computational complexity theory; complexity classes, such as P, NP, PSPACE, and so on; resource-limited computation; NP-completeness and other completeness concepts; oracle analogues of complexity classes; complexity-theoretic computational models such as automata, circuits; regular languages; ...

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9
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1answer
264 views

Can Shor's Algorithm be modified to run efficiently on a classical computer?

Shor's algorithm is an algorithm which factors integers in polynomial time on a quantum computer. If one tries to run it on a classical computer, one runs into the problem that the state vector that ...
0
votes
0answers
27 views

complexities aggregation [on hold]

I will ask a question, but I hope that it is not a stupid one, I have an algorithm, and I want to calculate its complexity. in fact it has two parts, the complexity of the first part is O(p) ...
3
votes
0answers
82 views

What is known about the complexity of this covering problem?

Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex ...
1
vote
0answers
48 views

FPTAS for approximating the permanent of a matrix

My question concerns approximating permanent of an $n$-by-$n$ matrix. Several approximation algorithms have been proposed in the literature for this purpose, whose time complexity depend on $n$ and ...
1
vote
2answers
154 views

Algorithm for fast factorization of polynomial over $\mathbb Z$ or over $\mathbb F_p$

I want to fast decompose polynomial over ring of integers (original polynomial has integer coefficients and all of factors have integer coefficients) and also over ring of integers modulo prime ...
1
vote
0answers
52 views

What's the complexity of the one sink directed subgraph isomorphism problem?

I am considering trying a new approach for the subgraph isomorphism problem in my PhD, but it just seems to work well for digraphs of one sink. By working well I mean some promise of not having to ...
0
votes
0answers
63 views

About expectation norms on graphs

Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= ...
0
votes
0answers
80 views

What is wrong with the argument that zero permanent is polynomial?

This Lecture summarizes some well known facts about $\#P$ completeness of permanent. Given a CNF formula $\phi$ on $n$ variables, they construct matrix $A$ such that: $$perm(A)=4^{3m} \#SAT(\phi)$$ ...
3
votes
1answer
84 views

What is the Complexity Class of the “Function Variant” of the Integer Factorization Problem?

I've been reading up a lot Prime Factorization and it's complexity, including a fair number of questions on this very site. However, I still feel there is a question still left unanswered. So, ...
2
votes
1answer
48 views

Polynomial degree comparison of Nullstellensatz and Positivstellensatz over real algebraic sets

Suppose we have a (finite) system of polynomials $P = \{ p_i \} \subseteq \mathbb{R}[x_1, \ldots, x_n]$. Then it is well known by the Nullstellensatz that either $P$ has a simultaneous zero over ...
1
vote
1answer
62 views

Connection between Barnette conjecture and hardness of cubic graph decomposition

Motivated by this post on cubic graphs decompositions and the connection to Barnette conjecture, I am interested in decomposing a connected bridgeless cubic graph into edge-disjoint paths of length 3 ...
0
votes
1answer
88 views

A particular argument in the review on expanders by Hoory-Linial-Wigderson

I am thinking about the third bullet point on page 455 here, http://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01126-8/ Can someone explain what is the argument there which seems to conclude ...
5
votes
1answer
289 views

When are (Abelian) Cayley graphs also expanders?

I want to ask the question in two parts, (1) Is there some fundamental distinguishing property between Abelian and non-Abelian Cayley graphs? (say some specific proof technique which distinguishes ...
2
votes
0answers
102 views

About the small set expansion conjecture

Given a graph $G=(V,E)$ and a $\delta > 0$ one wants to calculate $h(G,\delta)=min_{\vert S\vert \leq \delta \vert V \vert } \phi(S)$. ($\phi(S) = \frac{ E(S,\bar{S}) }{d min \{\vert S \vert , n - ...
6
votes
0answers
80 views

Is there a ``Ladner's Theorem" for the PH-vs-PSPACE scenario?

Like a statement of the kind, ``If the Polynomial Hierarchy (PH) $\neq$ PSPACE then there exists $L \in PSPACE \backslash PH$ which is not PSPACE-complete"? Or is there something else that states ...
11
votes
1answer
319 views

Is factorial definable using a $\Delta_0$ formula?

The factorial function is primitive recursive, and therefore definable by a $\Sigma_1$ formula. Is it also definable by a $\Delta_0$ formula (i.e. bounded quantifiers)? If not, why?
2
votes
1answer
202 views

Why a tensor product of $2\times 2$ unitaries cannot implement a $3\times 3$ unitary?

Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows: $$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
7
votes
0answers
118 views

Ricocheting pinball-like shot: Complexity?

Suppose one has $n$ perfect two-sided mirror segments in the plane $\mathbb{R}^2$. The segments are open, excluding their endpoints. They are disjoint as closed segments, i.e., no pair shares an ...
4
votes
1answer
194 views

Explicit bounds for transfer results in algebraic geometry

Assume you have an ideal $I\subseteq\mathbb{Z}[X_1,\ldots,X_n]$ of the polynomial ring in $n$ variables over the integers. For any field $\Bbbk$, I can consider the ideal ...
3
votes
0answers
61 views

Are all $k$th-longest-tour problems equally hard?

It is well known, that determining the shortest and, the longest Hamilton Cycle of a complete graph with real edge weights are algorithmically two sides of the same medal: one transforms to the other ...
0
votes
0answers
60 views

Confusion about reduction counting vertex covers to counting cycle covers

Cross-posted from cstheory This confuses me. One easy case of counting is when the decision problem is in $P$ and there are no solutions. A lecture show that the problem of counting the number of ...
6
votes
2answers
183 views

Geometric dominating set: NP-complete?

Let $G=(V,E)$ be a geometric graph, a graph embedded in the plane whose edge lengths are the Euclidean distance between its endpoint vertices. Say that a set of vertices $D \subseteq V$ is a geometric ...
10
votes
0answers
710 views

Primes and Parity

This problem is motivated by the polymath4 project. There, the aim was to find an efficient deterministic algorithm for finding a prime larger than $N$. The hope was to find a polynomial algorithm in ...
4
votes
1answer
156 views

Tiling with restricted overlap

Non-overlapping tilings of regions is a well-studied topic. I wonder if the following variant has been considered: A tile can be partitioned into several regions, where such regions from different ...
7
votes
2answers
218 views

Recent trends in effective analysis

The references listed at http://en.wikipedia.org/wiki/Computable_analysis have all been published 30-15 years ago. Are the approaches which these references expose still up-to-date and relevant to the ...
11
votes
5answers
1k views

Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from $1$

Found this on Complexity Zoo warning expired certificate check NP Over The Complex Numbers. [BCS+97] show the following striking result. For a positive integer $n$, let $t(n)$ denote the minimum ...
1
vote
1answer
103 views

Graph classes where finding explicit coloring have certificate that it is minumum

Graph coloring doesn't have certificate that smaller coloring doesn't exist in general. I am looking for graph classes where finding explicit coloring is not polynomial and have polynomially ...
2
votes
1answer
131 views

NP-hardness of finding maximum of minimum element in diagonal of a matrix

For $A = \{a_{ij}\} \in R^{n\times n}$, is finding $$ \max_{\sigma \in S_n}\min_{1 \le i \le n} a_{i,\ \sigma_i} $$ NP-hard?
0
votes
0answers
54 views

Surd Partition Problem

Could the following "Surd Partition" problem be NP complete? Note that if the square roots are omitted in the following then the problem is well known to have a polynomial solution. Surd Partition ...
1
vote
0answers
105 views

Determining strong base-orderability of a matroid

A matroid is said to be strongly base-orderable when for any two bases $B_1,B_2$ there exists a bijection $f:B_1 \mapsto B_2$ such that for any $X\subseteq B_1$ set $B_1 - X+ f(X)$ is also a base. ...
2
votes
0answers
110 views

Number of degree $k$ functions [closed]

Given a Boolean function $f:\{0,1\}^n\rightarrow\{0,1\}$, there is a real multivariate multilinear polynomial that is associated with in through interpolation. Example: ...
4
votes
2answers
186 views

Complexity of finding the maximum sum divided by product

What is the complexity of the following optimization problem? Problem. Given $n$ pairs of positive reals $(a_i,b_i)_{i=1}^n$, choose a subset $S \subseteq [n]$ to maximize $$ \frac{\sum_{i\in S} ...
2
votes
1answer
58 views

What is Known about Preprocessing for Stabbing Queries?

In a concrete setting, I have the following problem: given a fixed set of simple objects (e.g. disks or, convex polygons with few vertices), I need to quickly report the objects that are hit (i.e. ...
7
votes
0answers
141 views

Is there an infinite increasing sequence of naturals for which Landau's function can be efficiently computed?

Landau's function $g(n)$ is the largest order of an element of the symmetric group $S_n$. Equivalently, $g(n)$ is the largest least common multiple (lcm) of any partition of $n$. In general $g(n)$ is ...
2
votes
1answer
301 views

Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?

Not sure whether this question belongs here or math.stackexchange. You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be ...
7
votes
0answers
293 views

Evidence that Graph Isomorphism problem is not $NP$-complete

Graph isomorphism problem is one of the longest standing problems that resisted classification into $P$ or $NP$-complete problems. We have evidences that it can not be $NP$-complete. Firstly, Graph ...
3
votes
0answers
110 views

On the computational complexity of the Hilbert polynomial of numerical semigroup rings

Let $(R, \mathfrak{m}) = k[[X^a, X^b, X^c]]$, $a<b<c$, $gcd(a, b, c) = 1$, be a semigroup ring. We have $R$ is a Cohen-Macaulay local ring of dimension one. It is well known that ...
3
votes
1answer
99 views

Sorting interleaved sorted lists

By interleaving two lists I mean to combine them into a single list in any way that maintains the relative order of the elements coming from each list. For example, interleaving $(x_1,x_2,x_3)$ and ...
5
votes
1answer
162 views

Finding sparsest solution of a linear system

I want to find the solution with most zero-components for the following problem: $Ax=b$ for $A\in \mathbb{R}^{k\times n}, b \in \mathbb{R}^{k},k<n$, where $x$ is real and has no additional ...
2
votes
0answers
56 views

Recognizing sequences sortable by transpositions?

While reading the post, Probability of generating a desired permutation by random swaps, by Aaronson, and to continue my program I started in this post, How hard is reconstructing a permutation from ...
1
vote
0answers
56 views

Reducing enumeration of reduced words in general Coxeter groups to another #P-complete problem

There is a polynomial time algorithm that takes as inputs a Coxeter system $(W,S)$ with $S$ finite (but with $W$ not necessarily finite), say encoded as a Coxeter matrix, as well as two reduced words ...
-2
votes
2answers
175 views

Time estimate to determine if a number is prime [closed]

How long does it take to verify that a given number is a prime number, as a function of its number of digits, in a personal computer, say? How computationally hard is this?
12
votes
1answer
532 views

Is there a known primitive recursive upper bound on the nth “Zhang prime”

(This question is pure curiosity. Feel free to close it if you feel it is not appropriate for mathoverflow.) In 2013 Zhang showed that there are infinitely many pairs of primes which are less that ...
3
votes
1answer
80 views

What is Known About the Complexity of Calculating Minimal Surface Polyhedra?

I am currently ruminating about ways of generalizing Minimum Spanning Trees to Minimum Spanning "Hypertrees", where the cost is associated with simplex volumes and, where certain topological ...
0
votes
0answers
39 views

Is there any SMT solver which allows a call to another SMT solver?

I have a problem to solve which is likely to be $\mathbf{NP}^{\mathbf{NP}}$-hard. That is, I could solve it if I could take advantage of an instruction such as "assert UNSAT using ...
1
vote
0answers
106 views

NP hard problems on geometric graphs

I have posted this question before but i don't feel i expressed my confusion clearly enough. So i would like to try and explain again. This is a proof of the minimum vertex cover for unit disk graphs ...
2
votes
0answers
96 views

counting irreducible factors

In How hard is it to compute the number of prime factors of a given integer? a question was asked on computing number of prime factors of an integer. Suppose we have a polynomial $f(X)\in \Bbb Z[X]$ ...
28
votes
4answers
3k views

Massive cancellations

Let $A=\{a_1,\ldots,a_k\}$ be a fixed, finite set of reals. Let $S_A(n)$ be the set of all reals that are expressible as the sum of at most $2^n$ terms, where each term is a product of at most $n$ ...
2
votes
0answers
60 views

relationship of max-sat and min-cut in theory and practice

I have been using MAX-SAT solver to obtain the exact ground state of ising spin glass model: For 1D periodic model, for systems with 50 binary variables and interaction range of 15th nearest ...
2
votes
0answers
69 views

What is the computational complexity to compute the integral numerically?

Given $$\int_{\Delta}\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}$$ where $P_i$ is polynomial(that is $P_1(x_1,x_2,\dots,x_n), P_2(x_1,x_2,\dots,x_n)$ are polynomial) whose coefficients are ...