Why does the NP completeness of the Hartree-Fock method not lead to difficulty in practical calculation?

I read Computational Complexity of interacting electrons and fundamental limitations of Density Functional Theory. In appendix, it is claimed that

In the following, we show that approximating the ground state energy using the Hartree-Fock method is an NP-complete problem

As far as I know, if a problem being NP-hard, it is unlikely to be solved within polynomial time. If Hartree-Fock is NP-complete, computation for large system is unlikely to be possible. However, experience seems to suggest that Hartree-Fock could always be done within polynomial time. There are vast applications of Hartree-Fock method on various many-electron system: atoms, molecules, and solids. The claim from the arxiv paper is rather in contraction with my experience

My question is, why the Hartree-Fock method is applicable to wide class of many-electron system while it is NP-complete? Is that because its average complexity is low or any other reason?

• What are the practical consequences of any problem being NP-hard?
– Raphael
Jun 22 '14 at 16:32

There is plenty of precedent for problems that are NP-hard (in worst-case complexity) but can often be done in practice on many instances that arise in practice (e.g., because their average-case complexity is low, or for other reasons).

NP-hardness relates to the worst-case complexity of a problem. However, worst-case hardness isn't always representative of typical-case hardness. Sometimes typical instances are much easier than the worst-case instances. Therefore, NP-hardness results aren't always an indication that the problem can't be solved acceptably well in practice.

As Yuval Filmus explained so well in a comment:

I'm not sure there is any better answer than the fact that the algorithms work on real-world instances. NP-hardness is a worst-case concept, and sometimes the worst case is simply irrelevant. It is a challenging problem to characterize which properties of real-world instances render the problem solvable. Smooth analysis is one answer, and there might be others in the non-rigorous literature.

I recommend you take a look at Dealing with intractability: NP-complete problems for an overview of common methods for dealing with NP-hardness: i.e., for figuring out ways to solve a problem good enough for practical purposes, when we know it is NP-hard.

taking the wikipedia definition & working from that & taking a physics POV:

In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.

therefore this problem is closely connected to solving quantum-mechanical setups which is likely to be tightly coupled with the complexity of quantum computing. notice that even Feynman felt intuitively that classical systems could not solve quantum mechanical systems as fast as quantum mechanics itself, and this was some of the original rationale for quantum computing theory/foundations (while still widely held/believed, this is still currently an open conjecture spanning physics/CS).

another way to look at this is to study the transition point phenomenon in NP complete problems which is tightly connected with physical systems namely spin states in ferromagnetic systems. imagine transition points occurring in physical setups for Hartree-Fock equations & they likely will translate into the theoretical hardness variations.

another angle is to look up research into "creating hard instances for NP complete problems". it actually takes quite a bit of contrived logic at times to generate these instances, and there is not a large amount of research in this area (outside of the transition point in SAT). in other words, hard instances do not occur much at random (very rarely) and require some ingenuity to "artificially fabricate" algorithmically.

in other words visualize the problem as a large number of possible physical setups, a "parameter space", maybe many of which are solvable in P time, but a narrow section of "weird/ unusual/ pathological / contrived/ anomalous" (etc) cases that take abnormally long to solve.

• also consider the known, conjectured universal parameter of "constrainedness", $\kappa$. high or low constrainedness leads to "easy" solutions. there is a middle transition point of contrainedness in both physical and theoretical systems where hardness is maximized. constainedness in particle systems would probably relate to the "distinct paths/forces" a particle undergoes in moving toward other particles. if other particles are nearly equidistant, the constrainedness decreases (because the particle is equally attracted to a set of nearby particles).
– vzn
Jun 24 '14 at 3:09