**0**

votes

**0**answers

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

**0**answers

110 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

**0**answers

98 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

**4**answers

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

**0**answers

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

**0**answers

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 ...

**-2**

votes

**1**answer

97 views

### how to reduce 3-colorable graph to this? [closed]

suppose we have a finite set X and a set S of subsets of X and we want to determine is there a subset S' of S such that all members of X belong to exactly one set in S' I think the best problem to ...

**0**

votes

**0**answers

158 views

### Prove that the subset sum problem with fixed size and number reusability is NP complete

I'm trying to solve the following problem:
There are B lists of unspecified size containing integers. Pick a number from each list so that the sum of all the picks is exactly A. Prove that this ...

**1**

vote

**1**answer

143 views

### NP hard problems on UD graphs

I'm reading up on NP hard problems in Unit Disk graphs. I'd like to point out i'm fairly new to this NP hard stuff so i'm trying to get around how to prove something is NP hard.
...

**0**

votes

**1**answer

59 views

### Intersection graphs

Does anybody know of a paper which proves that finding the maximum independent set in geometric intersection graphs is NP hard? Even general intersection graphs?

**6**

votes

**0**answers

90 views

### Checking if a multiplication table represents a group [duplicate]

Given an $n \times n$ multiplication table,
can one check if it represents a group in $o(n^3)$ time?
All properties can be checked by mindless try-all possibilities loops:
Whether there is an ...

**4**

votes

**1**answer

175 views

### The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher

According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably ...

**8**

votes

**2**answers

162 views

### Polynomial-time algorithm for determining whether a polynomial is positive on $\mathbb{N}$

Does there exist a polynomial-time algorithm to determine whether a given polynomial $p(n)$ with integer coefficients is positive on $\mathbb{N}$, in the sense that $p(n) \geq 0$ for all ...

**2**

votes

**1**answer

51 views

### Length preserving rewriting system with NP-complete $u\to v$ problem

My question is related to Computational complexity of the word problem for semi-Thue systems with certain restrictions.
Is there a finite length-preserving string rewriting system $R$ (over say ...

**0**

votes

**1**answer

39 views

### Is it known whether Minimum Cost Multicut is APX-hard?

My questions is concerned with the following problem: Given an undirected graph $G = (V, E)$ and (edge costs) $c \in \mathbb{Z}^E$,
$$\min \left\{ \sum_{e \in E} c_e x_e\ \middle|\ x \in \{0,1\}^E \ ...

**3**

votes

**1**answer

62 views

### The link and equivalence between variant definition of computation model and computational complexity over reals

To unify the numerical computation and classic computability theory, or to pave a foundation for the numerical computation, mathematicians present variant computation model and computational ...

**2**

votes

**2**answers

55 views

### Deciding whether a given graph has an f-factor or not!

Given a graph $G$ with $n$ vertices and a function $f$ from $\{1,2,...,n\}$ to non-negative integers, Does there exist an efficient (for example polynomial time) algorithm, that decides whether $G$ ...

**8**

votes

**1**answer

202 views

### Can Schwartz-Zippel be formulated for commutative rings instead of fields?

The polynomials which occur in the Schwartz-Zippel lemma could be defined for any commutative ring, yet the lemma is restricted to fields. This makes it inapplicable for ...

**16**

votes

**2**answers

1k views

### “a shape that … lies halfway between a square and a circle”

An article in the
Notices of the AMS, Volume 61, Issue 10, 2014
(PDF download link),
on Khot's Unique Games Conjecture, says this:
Another group ... found a
shape that in a certain sense lies ...

**1**

vote

**1**answer

173 views

### The definition of computational complexity or complexity measure of computing reals [closed]

A real $r$ is computable if given any $i\in \mathbb{N}$, the $i$th bit can be outputed by a Turing Machine or an algorithm. So, what is computational complexity or complexity measure of computing the ...

**3**

votes

**1**answer

136 views

### Existence of subgraphs when given its degree sequence

For a given simple graph $G$ with $n$ vertices $v_1,v_2,\dots v_n$, the corresponding degree sequence is $d_1,d_2,\cdots,d_n$. My qusetion is:
How to determine whether there exist subgraphs in $G$ ...

**1**

vote

**0**answers

42 views

### Recognizing bridgeless cubic graph with special 2-factor

A 2-factor of graph $G(V, E)$ is a set of vertex-disjoint cycles that cover $V$. It is known that every connected bridgeless cubic graph contains a 2-factor (and a perfect matching).
I conjecture ...

**14**

votes

**2**answers

311 views

### Does a classification of simultaneous conjugacy classes in a product of symmetric groups exist?

Let the symmetric group $S_n$ on $n$ letters act on $S_n^d=S_n\times\cdots\times S_n$ by simultaneous conjugation, i.e. $\pi\in S_n$ acts on $(\sigma_1,\ldots,\sigma_d)\in S_n^d$ by ...

**13**

votes

**2**answers

381 views

### What is the minimal $C_k$, such that every $f\colon \{-1,1\}^n\to \mathbb{R}$ of degree at most $k$ satisfies $\|f\|_2\le C_k\|f\|_1$

Every $f\colon\{-1,1\}^n\to \mathbb{R}$ can be repsenented as a multilinean polynomial of the form $$f(x_1,x_2,\ldots ,x_n)=\sum _{S\subseteq [n]} \hat{f}(S)\prod_{i\in S} x_i $$ The degree of the ...

**2**

votes

**0**answers

53 views

### Computing basis of a lower set given basis of complementary upper set

In a poset $P$, $U\subseteq P$ is an upper set when for all $x\in U$, we have $y\ge x$ implies $y\in U$. Any subset of $P$ generates an upper set, and the basis of an upper set $U$ is the smallest ...

**1**

vote

**2**answers

145 views

### Combinatorial optimization problem involving infinite spin system

In material science research, I am developing an algorithm to solve an infinite combinatorial optimization problem which I believe is the most natural problem when the system size goes to infinity.
...

**2**

votes

**1**answer

103 views

### enumeration of connected blocks in finite size square

Given a square of size n by m, how many ways could we choose sites, such that all the sites are connected?
By "connected" we mean "connected" by adjacent sites. We will illustrate by example, say, we ...

**6**

votes

**0**answers

339 views

### Is simultaneous diophantine approximation (in a weaker sense) NP hard?

The traditional problem of simultaneous diophantine approximation is: Given a set of rational numbers $g_1,\ldots,g_d$, an integer $N$, and a rational $\gamma>0$, is there an integer $W$ with ...

**1**

vote

**0**answers

84 views

### Complexity of an algorithm to solve linear diophantine equations

A friend of mine ask me yesterday a problem, however, the interesting part for me it is not his problem, but what I will ask here.
I want to know the optimal complexity of an algorithm (I mean the ...

**5**

votes

**1**answer

114 views

### What is the (mixed strategies) equilibrium of this game?

Given a weight vector $w\in [0,1]^d$ such that $\sum w_i=1$, the game goes as follows:
Two players, $X,Y$ choose strategies $x,y\in [0,1]^d$ such that $\sum x_i = \sum y_i = 1$.
The utility (profit) ...

**0**

votes

**1**answer

72 views

### Generalized assignment problem with no integrality gap

Suppose I am solving the generalized assignment problem, so that I
am given matrices $U$ and $W$ and a vector $c$ (all three of which
have, say, positive entries), and I want to solve
...

**6**

votes

**0**answers

404 views

### Any reason to believe that $NP \neq P$ is unprovable in ZFC

We know $NP \neq P$ from a lot of point of view like empirical reason,or theoretical reasons such as finite model theory or descriptive complexity.Although we find so many reasons to believe $NP \neq ...

**8**

votes

**2**answers

199 views

### Is this problem on weighted bipartite graph solvable in polynomial time or it is NP-Complete

I encounter this problem recently and I want to know whether it is NP-Complete or solvable in polynomial time:
Given a undirected weighted bipartite graph $G = (V, E)$ where $V$ can be partitioned ...

**0**

votes

**1**answer

361 views

### Counterexample to Pólya's conjecture

It is known that Polya's conjecture is false and the smallest counter-example is about $10^9$.
Assuming that we are searching for a counter-example not knowing that it exists. What useful information ...

**1**

vote

**1**answer

86 views

### A possible minimal aperiodic set of corner Wang Tile

From one of my previous question Aperiodic set of corner Wang Tile (although it is put on hold), I realize there is a systematic way to construct aperiodic corner type of Wang tile from edge type ...

**5**

votes

**1**answer

137 views

### Aperiodic set of corner Wang Tile [closed]

There is quite some reference on aperiodicity of the edge-type of Wang Tile. But I could not yet find aperiodic corner type of Wang Tiles... Could someone provide me some instances (better with ...

**7**

votes

**2**answers

904 views

### Why are there so few zero-dimensional polynomial system solvers and is this because there is no real market for them?

My questions involve the quotes below from wikipedia regarding solving polynomial systems, which given the size of the market for Big Data & Predictive Analysis applications I find puzzling:
...

**2**

votes

**3**answers

222 views

### How to define the input of computable function or Turing machine over real numbers

Computation or computability over $\mathbb{N}$ can be extended to computation or computability over $\mathbb{R}$ or even computation or computability over $\mathbb{C}$.The following is a formal ...

**2**

votes

**0**answers

104 views

### Existence of roots of high order polynomial over finite fields

I want to solve the following question:
Consider a polynomial $f(x)=a_0+a_1*x^{e_1}+a_2*x^{e_2}+\cdots+x^{e_m}\in F_p[x]$ where $p$ is a prime such that $\log(p)\sim m$ and $e_m\sim 2^m$, I want to ...

**0**

votes

**1**answer

132 views

### Function that dominates everything in little o

I have a function $f(n)$ that satisfies the following property: for any function $g(n) = o(n^{-2})$, we have $f(n) = \Omega(g(n))$ (the implied proportionality constant in the $\Omega$ expression ...

**2**

votes

**0**answers

222 views

### NP-hard proof of optimization version of exact cover [closed]

Exact cover is NPC.
http://en.wikipedia.org/wiki/Exact_cover#Equivalent_problems
Given a collection $\mathcal{S}$ of subsets of a set $X$, an exact cover is a >>subcollection $\mathcal{S}^*$ ...

**8**

votes

**1**answer

231 views

### Compute an arbitrary decimal place of $\pi$

Is there a method to find the value of the $n$-th decimal place of $\pi$ which is more efficient than having to compute all decimal places before as well?

**2**

votes

**0**answers

161 views

### Both NP-hard but different [closed]

What's the fundamental difference between the Knapsack problem and the travelling salesman (TSP) problem both of which are NP-hard, while the reality is that TSP could be solved much much faster?

**2**

votes

**0**answers

205 views

### Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate
polynomial which counts the number of colorings of $G$
with $x$ colors for natural $x$.
Graph is not $k$ colorable iff $P(G,k)=0$.
The ...

**2**

votes

**1**answer

253 views

### How hard is a variant of graph automorphism problem?

I'm interested in a variant of graph automorphism problem (which is prime candidate for $NP$-Intermediate problem).
Restricted GA
Input: Given an undirected graph $G(E, V)$, and $\epsilon |V|/2$ ...

**0**

votes

**0**answers

53 views

### Complexity of graph isomorphism in $(P_4 \cup K_1,\overline{3K_2})$-free graphs

Related to this question where isomorphism preserving
transformation maps triangle-free graphs to $(P_4 \cup K_1,\overline{3K_2})$-free graphs.
What is the complexity of graph isomorphism in $(P_4 ...

**1**

vote

**1**answer

96 views

### A certain instance of the Set Covering problem

Is there any useful structure associated with the following instance of the Set Covering problem?
Let $G$ be a weighted graph and let $\mathcal{P}$ denote the set of all shortest paths between all ...

**1**

vote

**1**answer

64 views

### Implications of the impossibility of efficient sampling from random non-Hamiltonian graphs

Nisan's answer to this question shows the Impossibility of efficient sampling from random non-Hamiltonian graphs (unless $NP=coNP$). I am interested in the implications of this conjecture.
Does ...

**5**

votes

**2**answers

217 views

### Generating Hard Instances

Assume NP$\neq$P and let $L$ be an NP-complete language. Is there a polynomial time computable function $f:\{0\}^*\longrightarrow\{0,1\}^*$ with $|f(0^n)|=n$ for every $n$; such that L $=\{0^n: ...

**-3**

votes

**1**answer

186 views

### are all NP problems made up of P problems? [closed]

are all NP problems made up of P problems? that is, can NP problems be thought of as an accumulation of P problems? or can NP problems be divided up into a series of P problems?