Questions tagged [open-problem]
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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The hardness of active learning with fixed budget
I have been looking for theoretical papers studying this question of the hardness of PAC active learning algorithms. I found a few papers studying the problem from a fixed perspective (particular ...
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Exploding number of homomorphisms
I'm trying to tackle the following problem: given two graphs $A$ and $B$, if there exists a graph $D$ such that $\hom(A, D) > \hom(B, D)$ (i.e. there is more homomorphisms from $A$ to $D$ than from ...
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Open problem on *Finite Memory Clocks* by Tom Cover
This problem was proposed by Tom Cover in Open Problems in Communication and Computation (Cover and Gopinath, eds), 1987:
How does one tell time when the number of states in the clock is
insufficient ...
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"Refined" list of open problems in TCS
In the conference on learning theory (COLT), a list of open problems is published every year, for example, the list of 2019.
The open problems are being submitted and peer reviewed, which makes this ...
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Areas of research and open problems in functional programming [closed]
What are the major areas of functional programming that require more research and development? For example, I know a lot of people are asking for dependent types in Haskell, and someone at my uni is ...
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Open Problems About Nowhere-Dense Classes of Graphs
I'm writing a survey about nowhere-dense graphs. I would like to list some of the main open problems in the field. In particular I would like to list problems of the following form.
The problem has ...
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A conjecture related to the Cerny conjecture - counterexample/reference request
The Cerny conjecture is the statement that any synchronizing automaton with $n$ states has a synchronizing word of length at most $(n-1)^2$. The best current upper bound for the length of a ...
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Complexity of comparing extended integer power towers
Inspired by this stackexchange question, is it an open problem to compare two power towers of positive integers if we additionally allow numbers lower in the tower to themselves be represented by ...
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Reduction of irregular graphs, to regular graphs, while preserving hamiltonicity
I am wondering if this is a topic that has had research done...
If I could reduce irregular graphs to regular graphs (including replacing redundant node clusters with dummy nodes), while ensuring ...
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Open/unsolved problems in (computational) random matrix theory / matrix completion?
I was wondering what some open problems are in random matrix theory (especially those of interest to TCS people/so mainly non-asymptotic things, I imagine). Also, and relatedly, what are remaining/...
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List of (unsolved) complexity problems arising from PL
What are some major, open computational complexity problems that arise from programming languages, especially program analysis and compilation? I am looking for problems on the lines of "the time ...
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Noisy channel coding theorem in quantum information
Why Shannon's noisy channel coding theorem can't be used for quantum communication applications?
Schumacher proved the first Noiseless theorem and there are quantum error correction mechanisms out ...
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What progress is being made on $NP \cap coNP$ having complete problems? [closed]
I'm very curious about this and have a proof sketch that it does not in the answers below. The reasons I'm currently thinking about this is because of the new paper on the class $PTFNP$ which can be ...
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Does Memcomputing really solve an NP-complete problem?
I came across an article published in Science "Memcomputing NP-complete problems in polynomial time using polynomial resources and collective states", which makes some pretty astonishing claims.
...
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Does PEG contain CFG?
Despite their considerable expressive power, all PEGs can be parsed in linear time using a tabular or memoizing parser (8). These properties strongly suggest that CFGs and PEGs define incomparable ...
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In regards to the tautologies of a polynomially-bounded propositional proof system
In the book 'Logical Foundations of Proof Complexity', co-authored by Stephen Cook, the following definition is given:
A proof-system $F$ is said to be polynomially-bounded if there is a polynomial p(...
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Does there exist a hardest DCFL?
Greibach famously defined a language $H$, the so-called nondeterministic version of $D_2$, such that any CFL is an inverse morphic image of $H$. Does there exist a similar statement with DCFL, ...
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Why can't we have superlinear bounds on Boolean circuit size for an explicit function?
I am interested about the minimal size (number of gates) of a family of circuits (with negation) over a complete Boolean basis (with fanin 2) that computes some explicit Boolean function. (In other ...
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Log Rank Conjecture Collaborative Approach [closed]
Recently a post was made in Mathoverflow seeking possible avenues for collaborative projects.
I made a proposal for Log Rank conjecture in https://mathoverflow.net/questions/219638/proposals-for-...
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Massive online collaboration for solving open problem in theoretical computer science
In Polymath projects a large group work on an open problem.
What kind of problems seem to work best in this framework?
Are there any good candidates for a polymath project in theoretical computer ...
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Is it still open to determine the complexity of computing the treewidth of planar graphs?
For a constant $k \in \mathbb{N}$, one can determine in linear time, given an input graph $G$, whether its treewidth is $\leq k$. However, when both $k$ and $G$ are given as input, the problem is NP-...
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$\Delta = 57, d=2$ Moore Graph
I am looking into the last open question regarding the existence of Moore Graphs of diameter 2. A problem that has been open in combinatorics for more than 55 years.
You may recall that Hoffman and ...
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List of number theoretic or algebraic problems in various complexity classes
I am looking for a list about the known or unknown complexity of various number theoretic /algebraic problems. For example,
GCD in $NC^1$ is open,
factoring in $P$ is open,
computing sheaf ...
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Research problems in communication complexity
There have been many open challenges questions in this forum.
For instance,
Research and open challenges in Programming Language Theory
What are current open problems in compiler theory?
...
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Problems not known to be PSPACE-complete
What are problems with the following properties:
1) they are restriction of (possibly well known) problems that are PSPACE-complete;
2) the restricted versions are in PSPACE, but it is an open ...
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Are there any interesting open questions having to do with submodularity, specially in the intersection of theoretical machine learning?
I was interested in knowing about open research topics related with sub modularity, specially within its intersection with theoretical machine learning (and related topics).
I am particularly ...
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Integer linear programming in logarithmic number of variables
I read that integer linear programming is solvable in polynominal time if the number $n$ of variables is fixed, i.e. $n \in O(1)$. If the number of variables grows logarithmically, i.e. $n \in O(\...
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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 ...
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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)
...
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Is graph coloring complete for poly-APX?
Is the graph coloring problem complete for poly-APX under C-reductions
(alternatively, under AP-reductions)? For the graph coloring problem, speaking of a feasible solution means a proper coloring for ...
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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 don'...
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Identifying Reducible/Irreducible polynomials over $Z[x]$
It is well known LLL algorithm provides a fully polynomial algorithm to factor a reducible primitive polynomial over $\mathbb{Z}[x]$.
Say one only seeks to identify whether a given polynomial over $\...
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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 ...
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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 matrix)...
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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, $\...
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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 answer ...
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Deciding whether an NC${}^0_3$ circuit computes a permutation or not
I would like to ask about a special case of the question “Deciding if a given NC0 circuit computes a permutation” by QiCheng that has been left unanswered.
A Boolean circuit is called an NC0k circuit ...
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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 = ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 its rank ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 ...
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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 ...
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Efficiently computable function as a counter-example to Sarnak's Mobius conjecture
Recently, Gil Kalai and Dick Lipton both wrote nice articles on an interesting conjecture proposed by Peter Sarnak, an expert in number theory and the Riemann Hypothesis.
Conjecture. Let $\mu(k)$ ...
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