if there are instances that other hard problems reduce to. That is, only some instances of the problem have to be hard. NP hardness says nothing about "average" problem instances and even less about hand-picked ones. And that's what Sudoku puzzles are. They are made for humans to solve. They are specifically crafted to contain challenging, but possible, solution paths.
      • azornathogron 3y ago
        I was asking about why specifically it's surprising that humans are good at such problems. I think that to be surprising there must be some prior expectation about how good humans should be and then evidence that they're actually better than that. Both sides of that are interesting and I'd like to know more about it: What is the prior expectation of human capabilities here (and why), and how does it compare to actual observed capabilities. I'm not sure sudoku is a good example since the problem size there is so small that I don't have any intuitive sense that it should be something humans can't do.
  • cyberax 3y ago
    Hmm.... A neural network for register allocation? Might be an interesting experiment to try.
  • sp332 3y ago
    Sudoku puzzles are usually set by humans.
    • verve_rat 3y ago
      Not really, as I understand it. The big blow up of sudoku books full of puzzles came about because some guy figured out how to create them with a program, rather than by hand. Earned the guy millions I believe. Edit: Wayne Gould https://en.m.wikipedia.org/wiki/Wayne_Gould https://en.m.wikipedia.org/wiki/Wayne_Gould
      • mrloba 3y ago
        People who are more into it usually prefer human made puzzles since they often have a logical path that you're supposed to find, which can be quite satisfying. Generating sudoku puzzles is actually quite easy. Just put random numbers on the board until you have a unique solution. It runs surprisingly fast. The tricky part is deciding on the difficulty. I made a program that would identify all the different sudoku techniques needed to solve each puzzle (x wings, pairs, all the way up to chains), then set the difficulty based on what techniques were required. Code is here for anyone interested: https://github.com/magnusjt/sudoku/ https://github.com/magnusjt/sudoku/ Sadly I don't think anyone would pay millions for this anymore
  • mistercow 3y ago
    I’m not sure there’s necessarily going to be a satisfying general answer to find here though. I can invent an NP-complete problem with very easy instances by combining a linear time problem with an NP-complete problem, e.g. “Does this list of cities have a complete route shorter than X OR is one of the cities New York?” All of the “yes New York” solutions are easy, but there’s no deep, satisfying reason for that. It’s just a hard problem glued onto an easy problem. For all we know, a lot of NP-hard problems could be like that, and the structure of the easy instances could be essentially unrelated between problems.
    • foota 3y ago
      Maybe the answer then is to find what these hard cases are and use them in heuristics or approximate algorithms? If we could tell when part of the problem is hard, and maybe bound the error for them, and use an exact algorithm for other parts of the problem, you could get a better result in most cases without it blowing up on you.
      • kaba0 3y ago
        Well, you can just start an exact solver on the problem in one thread, and an approximate algorithm on another. If the former doesn’t finish in n seconds, you cancel it and use the result of the latter.
        • foota 3y ago
          The advantage of combining them though is that you might be able to treat different subparts of the problem differently (which is the hard part), so that you could use an approximate algorithm for the hard part of it, and an exact algorithm for the easy part. Thi of course assumes the problem can be divided in this way, which is fairly speculative.
  • kaba0 3y ago
    > Trained Humans are surprisingly good at these problems Are we? I don’t think we would even start working on problems with big enough `n` where the complexity actually ramps up. Like, optimally scheduling even just a couple of things will have a shitton of combinations, and I really doubt we would be good at it. Good at iteratively decreasing a cost function? Yeah, I guess with a good interface we could move around tasks to be scheduled on a timeline and optimize it. Finding the optimum? No chance.
    • hinkley 3y ago
      Our monkey brains can get within a few % of optimal on written TSP problems. For a lot of these problems, having a tight upper bound lets you narrow the search space, regardless of whether you're looking for the ideal answer or simply a less-bad one (Would you turn down saving $500,000 on fuel and labor in a year just because someone thinks $613,000 is the maximum savings achievable?) The more we can get automation to do approximations as well as humans, the better.
  • sacado2 3y ago
    It's often a modeling issue. Modeling languages are so abstract and general they give up a part of the problem's structure. The most famous example is the pigeonhole problem, when given to SAT solvers. Definition of the problem is: I have n pigeons and m pigeonholes, with n = m + 1. Can I put all n pigeons in a different pigeonhole? The answer is obvious to a human. If I have 20 pigeons and only 19 pigeonholes, I won't make it. But even state of the art SAT solvers will fail at solving this in a reasonable amount of time. Because of the way the problem is represented (a conjunction of boolean clauses). With just a slightly more expressive modeling language (such as pseudo-boolean representation / reasoning), you solve it with the blink of an eye. We humans are efficient because we recognize the underlying structure of the problem. You'll solve the pigeonhole problem in a split second. But if I encode the problem in CNF (the language of SAT solvers) and give it to you without more information, you won't be able to do it anymore.
• codeflo 3y ago
  Solvers aren't magic. But it turns out that many naturally occurring instances of NP-hard problems aren't the "really hard" instances that the NP-hardness proof depends on. Solvers can also fail for really tiny problems. You simply have to try to figure out how hard (or how well adapted to the solver's heuristics) your particular problem instance is.
  • ngruhn 3y ago
    I’m saying this because most people got from their CS degree that NP hardness is practically a death sentence. But it‘s really not true. Even if the problem is proven to be NP hard (or worse) and every previous approach has failed. There can still be some trick or technique (non heuristic!) that brings a breakthrough. Maybe you still wait 10^whatever years for a solution in some cases. But if you get an answer in seconds in 99.9% of cases then its not a big deal in practice. Even the famous SAT problem can almost be considered solved nowadays with the solvers we have.
    • fbdab103 3y ago
      It also depends on if you actually need the optimal solution. Getting a solution in your time budget, even if not guaranteed to be optimal, can be appropriate for many problems.
    • baq 3y ago
      The true benefit of being able to tell NP-complete/NP-hard from the garden variety bucket-o-bytes moving is not in giving up when you encounter them, as you correctly identified, but in knowing that even attempting an optimal solution is futile and proceeding to looking for approximate solutions to the business problem more or less instantly. People unfamiliar with the matter may attempt to solve the problem, might even get rewarded for effort and hard work if management is also unfamiliar with the matter. Maybe it's fine though...?
      • travisjungroth 3y ago
        Optimal is overblown for business problems in general. Knowing there’s a mathematically optimal solution, people want it. Even if it’s practically impossible to get. It feels like if you have the optimal, you don’t have to consider trade offs (not usually true). Having a solution within 1% of optimal with ten nines of confidence is usually indistinguishable. Anyone ever notice how these CS problems are rarely famous business problems? It’s because the pure problems don’t usually matter and the approximate versions are usually quite easy. You almost never need the shortest route for a salesman, or need to pick a secretary with no chance of going back.
• TimPC 3y ago
  Most instances of a class of problems that are NP-hard are in fact easy. Usually NP-hard is something resembling exponential blow-up in a problematic edge case.
  • hinkley 3y ago
    One of my favorite terms in this space is 'relaxation'. When you set yourself against the worst case scenario, you are destined to have a very bad time. But there are subsets of the problem where one or two details are treated as trivialities, while still keeping the rest of the problem 'interesting' or 'useful'. Compression cannot compress white noise. But it turns out humans don't find noise that interesting (except in the negative). We value signal, and meaning. Most of the communication we care to exchange has a much more straightforward message than the medium, and so we continue to find new ways to condense both the message, and the most valuable nuance, down.
• Xcelerate 3y ago
  > they can solve surprisingly large Traveling Salesmen instances to proven optimality. For someone who studies computational complexity theory, is the ability to solve some instances of NP-hard problems efficiently due more to these instances having lower average-case complexity than worst-case complexity, or because they possess structural properties that allow for more effective algorithmic exploitation? More formally, are certain NP-hard problems easier to solve because the expected time to solve a randomly selected instance from their language is polynomial? Or is it because real-world instances are not uniformly distributed across the language, and these instances possess some amount of Kolmogorov compressibility that leads to better-than-expected performance?
  • tooltower 3y ago
    The latter. The real-world instances are not uniformly distributed across the language. It is pretty easy to randomly generate problems that are hard for current solvers. Caveats: * "uniform distribution" is a tricky concept over infinite countable sets. Famously, it doesn't exist. You have to restrict the length or something. * I have no idea if there's a concrete result linking Kolmogorov complexity and solving ease. I suspect no, since even a complex but known problem can be solved rather quickly by special-casing.
    • Xcelerate 3y ago
      > I have no idea if there's a concrete result linking Kolmogorov complexity and solving ease I’ve only heard of the incompressibility method but don’t know too much about the details: https://core.ac.uk/download/pdf/301634196.pdf https://core.ac.uk/download/pdf/301634196.pdf
    • danbruc 3y ago
      I played quite a bit with vertex three coloring in the past and it has a surprisingly sharp phase transition. If you randomly generate graphs by including each possible edge with probability p, then the graph will have average degree p×n. Don't quote me on the exact numbers, but something like if the average degree is about 3, then the graph is usually hard to color. If it is only 2.9, then the graph is usually easy to color, if it is 3.1 then there is usually either no valid coloring at all or it is easy to find one. So of all the graphs with average degree from 0 to n, mostly only graphs with average degree in a narrow band around 3 are hard.