# Tagged Questions

P versus NP and other resource-bounded computation.

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### How is the VP=VNP question in char 2 different from other char? What is the current frontier in regards to this question?

What are the caveats one should be aware of when pursuing VP=VNP question in char 2 compared to other char? What is the current frontier in regards to this question?
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### The number of edges in the ith shortest path in a directed graph

$G$ - directed graph, $n$ - count of nodes According to Eppstein's Algorithm in this paper, the ith shortest path in a digraph may have $\Omega(ni)$ edges. Anybody can explain how this estimate is ...
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### Do we take these vertices? [on hold]

I am looking at an exercise about the vertex cover. We are given the undirected graph $G=(V,E)$ with $V=[10]$ and $E=\{(i, i+1)\mid i=1, \dots , 9\}$. Before I use the approximation algorithm, I ...
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### Is sparse embedding of a NP-complete problem in a polynomial problem NP-complete?

Consider the following problem P: Input is a finite graph G. If the number of vertices in G is 2^2^i for some integer i, then output a minimum vertex cover of G; otherwise output empty set. Can I say ...
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### Is there a linear space lower bound for streaming set equality?

Consider two streams. In each stream one string arrives at a time. A query asks: Is the set of strings that has arrived so far the same in both streams? Is there a linear space randomized lower ...
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### Computing Minima of the Projection of a Binary Cube

The problem is as follows: I want to compute the minima (with respect to the canonical partial order on vectors "$\leq$") of the linear projection of the extreme points of an $n$-dimensional $\{0,1\}$-...
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### Why is it impossible to work with polylog length encoding schemes for quantum circuits?

I am going through Quantum Computational Complexity by John Watrous. On page $12$, he said: The encoding disallows compression: it is not possible to work with encoding schemes that allow for ...
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### Is the infinitely-often version of Ladner's theorem known?

We say two languages $\;\;\; L\hspace{.02 in},\hspace{-0.02 in}L' \: \subseteq \: \{\hspace{-0.02 in}0,\hspace{-0.05 in}1\hspace{-0.03 in}\}^* \;\;\;$ agree infinitely-often with each other if and ...
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### Explicit Computational Complexity of the Shortest Weight Constrained Path Problem?

The Shortest Weight Constrained Path Problem is a known NP-Complete Problem (listed NPC in Garey and Johnson - ND30]. Thus, by definition the running time of the Problem is exponential in the worst ...
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### Confusing running time analysis for the Divide & Conquer algorithm of Hamiltonian Path problem

In the Hamiltonian Path problem we are given a graph $G=(V,E)$ and two distinct vertices $\{s,t\}$ and we ask if there is a path from $s$ to $t$ which traverses all other vertices exactly once. ...
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### Computational depth and p-time hard instances

After reading the nice results of the paper: "Worst-Case Running Times for Average-Case Algorithms" by Antunes and Fortnow, I was wondering about the existence of further results linking basic ...
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### $\text{P}^{\text{Mod}_k\text{P}}$ vs $\text{P}^{\#\text{P}}$?

$\text{Mod}_k\text{P}$ is the class of decision problems solvable by an NP machine such that the number of accepting paths is divisible by k, if and only if the answer is "no." For constant $k$, one ...
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### Complexity of testing if a hypergraph has generalized hypertreewidth $2$

A hypergraph $H = (V, E)$ consists of a set of vertices $V$ and a set $E$ of hyperedges, i.e., subsets of $V$. The generalized hypertreewidth (GHW) parameter is a measure of the cyclicity of a ...
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### States and Probability distributions that the 5-qubits IBM computer can produce

IBM has recently built a 5-qubits quantum computers based on superconducting qubits. It is even possible to make experiments over the cloud. The space of pure states for 5 qubits is the unit sphere ...
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### Combinatorial characterization of hypergraph Tseitin satisfiability

The Tseitin formulas are as follows: Given a connected graph and a function $\alpha: V \rightarrow \{0,1\}$. Associate each edge $e$ with a variable $x_e$. The Tseitin formula $G(\alpha)$ is defined ...
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### Counting points on curves

It is known (see "Counting curves and their projections" (free version) by von zur Gathen, Karpinski, and Shparlinski) that the problem of finding the number of $\mathbb{F}_q$-rational points on a ...
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### Testing Isomorphism of projective planes

Miller showed that isomorphism testing of projective planes can be done in $v^{O(\log \log v)}$. I would like to know whether Babai's techniques that led to the quasipolynomial time algorithm for GI ...
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### Complexity of $r$-sum as a function of integer sizes

Given $n$ distinct integers whose absolute value is of size $c_{n,k}=\lceil n^{1/k}\rceil$ bits ($c_{n,k}$th bit position is always $1$ for absolute value) we know that using dynamic programming we ...
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### For which regular expressions $\alpha$ is $\{ \beta \mid L(\alpha) = L(\beta) \}$ PSPACE-complete?

It is well known that the following problem is PSPACE-complete: Given regular expression $\beta$, does $L(\beta) = \Sigma^*$? What about determining equivalence to other (fixed) regular ...
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### Is a quadratic nondeterminism speed-up of deterministic computation plausible?

This is a follow up to nondeterministic speed-up of deterministic computation. Is it plausible that nondeterminism (or more generally alternation) would allow a general quadratic speed-up of ...
I have found that the problem of finding of dimension of algebraic varieties over $\mathbb{C}$ is $NP$-complete (https://pdfs.semanticscholar.org/a947/463a29ee512b89823176f6e8c9f9b2bb1a5e.pdf). Are ...