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I'm looking for interesting open questions in complexity theory that someone with an undergraduate degree in math and comp/sci could theoretically tackle. I have strong interest in the polynomial hierarchy, and the study of probabilistic classes like RP, co-RP, ZPP, BPP, and also their logarithmic counterparts. I also have some interest in quantum computing complexity theory, but I'm afraid my knowledge of quantum isn't that strong. (Only the basics from my modern physics class, which included some quantum). Important: must be new research or new expansion on previous research, a question that either hasn't been asked or hasn't been answered successfully that I might be able to say something interesting about, even if I don't ultimately solve it. I'm willing to learn any necessary programming languages/programs/methods/etc to the end of answering this question (or trying to)!

Note: this is for a senior year undergraduate thesis in math / comp sci.

Thanks for any suggestions!

Edit: An example might be to analyze the complexity of a particular problem that hasn't been (formally) analyzed yet but may yield interesting results.

Manny G
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  • I would begin by looking at the research level problems given in The Art of Computer Programming by Knuth. – user 85795 Oct 22 '15 at 05:18
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    Unfortunately, your question is not a good fit for the Stack Exchange format. We prefer questions that have objectively correct answers that will be useful both to the asker and others who have the same question in the future. What is or is not a suitable topic for study, projects or research is very much a matter of opinion and depends crucially on the interests and skills of the person who will be doing the work and the support that will be available to them. This is a question that you should be asking your professors. – David Richerby Oct 22 '15 at 07:31
  • Here is a post on open problems for Finite Automata: http://cstheory.stackexchange.com/questions/22493/are-there-any-open-problems-left-about-dfas/ – Michael Wehar Oct 22 '15 at 08:08
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    this question veers on the edge of SE "too broad" & "opinion-based" criteria. however the undergraduate scope significantly limits it. there are very many open problems but many are "phd level" or much harder. my suggestion would be to experiment with empirical analysis of algorithms which can be applied to nontrivial problems and still yield insights not found in the literature. may write this up later. also suggest drop by [chat] for more discussion. – vzn Oct 22 '15 at 15:12

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Do something interesting with visibly languages. They are a relatively recent topic with many useful practical applications. For example, I would find it interesting to better understand the role of synchronization here, compared to the role of synchronization for the theory of codes. Already the simplest case could be interesting, where the visibly language just distinguishes between inner symbols and separator symbols, and the considered codes are just bifix code.

David Richerby is right that this questions allows opinion biased answers. The origins of the opinions voiced in this answer can be found here.

Community
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Thomas Klimpel
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  • If you agree that the question is inappropriate for the site, please don't answer it. – David Richerby Oct 22 '15 at 08:37
  • @DavidRicherby Well, "Many good question generate some degree of opinion based on expert experience...", and my answer is not "almost entirely based on opinion", because it also contains facts and references. These references (to visibly languages) are the reason why I decided to answer this question. But everything after the first two sentences is primarily opinion based, so I found it appropriate to include a disclaimer. – Thomas Klimpel Oct 22 '15 at 08:55
  • Thank you for the suggestion, @ThomasKlimpel. I'll definitely take a look at this! – Manny G Oct 22 '15 at 15:57
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P = NP problem is a major unsolved problem in computer science (in complexity theory ). Details can be found https://en.wikipedia.org/wiki/P_versus_NP_problem

Pranjal
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    "that someone with an undergraduate degree in math and comp/sci could theoretically tackle", experience shows that this problem isn't likely to answer this requirement. – Ariel Oct 22 '15 at 03:59
  • well, I don't think the unsolved problems can be classified on the basis of what current degree or education the person has. He/she has to extend further to solve them – Pranjal Oct 22 '15 at 04:01
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    This is definitely outside the scope of what I could study. I'm looking for a question I have a good shot of answering. I think what @Ariel means is that there's little (if anything) I could say on the subject that no other (far more experienced) computing theorist has already said. – Manny G Oct 22 '15 at 04:38
  • But I thank you for the suggestion, regardless (: – Manny G Oct 22 '15 at 04:38
  • @MannyG Thank you. But I want to tell you that if you want to start research or solve problems at this stage, do some literature survey of latest CS papers. Analyze them and try to add something to them. Given your current education/knowledge, it is very difficult to solve an unsolved problem. Much better way is to help a professor who is solving a problem. – Pranjal Oct 22 '15 at 04:56
  • That's good advice; and I've heard similar sentiments echoed in other forums I've posted this question in. Do you have any recommendations for where I could go to survey recent CS papers? – Manny G Oct 22 '15 at 05:11
  • @MannyG try ieee xplore, acm library etc – Pranjal Oct 22 '15 at 07:02
  • agreed P vs NP is one of the hardest open math problems. however there are basic ways to attack it that can yield insight & new knowledge (eg empirically) while still short of solving it. – vzn Oct 22 '15 at 15:14
  • @Pranjal Thank you; I will definitely explore those resources! – Manny G Oct 22 '15 at 15:54