# Major unsolved problems in theoretical computer science?

Wikipedia only lists two problems under "unsolved problems in computer science":

What are other major problems that should be added to this list?

Rules:

1. Only one problem per answer
2. Provide a brief description and any relevant links
• Since you're asking for a list and there is no single answer, this may work better flagged as a community wiki. – Daniel Apon Aug 17 '10 at 17:23
• One unsolved problem per answer, please; then we can easily rank the answers by voting up/down! – Jukka Suomela Aug 17 '10 at 19:36
• Why only complexity results? There's more to TCS than complexity! No open problems in type theory? programming languages? – Jacques Carette Aug 17 '10 at 21:44
• add them in, Jacques :). – Suresh Venkat Aug 18 '10 at 5:05
• I think we should distinguish between major open problems that are viewed as fundamental problems, like $P\neq NP$, and major open problems which will constitute a technical breakthrough, if solved, but are not necessarily as fundamental, e.g., exponential lower bounds on $AC^0(6)$ circuits (i.e., $AC^0+\mod 6$ gates). So we should possibly open a new community wiki entitled "open problems in the frontiers of TCS", or the like. – Iddo Tzameret Aug 19 '10 at 22:18

Sensitivity versus block sensitivity

Boolean sensitivity is interesting because block sensitivity, a close relative, is polynomially related to several other important and interesting complexity measures (like the certificate complexity of a boolean function). If sensitivity is always related to block sensitivity in a polynomial way, we have an extremely simple characteristic of boolean function that's related to so many others.

One might read Rubinstein's "Sensitivity vs. block sensitivity of Boolean functions" or Kenyon and Kutin's "Sensitivity, block sensitivity, and l-block sensitivity of boolean functions."

• That's a great open problem, but I think that it can't be placed together with the fundamental open problems of the field (e.g., P vs NP) in the same league of "the major problems of TCS". – Dana Moshkovitz Oct 21 '10 at 20:22
• Now resolved! (should be removed?) – Joshua Grochow Sep 15 '19 at 18:03

Is BQP in PH (polynomial hierachy)?

Does there exist any hypothesis class that is NP-Hard to (improperly) PAC learn?

This has some possible implications for complexity, and I think the best progress on this question is here: http://www.cs.princeton.edu/~dxiao/docs/ABX08.pdf

Are NP-completeness in the sense of Cook and NP-completeness in the sense of Karp different concepts, assuming P $\neq$ NP?

The "P vs NP" question extends naturally to polynomial-time hierarchy (PH): "Whether PH has infinite levels, or it collapses to some finite level?"

I think this question is (or should be considered as) the most intriguing question of the computer science: If PH has infinite levels, then $\mathbf{P} \neq \mathbf{NP}$. In addition, several researchers have shown that if Graph Isomorphism is NP-complete, then PH collapses to the 2nd level. Therefore, if PH has infinite levels, then Graph Isomorphism is provably not NP-complete.

Several other results follow from the infiniteness of the levels of PH.

What about proving BPP is contained in NP? (Unconditionally; we already know that BPP=P assuming pretty reasonable complexity assumptions)

Another open problem for the lambda calculus (from TLCA list of open problems; PDF version ).

Problem #22 on the list:

Is there a continuously complete CPO model of the $\lambda$-calculus whose theory is precisely λβη or λβ ?

Is there a best algorithm for integer multiplication and matrix multiplication (MM), or for that matter any other familiar problem? Manuel Blum has suggested these are good candidates not to have a best algorithm. Among bilinear identities such as Strassen's there is no best one according to Coppersmith and Winograd (1982). If the conjectures of Umans et al are correct, then there is no best algorithm of the type they study. For relevant articles Google "Speedup for Natural Problems".

• Can you be more specific on you mean by a "best algorithm"? (There is a general linear speed-up theorem.) – Kaveh Nov 16 '10 at 4:31

The word "major" is a bit frightening and takes us to the P/NP and related questions. Among the almost-major problems which might be feasible, one that I like is the question of randomized decision trees for graph properties. Is it true that for every non-trivial monotone graph property for graphs with n vertices the expected number of queries that you need to ask in order to know if the graph satisfy the property is constant n^2.

This conjecture is known as the Aanderaa-Karp-Rosenberg conjecture.

Proving the existence of hard-on-average problems in NP using the P≠NP assumption.

Bogdanov and Trevisan, Average-Case Complexity, Foundations and Trends in Theoretical Computer Science Vol. 2, No 1 (2006) 1–106

• Actually, proving the existence of hard-on-average problems in NP based on any reasonable assumption would be great. – Or Meir Jun 8 '12 at 23:56
• Is this related to the existance of one-way functions? – Zsbán Ambrus Feb 5 '15 at 7:32

K-server conjecture and randomized K-server conjecture.

Definition according to wikipedia

An online algorithm must control the movement of a set of k servers, represented as points in a metric space, and handle requests that are also in the form of points in the space. As each request arrives, the algorithm must determine which server to move to the requested point. The goal of the algorithm is to keep the total distance all servers move small, relative to the total distance the servers could have moved by an optimal adversary who knows in advance the entire sequence of requests.

Unconditional derandomization of Arthur-Merlin games. It is also known that under hardness assumptions AM = NP. The question is, can we prove unconditionally that AM is a subset of sigma sub 2: http://cs.haifa.ac.il/~ronen/online_papers/online_papers.html

• Could you link to one paper instead of a collection? – domotorp Oct 27 '17 at 3:38

Is there an algorithm to compute the generalized star-height of a given regular language?

Generalized regular expressions are defined like regular expressions, but they allow the complement operator. The generalized star height (gsh) of a regular language is the minimum nesting depth of Kleene stars needed to represent the language by a generalized regular expression. Regular languages of gsh 0 (also known as star-free languages) have two nice characterizations: Schützenberger gave an algebraic characterization (their syntactic monoid is aperiodic) and McNaughton showed they correspond to FO[<].

It follows that there are languages of gsh $1$, like $(aa)^*$, but no language of gsh $> 1$ is known! Thus a subproblem would be first to find such a language, or to prove that all regular languages have gsh 1. See also http://www.liafa.univ-paris-diderot.fr/~jep/Problemes/starheight.html

• Welcome to cstheory.stackexchange! Very interesting that no language of gsh > 1 is known. Are there even candidate languages that people think might not have gsh = 1? Or a regular language whose gsh is unknown? Or is it that all regular languages that anyone has ever checked have gsh$\leq 1$? – Joshua Grochow Aug 8 '13 at 13:58
• @JoshuaGrochow It is likely that complicated examples can be found as follows: take a large permutation group, say $S_9$ generated by a cycle $a$ and a transposition $b$. Now consider the (regular) language of all words whose value in $S_9$ is the identity. One can hope that this language has gsh $> 1$. On the other hand, some people believe that every regular language has gsh $\leqslant 1$. Actually, one could define an "intermediate star-height" by replacing complement by intersection, but even so, I don't know of any language of intermediate star-height $> 1$. – J.-E. Pin Aug 9 '13 at 22:54

Open Problem Garden hosts a number of unsolved problems in theoretical computer science.

• Could you find examples of major open problems that we don't have on this list yet, and post them as separate answers? – Jukka Suomela Sep 5 '10 at 10:09
• Sure Jukka you got it. – Vincent Russo Sep 5 '10 at 13:53

Proving that BPP is in NP is harder than separating NEXP from BPP

That BPP is in NP implies that the polynomial identity problem is in NP, which separates NEXP from P/Poly (BPP is in P/Poly).

• Indeed, proving "BPP is in NP" necessarily "separates BPP from NEXP", because NP and NEXP are separated. But BPP is contained in NP (or in NTIME(..somewhat more than poly..)) is a concrete inclusion I would really like to see proven in the near future. – Dana Moshkovitz Oct 18 '10 at 11:13

### Open Problem Lists

• The number of answers spans over a page. I think a list of responses (similar to other big-list questions) would be useful. – Kaveh Oct 11 '15 at 19:40

Algebraic dichotomy conjecture (Bulatov, Jeavons and Krokhin): Assuming ETH, every constraint satisfaction problem is either in $P$ or requires $2^{ \Omega(n)}$ time.

• Note that the dichotomy only applies to CSPs of a particular form, those consisting of all instances expressible with a fixed set of constraint relations. Moreover, there is a further technical issue, whether every set of constraint relations that is NP-complete can be equivalently represented by a finite set of relations. – András Salamon Nov 13 '10 at 20:46

The Open Problems Project: http://cs.smith.edu/~orourke/TOPP/

Is Quasi-Polynomial Time in PSPACE?

It seems strange to me that almost all the answers are about computational complexity, while the question asks for problems in all computer science.

To counter-balance a little bit:

Decidability of the dot-depth hierarchy: Given a first-order formula on finite words and an integer $k$, is there an equivalent first-order formula with only $k$ quantifier alternations?

Recent progress has been made, it has been showed decidable for $k=2$ in a 2014 paper by Thomas Place and Marc Zeitoun, but the general problem is still wide open.

Can we multiply two arbitrary $$n$$-bit numbers in $$O(n)$$ time? There is a trivial lower bound of $$\Omega(n)$$, but no better lower bound is known. Currently, the asymptotically fastest algorithm is $$O(n \log n)$$ - a recent breakthrough by Harvey and van der Hoeven 2019. The previous best took time $$O(n \log n 2^{\log^* n})$$ due to Martin Fürer, building off of the original algorithm by Schönhage–Strassen which ran in $$\Theta(n \log n \log \log n)$$ time. Regan and Lipton showed that a super-linear lower bound would follow from the Hartmanis-Stearns Real-Time Computability Conjecture.

A recent result of Afshani, Freksen, Kamma, and Larsen established the following conditional lower bound: assuming the network coding conjecture, any constant-degree Boolean circuit for multiplication must have size $$\Omega(n\log n)$$.

• This is a clearer exposition than an earlier version of the question by J.A from 2012. – András Salamon Mar 3 '14 at 15:03
• Maybe "asymptotically fastest" was a typo when referring to Schonhage-Strassen (since you then cite Furer's algorithm which is asymptotically faster...)? – Joshua Grochow Sep 18 '14 at 19:52
• Wouldn't it be good to mention the recent (conditional) lower bound based on the network coding conjecture? @JoshuaGrochow – Clement C. Sep 15 '19 at 19:34
• @ClementC. Is network coding complexity theoretic? – T.... Sep 16 '19 at 2:09
• @T.... The lower bound is. "Assuming [conjecture], any constant degree boolean circuit for multiplication must have size $\Omega(n\log n)$." – Clement C. Sep 16 '19 at 2:11
• Do a pseudo-randomized numbers generator exist?
• If one way function exists, then PRG exist. That is a very well known results from late 80s. So this doesn't make it a independent unresolved problem because if you start counting this, then so does most of the cryptographic primitives that can be constructed by OWF. – Jalaj Jan 27 '12 at 5:12

If P != NP, Does the polynomial hierarchy collapse?

(Because if P = NP then it completely collapses, of course)

Finding natural SampNP-complete distributional problems.

Informally, Samp-NP is the class of NP problems restricted to distributions that are samplable in polynomial time ("On the Theory of Average Case Complexity", Ben-David, Chor, Goldreich and Luby, JCSS 1992, doi:10.1016/0022-0000(92)90019-F). This class aims to capture the complexity of solving NP on real life instances. While it is known that this class has complete problems, we do not know of any natural complete problems for this class. Finding such a problem would yield the first natural problem for which we have good theoretical reasons to believe that it is hard on average.

Some open problems in complexity theory lower bounds, together with their relationships, are mapped here.

• nice, useful, detailed, how about more of a summary/abstract on the misc relationships covered (which isnt really in the abstract either) – vzn Mar 3 '14 at 1:07

Getting an O(1) factor approximation algorithm in polytime for the Maximum Independent Set of Rectangles.

This is one of the biggest open problems in Computational Geometry. Recently, Anna Adamaszek and Andreas Wiese [1] have given a QPTAS for this problem, which shows the existence of a PTAS assuming standard complexity theory conjectures. However even a constant factor approximation is not known yet that can be achieved in polytime. The best known polytime approximation factor is $O(\log \log n)$ [2]. More recently, Abed et al. [3] have given a constant factor approximation based on a conjecture.

Conjunctive query containment over bag semantics

In a GoogleFight between two search queries, can one tell if the first query always wins, without looking at the data?

This 1993 question from database theory [1] asks whether it is possible to decide if an SQL query (more precisely, a conjunctive query) always yields at least as many answers as another conjunctive query, over all possible databases. It would be nice to answer this question to help with query optimization.

One can also formalise the question without referring to databases (see also [2]):

Homomorphism Domination
Input: finite relational structures $S$ and $S'$.
Question: is it true that for any finite relational structure $T$, there are at least as many relational structure homomorphisms from $S$ to $T$ as there are from $S'$ to $T$?

Since the quantification is over an infinite set of structures $T$, this may be undecidable. It is known to be NP-hard [3] as a special case of a more general question over positive semirings (the non-negative integers with addition and multiplication is a semiring); $\Pi^P_2$-hardness was claimed two decades ago but remains unclear. If instead of conjunctive queries, slightly more general queries are allowed, then the problem does become undecidable, via reductions from Hilbert's 10th problem [4,5].

What makes this question interesting is that for most positive semirings of interest the general question is decidable, and actually in $\Pi_2^P$. In fact, for the Boolean semiring case the question becomes: given two conjunctive queries, is it always true that when the first query has an answer then so does the second? Ashok Chandra and Philip Merlin showed in 1977 [6] that this is equivalent to checking whether there exists a homomorphism between the queries, which is in NP. Moreover, in typical databases the queries are usually small or even fixed, while the data is large and changes frequently. This means that even brute force search for a homomorphism between the input queries may be worthwhile.

So it might be a good idea to look quite closely at two small fixed conjunctive queries to decide which is the better one to use. Yet we don't know if such queries can be compared based on the number of answers they generate.

Edit: added some key references as requested by Sylvain.

1. Surajit Chaudhuri and Moshe Y. Vardi, Optimization of Real Conjunctive Queries, PODS 1993, 59–70. doi:10.1145/153850.153856
2. Swastik Kopparty and Benjamin Rossman, The homomorphism domination exponent, EJOC 2011, 32(7) 1097–1114. doi:10.1016/j.ejc.2011.03.009
3. Egor V. Kostylev, Juan L. Reutter, and András Z. Salamon, Classification of Annotation Semirings over Containment of Conjunctive Queries, TODS 2014, 39(1) 1:1–39. doi:10.1145/2556524
4. Yannis E. Ioannidis and Raghu Ramakrishnan, Containment of conjunctive queries: beyond relations as sets, TODS 1995, 20(3) 288–324. doi:10.1145/211414.211419
5. T.S. Jayram, Phokion G. Kolaitis, and Erik Vee, The containment problem for Real conjunctive queries with inequalities, PODS 2006, 80–89. doi:10.1145/1142351.1142363
6. Ashok K. Chandra and Philip M. Merlin, Optimal Implementation of Conjunctive Queries in Relational Data Bases, STOC 1977, 77–90. doi:10.1145/800105.803397
• Nice one. Could you provide more links to the literature, on decidable cases etc. ? – Sylvain Feb 28 '14 at 21:44
• @Sylvain: as is usual here, I have avoided linking directly to my own work but our TODS 2014 paper has all the key links. – András Salamon Feb 28 '14 at 23:36
• Since I'm not bound by such usage habits, here is a link for others who might be interested: tods.acm.org/accepted/2013/SalamonClassification.pdf. You could however provide a link to the PODS 1993 Chaudhuri & Vardi paper at dx.doi.org/10.1145/153850.153856 without being suspected of self-promotion. – Sylvain Mar 1 '14 at 15:52

Is $\mathsf{L}=\mathsf{NL}$? What about non-uniform versions?

• Links or references, please. When you edit, would you consider make it standard fonts like everybody else? – scaaahu Jun 8 '12 at 12:09
• Please use regular fonts. and run a spell checker – Suresh Venkat Jun 9 '12 at 0:12