Problems known to be open in the literature and any problem that, after being posed, is decided to be open by the community.

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6
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0answers
83 views

Examples of open problems solved through application of a theorem already known

Are there good examples of reasonable open problems in TCS that had an 'obvious' solution via application of a theorem found in mathematics probably found a few decades earlier but went unnoticed in ...
0
votes
0answers
60 views

Is inclusion of DCFL (or even DCSL) decidable?

Given two deterministic context-free languages $L_1$ and $L_2$, is $L_1 \subset L_2$ decidable? If so, in what complexity class may such an algorithm be? It is clear that the same problem for ...
1
vote
2answers
195 views

What is the asymptotic time complexity of the number of steps of “Half Or Triple Plus One” ( HOTPO)?

The "Half Or Triple Plus One" process goes as follows: start with $x=n$ for some value of $n$ if ($x$ is odd) $x = 3x+1$ else $x = \frac{x}{2}$ if ($x$ > 1) goto (2) ...
47
votes
9answers
5k views

Are there any open problems left about DFAs?

After studying deterministic finite state automata (DFA) in undergrad, I felt they are extremely well understood. My question is whether there is something we still don't understand about them. I ...
17
votes
4answers
3k views

Research and open challenges in Programming Language Theory

In the spirit of some general discussions like this one, I'm opening this thread with the intention to gather opinions on what are the open challenges and hot topics in research on programming ...
10
votes
1answer
246 views

Why does the log-rank conjecture use rank over the reals?

In communication complexity, the log-rank conjecture states that $$cc(M) = (\log rk(M))^{O(1)}$$ Where $cc(M)$ is the communication complexity of $M(x,y)$ and $rk(M)$ is the rank of $M$ (as a ...
9
votes
1answer
224 views

On $\mathcal L$, $\mathcal{N\!L}$, $\mathcal L^2$, $\mathcal P$ and $\mathcal{N\!P}$

We know that $\mathcal{L}\subseteq \mathcal{N\!L}\subseteq\mathcal{P}\subseteq\mathcal{N\!P}$. From Savitch's Theorem, $\mathcal{N\!L}\subseteq\mathcal{L}^2$, and, from Space Hierarchy Teorem, ...
5
votes
1answer
338 views

Do we know that the P vs. NP question isn't affected by Gödels incompleteness theorem? [duplicate]

Possible Duplicate: Implications of unprovability of $P\neq NP$ I shortly came across Gödels incompleteness theorem again and I wondered, since so much time has been spent on trying to ...
8
votes
0answers
180 views

What are the most recent developments in small-depth quantum circuits?

Back in 2005, Scott Aaronson posted a list of 10 "semi-grand" challenges for quantum computing theory which contained the following challenge: The power of small-depth quantum circuits. Is $BQP = ...
9
votes
0answers
146 views

Learning with (Signed) Errors

$\underline{\bf Background}$ In 2005, Regev [1] introduced the Learning with Errors (LWE) problem, a generalization of the Learning Parity with Error problem. The assumption of this problem's ...
9
votes
1answer
1k views

What is the “nearest” problem to the Collatz conjecture that has been successfully resolved?

I am interested in the "nearest" (and "most complex") problem to the Collatz conjecture that has been successfully solved (which Erdos famously said "mathematics is not yet ripe for such problems"). ...
12
votes
2answers
1k views

Optimal algorithm for finding the girth of a sparse graph?

I wonder how to find the girth of a sparse undirected graph. By sparse I mean $|E|=O(|V|)$. By optimum I mean the lowest time complexity. I thought about some modification on Tarjan's algorithm for ...
0
votes
2answers
374 views

largest language class for which inclusion is decidable

am wondering what is the largest language class that is known for which set inclusion is decidable, ie a class such that if $A, B$ are in that class then $A \subset B$ is decidable. am also ...
1
vote
1answer
254 views

Questions about computing matrix rigidity

Matrix rigidity was introduced by Valiant in 1977: The rigidity $Rig_M(r)$ of boolean matrix $M$ over GF(2) is the smallest number of entries of $M$ that must be changed in order to reduce ...
-1
votes
2answers
595 views

What progress has been made to prove whether or not p=np? [closed]

I know that it is still one of the biggest mysteries of computer science whether non-deterministically polynomial problems can be solved in polynomial time. I am curious to know what makes this ...
9
votes
0answers
192 views

Complexity of the min edge-colored cut problem

Given an undirected graph $G=(V,E)$ with a color on each edge, the problem is to find a 2-partition $(V_1,V_2)$ of $V$ s.t. the number of colors used by the edges $uv, u \in V_1, v \in V_2$ is ...
13
votes
1answer
429 views

Given a graph, decide if its edge connectivity is at least n/2 or not

Chapter 1 of the book The Probabilistic Method, by Alon and Spencer mentions the following problem: Given a graph $G$, decide if its edge connectivity is at least $n/2$ or not. The author mentions ...
4
votes
0answers
293 views

Complexity of balanced graph partition problem

Wagner and Wagner, in "Between min cut and graph bisection" (MFCS 1993), studied a variant of minimum bisection problem where we seek a cut with minimum size such that each partition has at least ...
12
votes
0answers
505 views

Online algorithms: open problems

Recently the long-standing k-server problem has been solved by Nikhil Bansal, Niv Buchbinder, Aleksander Mądry and Seffi Naor (to appear in FOCS 2011). I'm interested in knowing other open problems in ...
14
votes
2answers
390 views

Quantum PAC learning

Background Functions in $AC^0$ are PAC learnable in quasipolynomial time with a classical algorithm that requires $O(2^{log(n)^{O(d)}})$ randomly chosen queries to learn a circuit of depth d [1]. If ...
34
votes
1answer
875 views

Efficiently computable function as a counter-example to Sarnak's Mobius conjecture

Recently Gil Kalai and Dick Lipton both wrote a nice article on an interesting conjecture proposed by Peter Sarnak, an expert in number theory and Riemann Hypothesis. Conjecture. Let $\mu(k)$ be ...
23
votes
2answers
824 views

Finding a prime greater than a given bound

Is a deterministic polynomial-time algorithm known for the following problem: Input: a natural number $n$ (in binary encoding) Output: a prime number $p > n$. (According to a list of open ...
10
votes
2answers
1k views

What are current open problems in compiler theory?

Compiler theory seems to be a pretty vetted subject. What are some open problems or current research happening in the field?
11
votes
2answers
980 views

Space-time tradeoff and the best algorithm

Consider some language $L$ such that: $L \in DTIME(O(f(n))) \cap DSPACE(O(g(n)))$ and so that $L \not\in DTIME(o(f(n))) \cup DSPACE(o(g(n)))$ In other words, the fastest machine $M$ computes $L$ ...
22
votes
2answers
939 views

Protocol partition number and deterministic communication complexity

Besides (deterministic) communication complexity $cc(R)$ of a relation $R$, another basic measure for the amount of communication needed is the protocol partition number $pp(R)$. The relation between ...
21
votes
1answer
618 views

Complexity of computing shortest paths in the plane with polygonal obstacles

Suppose we are given several disjoint simple polygons in the plane, and two points $s$ and $t$ outside every polygon. The Euclidean shortest path problem is to compute the Euclidean shortest path ...
5
votes
0answers
376 views

Contained in NP and Turing-reduction from an NP-complete problem $\Rightarrow$ NP-complete under Karp reductions? [duplicate]

Possible Duplicates: Do many-one reductions and Turing reductions define the same class NPC Many-one reductions vs. Turing reductions to define NPC Let $P,Q \subseteq \Sigma^*$ be ...
14
votes
1answer
767 views

Status of Cerny Conjecture?

A DFA has a synchronizing word if there is a string that sends any state of the DFA to a single state. In ‘The Cerny Conjecture for Aperiodic Automata” by A. N. Trahtman (Discrete Mathematics and ...
48
votes
9answers
5k views

One Stack, Two Queues

background Several years ago, when I was an undergraduate, we were given a homework on amortized analysis. I was unable to solve one of the problems. I had asked it in comp.theory, but no ...
11
votes
2answers
1k views

Is $coNP^{\#P}=NP^{\#P}=P^{\#P}$?

By http://www.cs.umd.edu/~jkatz/complexity/relativization.pdf If $A$ is a PSPACE-complete language, $P^{A}=NP^{A}$. If $B$ is a deterministic polynomial-time oracle, $P^{B}\ne NP^{B}$ (assuming ...
11
votes
2answers
512 views

Is the 3-sphere recognition problem NP-complete?

It is known that determining whether or not a given triangulated 3-manifold is a 3-sphere is in NP, via work by Saul Schleimer in 2004: "Sphere recognition lies in NP" arXiv:math/0407047v1 [math.GT]. ...
14
votes
0answers
252 views

Any example of an unsatisfiable integer program with non constant Rank Lower bounds for LS+ cuts but with short LS+ refutations?

Assume we want to refute an unsatisfiable CNF. We can interpret it as an integer program, thus a refutation can be done by applying Lovasz-Schrijver semidefinite cuts ($LS_{+}$ cuts) to its linear ...
23
votes
3answers
1k views

Is it NP-hard to play international draughts correctly?

Is the following problem NP-hard? Given a board configuration for $n\times n$ international draughts, find a single legal move. The corresponding problem for $n\times n$ American checkers (aka ...
33
votes
7answers
2k views

What is the oldest open problem in TCS?

This problem is inspired by this MO question, which I thought was very interesting. What is the oldest open problem in TCS? Clearly this question needs some clarification. First, what is TCS? ...
15
votes
2answers
900 views

Positive topological ordering, take 3

Suppose we have an n by n matrix. Is it possible to reorder its rows and columns such that we get an upper-triangular matrix? This question is motivated by this problem: ...
46
votes
1answer
1k views

A combinatorial version for the polynomial Hirsch conjecture

Consider $t$ disjoint families of subsets of {1,2,…,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ . Suppose that (*) For every $i \lt j \lt k$ and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$, ...
17
votes
2answers
649 views

Problems between NC and P: How many have been resolved from this list?

In the paper "A Compendium of Problems Complete for P" by Greenlaw, Hoover and Ruzzo (PS) (PDF), there is a list of problems in P that are not known to be in NC and not known to be P-complete either. ...
19
votes
2answers
1k views

Polynomial time approximation algorithms for machine scheduling: how many open problems are left?

In 1999, Petra Schuurman and Gerhard J. Woeginger published the paper "Polynomial time approximation algorithms for machine scheduling: Ten open problems". Since then, to the best of my knowledge, ...
9
votes
1answer
503 views

Projective Plane of Order 12

Objective: Settle the conjecture that there is no projective plane of order 12. In 1989, using computer search on a Cray, Lam proved that no projective plane of order 10 exists. Now that God's ...
44
votes
13answers
5k views

Open problems on the frontiers of TCS

In the thread Major unsolved problems in theoretical computer science?, Iddo Tzameret made the following excellent comment: I think we should distinguish between major open problems that are ...
-3
votes
2answers
649 views

At last P != NP or not [duplicate]

Possible Duplicate: Is the recent proof that P != NP correct? some weeks ago I heard a news that some one proof that P != NP (link1 - link2) andsome days later I heard that he was wrong (I ...
35
votes
6answers
2k views

Grid $k$-coloring without monochromatic rectangles

Update: The obstruction set (i.e. the NxM "barrier" between colorable and uncolorable grid sizes) for all monochromatic-rectangle-free 4-colorings is now known. Anyone feel up to trying 5-colorings? ...
33
votes
2answers
3k views

Is optimally solving the n×n×n Rubik's Cube NP-hard?

Consider the obvious $n\times n\times n$ generalization of the Rubik's Cube. Is it NP-hard to compute the shortest sequence of moves that solves a given scrambled cube, or is there a polynomial-time ...
34
votes
1answer
2k views

Multiplying n polynomials of degree 1

The problem is to compute the polynomial $(a_1 x + b_1) \times \cdots \times (a_n x + b_n)$. Assume that all coefficients fit in a machine word, i.e. can be manipulated in unit time. You can do $O(n ...
5
votes
4answers
391 views

Are there alternatives to using polynomials in defining the different notions of efficient computation?

Is invoking polynomials in defining the different notions of efficient computation the real obstacle to resolve the P vs NP problem? Do we need a paradigm shift by redefining what constitute an ...
18
votes
1answer
401 views

Approximating the sign rank of a matrix

The sign rank of a matrix A with +1,-1 entries is the least rank (over the reals) of a matrix B which has the same sign pattern as A (i.e., $A_{ij}B_{ij}>0$ for all $i,j$). This notion is important in ...
164
votes
58answers
50k views

Major unsolved problems in theoretical computer science?

Wikipedia only lists two problems under "unsolved problems in computer science": P = NP? The existence of one-way functions What are other major problems that should be added to this list? Rules: ...
102
votes
11answers
8k views

How hard is unshuffling a string?

A shuffle of two strings is formed by interspersing the characters into a new string, keeping the characters of each string in order. For example, MISSISSIPPI is a ...