Unanswered Questions
2,636 questions with no upvoted or accepted answers
45
votes
0answers
1k views
Monotone complexity of s-t connectivity
In the problem CONN, we obtain a directed $n$-vertex graph (encoded as a boolean string of $n^2$ bits, one for each potential edge), and want to decide
whether there is a path between all $n^2$ pairs $...
43
votes
0answers
929 views
Problem unsolvable in $2^{o(n)}$ on inputs with $n$ bits, assuming ETH?
If we assume the Exponential-Time Hypothesis, then there is no $2^{o(n)}$ algorithm for $n$-variable 3-SAT, and many other natural problems, such as 3-COLORING on graphs with $n$ vertices. Notice ...
30
votes
0answers
755 views
Is BPP= P known for ANY uniform model of computation?
Many believe that BPP $=$ P "should" hold for Turing machines. We even have some "witnesses" for this: otherwise some "strange" things would happen; see e.g. this paper by Implagliazzo and Wigderson.
...
30
votes
0answers
4k views
Combinatorics of Bellman-Ford or how to make cyclic graphs acyclic?
Roughly speaking, my question is:
How costly is to make a cyclic graph
acyclic while preserving all simple $s$-$t$ paths?
Let $K_n$ be a complete undirected graph on vertices $\{0,1,\ldots,n+1\}$.
(...
29
votes
0answers
925 views
Does $EXP\neq ZPP$ imply sub-exponential simulation of BPP or NP?
By simulation I mean in the Impaglazzio-Widgerson [IW98] sense, i.e. sub-exponential deterministic simulation which appears correct i.o to every efficient adversary.
I think this is a proof: if $EXP\...
27
votes
0answers
633 views
The complexity of checking whether two DAG have the same number of topological sorts
This problem is highly related to the CNF one.
Here is the problem: given two DAG (directed acyclic graphs), if they have the same counting of topological sorts, answer "Yes", otherwise, answer "No".
...
26
votes
0answers
507 views
Rank mod 6 vs rank over the reals
Let $A$ be a boolean matrix (eg with $0,1$ entries). Assume that $A$ has rank $\le r$ both over $\mathbb{F}_2$ and over $\mathbb{F}_3$. Does this imply that $A$ has low rank over the reals? This seems ...
26
votes
0answers
646 views
Is Hankelability NP-hard?
I asked this question on SO on April 7 and added a bounty which has now expired but no poly time solution has been found yet.
I am trying to write code to detect if a matrix is a permutation of a ...
26
votes
0answers
438 views
Adiabatic quantum computing with level crossings
Question.
In adiabatic evolution, to ensure that the ground state high overlap with the unique ground state of the system (i.e. to achieve arbitrarily small error) using adiabatic theorems, it is ...
24
votes
0answers
714 views
What are consequences of the collapse of CH?
I don't grasp the full complexity of the counting hierarchy $CH$. I understand $CH$ is in $PSPACE$, and contains $PH$ within its second level, due to the Toda's theorem. But, what would be important ...
24
votes
0answers
991 views
Counting Isomorphism Types of Graphs
Polya's counting theorem leads to an algorithm for counting (precisely) the number of isomorphism types of graphs with $n$ vertices in $\exp (\sqrt n )$ steps. From Polya theorem you get a formula ...
21
votes
0answers
573 views
What is the power of general poly-size permutation branching programs?
Call $\mathsf{PPBP}$ the class of languages decided by poly-size families of permutation branching programs, which are layered branching programs (i.e., the ones defined here) whose transitions ...
Kristoffer Arnsfelt Hansen
3,9052 gold badges22 silver badges34 bronze badges
21
votes
0answers
792 views
$\Delta = 57, d=2$ 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 ...
21
votes
0answers
667 views
Partial circulant matrices: Is there a non-zero vector $v\in \{-1,0,1\}^n$ such that $Mv=0$?
The following question arose as a side product of some work I have been part of recently.
An $m$ by $n$ $(0,1)$-matrix $M$ is partial circulant if it can be formed by taking the first $m$ rows of a ...
21
votes
0answers
765 views
Longest geometrically increasing subsequence
Given a sorted array of $n$ positive integers, the problem is to find the longest subsequence so that the progression of differences between consecutive elements of the subsequence is geometrically ...
