# All Questions

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### 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". ...
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### 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 ...
558 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
971 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 ...
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### Fine-grained complexity of BPP

If E does not have i.o.-$2^{o(n)}$ circuits, then P=BPP, but this does not tell us about the fine-grained containments between $\mathrm{Time}(n^a)$ and $\mathrm{BPTime}(n^b)$. Are there reasonable ...
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### 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 ...
681 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 ...
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### 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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### Why is the Pumping Lemma sometimes called Bar-Hillel's Lemma?

There are several papers in the literature that refer to the Pumping Lemma for context free languages as Bar-Hillel's Lemma (for example, here, here, and on the Wikipedia page). However, the first ...
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### Exact algorithm for NAE-3SAT

The NAE-3SAT problem is to determine whether a given 3CNF formula has a satisfying assignment that gives each clause at least one false (and at least one true) literal. The problem is NP-complete. One ...
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### Descriptive complexity characterization of TimeSpace classes

Are there descriptive complexity characterizations for TimeSpace complexity classes like $\mathsf{SC^i}= \mathsf{DTimeSpace}(n^{O(1)},O(\lg^i n))$?
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### Weighted Hamming distance

Basically my question is, what kind of geometry do we get if we use a "weighted" Hamming distance. This is not necessarily Theoretical Computer Science but I think similar things come up ...
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### Are monotone Boolean functions in P well-approximated by monotone polynomial-size circuits?

Question 1: Is it true that for every polynomial $p(n)$ and $\epsilon >0$ there is a polynomial $q(n)$ such that every monotone Boolean function on $n$ variables that can be expressed by a Boolean ...
427 views

### Courcelle's theorem for bounded clique-width graphs

Courcelle's theorem states that "Every graph property which is expressible in monadic second order logic is decidable in linear time for bounded tree-width graphs". Later it was extended to bounded ...
526 views

### Complexity of the densest $k$-subgraph problem on planar graphs

In the densest $k$-subgraph problem, one is given an undirected graph $G$ and wants to find a set of vertices $N$ with $|N| = k$ such that the number of edges in the subgraph of $G$ induced by $N$ is ...
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### Descriptive complexity of communication complexity classes

It is well known that some major complexity classes, like P or NP, admit a full logical characterization (e.g NP = existential 2nd order logic by Fagin's theorem). On the other hand, one can also ...
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### Complexity of interval cover problem

Consider the following problem $Q$: We are given an integer $n$, and $k$ intervals $[l_i,r_i]$ with $1\leq l_i\leq r_i\leq 2n$. We are also given $2n$ integers $d_1,…,d_{2n}\geq 0$. The task is to ...
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### Did von Neumann answer to Gödel's letter?

On 20 March 1956, Kurt Gödel wrote a famous letter to John von Neumann, in which he formulated the P versus NP question. Here is a link to that letter: [pdf of letter] I cant seem to find John von ...
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### Intermediate problems between PSPACE and EXPTIME

Intermediate problems between P and NP are quite famous, and are sometimes considered as complexity classes by themselves. Do you know of any problem that is known to be PSPACE-hard and in EXPTIME, ...
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### Does Factoring have a Statistical Zero Knowledge Proof?

The title should be pretty self-explanatory, but to be more precise, consider the decision version of factoring, which is given input $(x,k)$, where $x$ and $k$ are binary encodings of integers, to ...
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### Quantum Hardness of Finding Nash Equilibria

This question is inspired by the recent, beautiful work On the Cryptographic Hardness of Finding a Nash Equilibrium by Bitansky, Paneth, and Rosen. Their main result is that the existence of ...
459 views

### Can short-distance connectivity be harder than connectivity?

Has anybody seen the following (or similar) question being considered: Can it be easier to determine the presence/absence of $s$-$t$ paths than to determine the presence/absence of short $s$-$t$ ...
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### Practically Good Algorithms of a Very Low Computational Complexity Class

I am looking for one (or more) examples of a parametric class of algorithms $P_t$ for approximately solving a class $\cal A$ of algorithmic questions with the following properties: 1) Solving the ...
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### What if an $\mathsf L$-complete problem has $\mathsf{NC}^1$ circuits? More generally, what evidence is there against $\mathsf{NC}^1=\mathsf{L}$?

Edit: let me reformulate the question in a more specific way (and change the title accordingly). A slightly edited version of the original question follows. Is there a result comparable to the Karp-...
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### Lower bounds on single-source shortest paths in directed graphs

Are there any non-trivial lower bounds on the complexity of single-source shortest paths (SSSP) in a directed graph, where all edges have non-negative edge weights? Can we rule out the possibility of ...
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### Tiling a rectangle with the fewest squares

Consider this problem: Find a tiling of an $m \times n$ rectangle by minimum number of integer-sided squares. Is there any polynomial time (in $m$ and $n$) algorithm to do this? What is the best ...
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### Problem-Dependent Derandomization

The famous result of Impagliazzo and Wigderson in '97 cemented our belief that BPP is most likely the same as P; that is, problems that can be efficiently solved with randomness can also be ...
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