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Unanswered Questions

43
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 $...
40
votes
0answers
813 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 ...
29
votes
0answers
625 views

Eilenberg's rational hierarchy of nonrational automata & languages — where is it now?

In the preface to his very influential books Automata, Languages and Machines (Volumes A, B), Samuel Eilenberg tantalizingly promised Volumes C and D dealing with "a hierarchy (called the rational ...
29
votes
0answers
845 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
594 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. ...
27
votes
0answers
598 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
416 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
3k 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 $\{0,1,\ldots,n+1\}$. (...
26
votes
0answers
401 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 ...
25
votes
0answers
613 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 ...
24
votes
0answers
686 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 ...
23
votes
0answers
908 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 ...
22
votes
0answers
642 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 ...
22
votes
0answers
509 views

Regularity Lemma for Sparse Graphs

Szemeredi's Regularity Lemma says that every dense graph can be approximated as a union of $O(1)$ many bipartite expander graphs. More accurately, there's a partition of most vertices into $O(1)$ sets ...
21
votes
0answers
726 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 ...

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