Often, the reduction from your problem to a known NP-complete problem is obvious. When it isn't is when things get interesting. If you have trouble proving a problem NP-complete, it is often - but certainly not always - because the problem in fact isn't NP-complete. After you've done a lot of these proofs, you start to get a feel for how many degrees of freedom in the solution space might still be apprehended in polynomial time, and how many will probably lead to NP-completeness. Often, the magic boundary is between 2 and 3 degrees of freedom: 2-coloring a graph can be done efficiently, while 3-coloring is NP-complete, and likewise bipartite matching is in P while three-dimensional matching is NP-complete. However, some of the NP-completeness proofs that have been devised are incredibly complicated and nonobvious, so don't take your difficulty in proving NP-completeness as strong evidence that the problem is not NP-complete.

A productive approach is often to bounce back and forth between trying to devise an efficient algorithm and trying to prove NP-completeness. Your failures at proving NP-completeness can produce inspirations about how to attack the problem algorithmically, and your failures at devising an efficient algorithm can lead to insights into how to establish NP-completeness. One of my favorite examples from my own work was to solve a system of rectilinear orientation constraints. You had a bunch of objects in two dimensions that could have one of the eight axis-parallel orientations, and a set of constraints between pairs of them: That they must have the same orientation, have orientation mirrored about one or the other axis, have orientation rotated 90 degrees from one another, or any compatible combination of these. My initial attempts to prove it NP-complete focused on looking at it as an 8-coloring problem, but the problem didn't seem quite general enough to encode a generic graph coloring problem as a system of these constraints. Ultimately these failed efforts to prove NP-completeness led to the crucial insight to solving the problem efficiently: That the constraints could be viewed as three *independent* systems of constraints with respect to the three elements of the orientations: mirroring about *x*, mirroring about *y*, and 90-degree rotation. Each of those could be formulated as a graph 2-coloring problem independent of the other two problems, and thus efficiently solved. The independence of these three 2-coloring problems was the source of the difficulty in proving NP-completeness, and I'm certain I made that cognitive leap as a result of my attempts to prove the problem NP-complete.

Once you've determined that your problem is NP-complete, that still doesn't necessarily mean it is hard in practice. NP-completeness means that the problem is hard *in the worst case* and *for general inputs*. There are some NP-complete problems for which algorithms have been devised that, while technically heuristics -- they cannot guarantee to find optimal solutions for all problem inputs -- nonetheless in practice almost always do find optimal solutions to problem instances that arise in the real world. This is true, for example, of the satisfiability problem: Determining whether there is an assignment of values to boolean variables that makes a particular logic formula evaluate to *true*.

Further, it is often the case that the inputs your problem will be presented with in whatever real-world situation you are trying to solve will not, in fact, be fully general. Often, an NP-complete problem is no longer NP-complete if the inputs are known to have some kind of structure more restricted than the general case. A column by David Johnson, for example, addresses what graph problems turn out to be efficiently solvable if the input graphs are restricted to specific classes of graphs, e.g. trees, planar graphs, chordal graphs, bipartite graphs, and many more. If you know that your inputs are restricted in some way, sometimes a problem that is NP-complete in general will turn out to be efficiently solvable for those kinds of inputs.

Subsequent posts will address what to do if your problem does turn out to be hard in practice for the kinds of inputs you really see.

Labels: algorithms, np-completeness, theory

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