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NP-Complete isn't (always) Hard

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A common assumption I see on the ‘net is that NP-complete problems are impossible to solve. I recently read that dependency management in Python is hard because package resolution is NP-complete. This is true in principle, but the reality is more complicated. When we say “NP-complete is hard” we’re talking about worst-case complexity, and the average-case complexity is often much more tractable. Many industry problems are “well-behaved” and modern SAT solvers can solve them quickly.

Okay I’ve probably lost 40% of my readers by this point so let’s back up and explain “NP-complete” and “SAT solver” and stuff.

Terms terms terms

(If you’re not familiar with computational complexity and O(n) and what “solvable in polynomial time” means, this is a pretty good overview.)

NP is the set of all problems that are solvable in polynomial time by a “nondeterministic Turing Machine” (NTM). Very roughly, that’s a TM that, when faced with a choice, can create parallel timelines and try a different choice in each timeline. This is equivalent to the set of all problems that can be verified in polynomial time, which is the definition most non-CS people learn.1

All problems in P are also in NP. It’s an open question whether NP problems can’t be solved in deterministic polynomial time, aka the P=NP problem. If P!=NP, then some NP problems are harder than others. There’s also an upper bound to “how hard” an NP problem can be. This set of “most difficult” NP programs (which again, might just be P) form the NP-complete complexity class. There are also problems that can’t even be verified in polynomial time. NP-hard is the set of all problems that are NP-complete or harder.2 People often conflate NP-complete and NP-hard.

The quintessential NP-complete problem is Boolean SATisfiability, or SAT. A SAT problem consists of a set of boolean variables and a set of clauses, like this:

A | C
!A | D | !E
B | !D | E | F
…

The goal is to find an assignment of variables that makes every clause true. Checking an answer is obviously in P: just plug the proposed assignments into the clauses and see if they all pass. We don’t have a polynomial time algorithm for finding assignments, though, and mathematicians are confident there isn’t one. It’s NP-complete. By the Cook-Levin Theorem, any other NP-complete problem can be converted into a SAT problem, and in fact “convert NP-complete problem to SAT” is in P.

That’s a lot of words about why NP-complete is hard, so let’s talk about why it isn’t.

Why it’s not hard

When we talk about computational complexity, we’re almost always talking about worst-case complexity. The hardest-possible SAT problem with 19,203 variables will take twice-ish as long to solve as the hardest-possible SAT problem with 19,202 variables.3

However, this says nothing about the average problem. Imagine if 99.99% of SAT problems could be solved instantly, while the remaining 0.01% took exponential time. Then SAT would still be considered NP-complete, even though the “typical” case was trivially solvable.

Of course life isn’t that easy and randomly generated SAT problems tend to be intractable.4 But a lot of industrial problems map to the subset of SAT problems that are tractable (why this is is an open question), meaning you can some pretty big gains from making fast SAT solvers. In principle, that’s all you need to realistically solve NP-complete problems. Any NP-complete problem can be converted to SAT, and since most industrial problems are well-behaved, this means that solving NP-complete problems is often quite viable.

Walking the walk

I grabbed one SAT solver, Glucose, and tried it on a Sudoku, from my how to solve Sudoku satire. That has 729 variables with 12,000 clauses, and Glucose could solve it in a tenth of a second.5

Next, I downloaded the SAT Race 20196 problems and picked out a typical example. CarSequencing-90-02_c18 is an example of the “Car Sequencing” problem, which is described and shown to be NP-complete in this paper. This particular problem has 48,491 variables and 550,000 clauses, so we’d expect it to take about 1.4^47762 times longer, well above the lifetime of the universe. But Glucose finds a solution in just 25 seconds.

On the other hand, the problem 58bits_dimacs (I think it’s an integer factorization problem?) has just 1,019 variables and 23,771 clauses. I let it run overnight and it didn’t finish. Checking the scoreboard, only 4 out of 55 SAT solvers were able to solve it in under two hours. That’s closer to a worst-case scenario where the blowup is exponential. RSA is still safe!

Problem Variable Ratio Clause Rat. Time Rat. Time/Var
Sudoku 1 1 1 1
Car Sequencing 66 46 250 4
58bits 1.4 2 >500,000 >500,000

So Why Aren’t We Using SAT Solvers?

When I started this post, I thought that SAT solvers were widely used in package dependency management. Turns out they’re not! Most dependency resolution uses PubGrub or an ad-hoc solver. Why is that?

I have four guesses:

  1. Converting problems to SAT is intellectually difficult. I felt real clever when I figured out how to convert Sudoku, but I wouldn’t be able to convert much else without a lot more experience.
  2. Converting problems introduces overhead. SAT clauses are a restrictive format, so you have to add lots of auxiliary booleans to track various aspects of your original problem. And there’s often several ways to convert a problem to SAT with different performance characteristics. The Car Sequencing folk came up with three different conversions, and none were strictly “the best”.
  3. Dedicated solvers can exploit properties in the problem domain that are lost by conversion to SAT. Integer Programming is an NP-complete problem but solvers just work with problems directly.
  4. Often you want to pull metadata from the problem, like return which specific dependency requirements conflict. That’s a lot harder to do (if at all possible) after you’ve converted to SAT.

Further Reading

If you’re interested in learning more about SAT solvers:

If you’re interested in learning more about NP-completeness:

Thanks to Alex Koppel and Jimmy Koppel (no relation) for feedback.

  1. Very-rough-sketch-of-proof: 1) If you can check a solution of length N in polynomial time, just branch on every single possible input of length N. 2) If you can solve a problem with branching, have the NTM also track the history of every step. Then a valid solution will also have a linear set of steps. [return]
  2. Finding problems that are NP-hard but not NP-complete is really annoying, because a lot of mathematicians are smartasses and say “the halting problem!” I think most bounded model checking falls under this category, though. [return]
  3. In practice it’s more like 1.4ish times harder, so adding 10 variables “only” makes it 30x harder (as opposed to 1000x). [return]
  4. Early research showed that there’s a phase transition in the ratio of clauses to variables: there’s a band of ratios where the problem is really hard, and it’s otherwise pretty easy. Later research is challenging this idea, though. [return]
  5. All of these are done on an i9-12900K with 64gb ram. [return]
  6. Oh yeah, there’s an annual SAT competition. The 2019 problem set is 3.5 gigabytes (zipped). [return]