How can we collaboratively investigate open problems?

Question

This question builds on a comment of Shane's in a previous meta post on what can be achieved using a theory Q&A site.

I've spent a good deal of time on Jeff Erickson's lovely problem about unshuffling a string, and during the proposal phase, on his question about the complexity of minimum sequences in generalized Rubik's cube. However, I haven't posted any responses to those questions because I didn't end up with an answer.

Most of what I've done is discover dead ends and approaches that don't work. These might be useful for someone to know so that they don't spend time on the same issue or if they'd like to pick up where I left off. Unfortunately, the Q&A format doesn't seem naturally suited to this style of inquiry. A wrong answer isn't an "answer" per se, and a detailed discussion won't fit in a non-nested comment thread.

The question: how can the CSTheory Q&A format better be used to support these kinds of parallel group investigations into problems?

One idea is to adopt the convention that "answers" to open questions are to be regarded as a discussion of possible approaches, rather than literal answers or partial answers to the question, then to encourage development of these approaches in the comments. The downside is that these comments are not nested.

Another possibility is to fork new questions concerning particular avenues of inquiry with links back to the parent question, perhaps with a dedicated "subproblem" tag; this might lead to a messy proliferation of subquestions.

Has any better format been discovered for collaborative investigation of open problems on other StackExchange sites, in particular, MO? Are there other features in StackExchange we can activate to support this process? Or is it better to instead restrict these kinds of investigations to blogs and wikis, as was done in Polymath?

Answer 1

I am skeptical whether this format is suitable for the polymath-style discussions you're interested in. Threaded comments appear to be critical, as you mention, as well as managing the threads of ideas as they branch out.

Answer 2

I think this site could be used as a supplement to Polymath-style wikis. If the discussion in one of the wiki threads leads to a concrete question that seems fairly innocent (= it looks like it should have been already solved in 1960s; the wiki participants just can't find the right reference and wouldn't like to spend too much time on that part), then they could post the question here (with a pointer to the relevant Polymath thread).

Answer 3

Many open problems (and some experiments, like hard IMO problems) are handled on the polymath blog. MO questions inspired by polymath discussions are tagged as such. JeffE's questions would sit well in a 'CS polymath' blog.

Edit: In other words, I think it's better to start a blog associated with this site, instead of discussing open problems here.

Answer 4

Yes, within the framework of this Q&A, but it requires a little more judgement of what is interesting than the usual manner of use.

Each dead-end you found started as an avenue worth exploring. What question characterises that avenue?

The risk is that these questions are not very interesting. There's a close reason "too localized" that may be invoked if a question concerns, among other alternatives, an extraordinary narrow situation. Looking for an interesting generalisation of the subproblem might help.

If you do craft a nice question, there's a bonus: the motivation for your question is likely to already have interested other posters.

There was a nice instance of a question on SO, Circles and triangles problem, whose answers were in fact a collaborative investigation; all the subquestions were resolved in the Q&A thread over six days, resulting in a good algorithm that was not there shown to be optimal.

Answer 5

This is something I am very interested in too, however a stack exchange site does not seem ideally suited for dealing with open problems. Perhaps some kind of wiki + voting system would be good. Either way, it would seem to be something that needs to be done outside of the stack exchange framework, as I don't think it is the type of use that the owners have in mind for stack exchange. Additionally, you don't need as many users to make such a site useful, as even two or three people collaboratively working on a problem may make more progress than a single person on their own. Perhaps a companion site to this could be set up to deal with open questions? I guess a bunch of us already have our own hosting, so I doubt it would be hard to find a home.