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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2
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63 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 $f(X)\in \Bbb Z[X]$ ...
18
votes
4answers
2k 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
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0answers
41 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
58 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
1answer
77 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
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0answers
37 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
1answer
105 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
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1answer
54 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
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0answers
89 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
1answer
124 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
2answers
153 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
1answer
47 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
1answer
31 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
1answer
57 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
2answers
49 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 $G$ ...
7
votes
1answer
142 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 ...
14
votes
2answers
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
1answer
169 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
1answer
97 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
0answers
36 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
2answers
296 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
2answers
367 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
0answers
48 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
2answers
125 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
1answer
101 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
0answers
249 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 ...
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vote
0answers
43 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
1answer
105 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
1answer
46 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
0answers
354 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
2answers
183 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 ...
1
vote
0answers
93 views

reference on aperiodicity and cluster [closed]

From this image: I believe there is a message relating those clusters drawn in picture and aperiodic tiling. Does anyone have some reference on this? Thank you :)
0
votes
1answer
351 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
1answer
57 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
1answer
118 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
2answers
860 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
3answers
186 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
0answers
86 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
1answer
130 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
0answers
142 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 $\mathcal{S}^*$ ...
7
votes
1answer
225 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
0answers
150 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
0answers
173 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
1answer
230 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
0answers
44 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
1answer
85 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
1answer
61 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
2answers
205 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
1answer
185 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?
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0answers
53 views

Polynomial problems in graph classes defined by forbidden induced cyclic subgraphs

Let $C$ be a graph class defined by a finite number of forbidden induced subgraphs, all of which are cyclic (contain at least one cycle). Are there graph problems that can be solved in ...