Comment by preygel | Hacker News Reader

Comment by preygel | Hacker News Reader

A bit tangentially, this is also a great display of just how good readily available approximate solvers have gotten for a wide range of combinatorial optimization problems (like TSP).

The LKH solver used here has rather impressive performance. From http://webhotel4.ruc.dk/~keld/research/LKH/: "LKH has produced optimal solutions for all solved problems we have been able to obtain," and the studies linked there show that it really does find optimal solutions "with an impressively high frequency."

While the point here is to use an off-the-shelf solver, it can be nice to have visibility into what's actually happening:

- This is a reasonable example for explaining some of the local search TSP heuristics. For instance a "2-opt" move corresponds to picking a contiguous range of columns and flipping them.

- The previous HN post used simulated annealing, which would not be an outright terrible approach to TSP itself -- were it not for the better Lin-Kernighan-based approaches (like LKH).

- If we want, we can tell LKH to start from Sangaline's "nearest-neighbor" approach ("INITIAL_TOUR_ALGORITHM = NEAREST-NEIGHBOR"). This does not make a difference here though.

Not so long ago, we’d say “That’s an NP-complete problem” with the implication that it was hopeless to expect an efficient solution. Now (to exaggerate a little) it’s more like a sign that there probably is an efficient solution using some powerful solver.

LKH is a lovely piece of work. I used it a couple of years ago to find a counterexample to an unimportant but fairly long-standing conjecture in combinatorics[0], and the list of scientific applications[1] is impressively diverse.

The best SMT solvers are also hugely impressive. Z3 has an excellent interactive tutorial and online solver.[2]

[0] https://arxiv.org/abs/1408.5108 [1] http://webhotel4.ruc.dk/~keld/research/LKH/ScientificApplica... [2] http://rise4fun.com/z3/tutorial

Absolutely!

Not to mention the often impressive performance of "general" MIP solvers. It is only a shame that the best ones there are commercial (Gurobi followed by Cplex). That said, Cbc is lovely in a wide range of cases, and is open-source.

I always thought "NP-complete/NP-hard" metaphor only meaningful in theoretical work. In practice, it does not make much sense in guiding what/how the real world problem is to be solved.

I don't know. I find it useful in practice. Apart from anything else it means there's (probably) no point looking for an exact polynomial-time algorithm, which can save a lot of wasted effort. But it also gives useful clues about what sort of algorithmic approaches might be useful, and allows you to relate your problem to a huge amount of existing research.

It's really basic taxonomy: what sort of problem are we dealing with? If you don't know that, how could you even begin to design a reasonable algorithm for it?

Maybe you are heavily on huge algorithmic work. In practice, most algorithms are polynomial-time, and we are looking for under-polynomial-time alternative - log(n). For exponential-time, we are just looking for an approximate alternative.

Z3 (and other constraint solvers) are very impressive. The most interesting application I've seen is in binary exploitation and reverse-engineering through symbolic execution.

Angr is a very popular framework which allows for symbolic execution of binaries using Z3 as a backend. It's made a whole class of common CTF challenges almost trivial:

https://docs.angr.io/docs/examples.html

How is it determined that optimal solutions have been produced? Does an exact solver have to find the same solution, or is there a more analytic / less computationally-intensive method?

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