
Open Problems in Theoretical Computer Science - p4bl0
http://a3nm.net/work/research/questions/
======
danharaj
Open problems in Homotopy Type Theory:
[https://ncatlab.org/homotopytypetheory/show/open+problems](https://ncatlab.org/homotopytypetheory/show/open+problems)

------
crb002
#1 What is the complexity of boolean matrix multiplication?

#2 Is there a parallel algorithm for GCD?

------
a3_nm
Author here, thanks for the comments! I added two problems to
[http://a3nm.net/work/research/questions/#well-known-
conjectu...](http://a3nm.net/work/research/questions/#well-known-conjectures)
following ideas here.

~~~
chaoxu
I have just answered one of those problems here.
[http://cstheory.stackexchange.com/questions/20245/subset-
of-...](http://cstheory.stackexchange.com/questions/20245/subset-of-a-
bipartite-graph-with-maximal-number-of-minimal-unmatched-vertices) I hope I
didn't misunderstood the problem.

~~~
a3_nm
Thanks a lot! I think you understood the problem correctly, I checked your
argument and it seems correct to me, see
[https://cstheory.stackexchange.com/questions/20245/subset-
of...](https://cstheory.stackexchange.com/questions/20245/subset-of-a-
bipartite-graph-with-maximal-number-of-minimal-unmatched-
vertices#comment77255_33603)

We have no specific plans for this result but it's always comforting to have
an answer after all and know that it's in P. Plus this way I learnt about the
existence of submodular functions. :)

List updated ([http://a3nm.net/work/research/questions/#complexity-of-an-
as...](http://a3nm.net/work/research/questions/#complexity-of-an-assignment-
problem-with-subsets)). Thanks again for making me remove the first problem
from this list!

~~~
chaoxu
np. There might be ways to speed up the running time for this special case.
Naive algorithm using submodular minization + maximum matching as an oracle
would take around O(n^c) where c is probably more than 10.

------
Ace17
Fun fact: the problems are a lot more hard to understand if you're browsing
with Javascript disabled.

~~~
a3_nm
Sorry about this, it should now be fixed, and I submitted it upstream:
[https://github.com/mitya57/python-markdown-
math/pull/10](https://github.com/mitya57/python-markdown-math/pull/10)

Thanks for reporting!

------
Animats
One of the big problems is to find a lower bound on some cryptographic
algorithm. We think that factoring is hard, but there is no proof of this. The
knapsack problem, used in the first public key system, was thought to be hard,
but isn't.

~~~
_asummers
My understanding is that factoring is NOT believed to be NP-hard. Can you
point to literature that suggests it is? Researchers and professors I've
spoken to seem to think it lacks the inherent structure that many NP-complete
problems seem to have. My (nonexpert) personal belief is that it will be one
of the next problems to fall.

~~~
kirrent
Factoring is almost certainly not NP-hard. That would mean that NP is
contained in BQP which would be a staggering result. While there are some NP
problems that are in BQP the idea that somehow NP-complete problems are
solvable in polynomial time on a quantum computer seems kinda ridiculous to
me. I'd be happy to be proven wrong some time in the future though.

