Questions tagged [polynomial-time]

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Are classes of graphs represented by adjacency matrix ordered structures?

We know that FO[LFP] captures PTIME on the class of ordered structures. However, I have difficulties interpreting this result. From what I understand, it means that, given a constant, finite alphabet $...
4 votes
0 answers
72 views

distinguishments between query complexity of membership oracles and standard time complexity

Many combinatorial optimization problems can be described as follows. We are given a set system $(E,I)$, where $I \subseteq 2^E$ and a weight function $w: E \rightarrow \mathbb{N}$. The goal is to ...
1 vote
0 answers
53 views

Find linear combination with small support

Let $v_1,\dots,v_n$ be a basis of a vector subspace of $\Bbbk^N$, say for $\Bbbk$ a finite field. I would like an algorithm to find a linear combination of the $v_i$'s with small support, i.e. with ...
2 votes
0 answers
69 views

Resource bounded Kolmogorov complexity hardness on average over a non uniform distribution of inputs

$K^{poly}$, as well as other related problems such as $MCSP$, is believed to be hard on average [1, 2] when the input is sampled from a uniform distribution (since otherwise one way functions, pseudo-...
6 votes
1 answer
244 views

Fast algorithms for time bounded Kolmogorov complexity

For a universal Turing machine $U$, the time bounded Kolmogorov complexity of a string $x$ is silmilar to the usual Kolmogorov complexity but limited to programs $p$ running in time at most $t(|x|)$: $...
7 votes
1 answer
209 views

Is Circuit Minimization $P$-hard under logspace reductions?

By Circuit Minimization, I am referring to the following decision problem. Circuit Minimization Input: A bit string $x$ and a number $k$. Question: Does there exist a Boolean Circuit $C$...
6 votes
2 answers
331 views

What are the consequences of $BPP \neq P$?

I have seen a lot of people assume, $BPP = P$ . But to me, this seems false intuitively.(Though math is not without unintuitive results) And, to my admittedly limited understanding of the topic, the ...
13 votes
1 answer
260 views

Complexity of 1-or-3-in-3-SAT (odd-3-SAT)

Consider a 3-CNF formula $\Phi$, i.e., a conjunction of clauses of 3 literals. I call odd-SAT (or 1-or-3-in-3-SAT) the problem of checking whether there is an assignment of the variables such that ...
-2 votes
1 answer
116 views

Polynomial vs. Exponential Time Complexity [closed]

Does $2^{log_2{n}}$ grow faster than a polynomial? I know that $2^{log_2{n}}$ can be simplified as $n$ but can it be considered as an exponential?
3 votes
1 answer
103 views

Running time analysis of problems with a variable in problem definition

I am a research scholar in the field of algorithms and complexity theory. The problem that I am currently working is the $[1,j]$-domination problem. Given a graph $G = (V, E)$, $n = |V|$, the problem ...
2 votes
0 answers
105 views

Prize Box Ordering Problem With Position Constraints: Easy or Hard

I have a problem where we have $n$ boxes, each box $i$ have probability $p_i$ of containing a prize with value $h_i > 0$, and remaining probability of containing nothing. Now we are asked to order ...
1 vote
2 answers
101 views

Is Maximum Independent Set polynomial-time solvable in $(p,1)$-colorable graphs for general $p$?

It is well-known that Maximum Independent Set (MIS) in bipartite graphs is polynomial-time solvable. What happens if we generalize the input graphs by replacing the vertices in one partite with ...
2 votes
1 answer
242 views

6-regular graph without small 3-regular subgraph

My name is Balchandar Reddy. I am a research scholar and am currently working on graph algorithms. I am looking to find a 6-regular graph that does not have small 3-regular subgraphs. For example, I ...
2 votes
0 answers
118 views

Computing real numbers with Turing Machines

Consider the following decision problem: Given a two integers $n$ and $k$, decide whether $k=\lfloor n\pi\rfloor$ Question: Is this problem known to be in $P$? Although this may look like a stupid ...
4 votes
1 answer
172 views

Computation with cellular automata in practice

It is well known that certain cellular automata (CA) are computationally universal, such as Conway's game of life in 2 dimensions or the rule 110 in 1 dimension. As far as I know, they can emulate ...
17 votes
3 answers
2k views

Does P contain languages whose existence is independent of PA or ZFC? (TCS community wiki)

Answer: not known. The questions asked are natural, open, and apparently difficult; the question now is a community wiki. Overview The question seeks to divide languages belonging to the complexity ...
4 votes
1 answer
185 views

Finding a "typical" path

Consider an undirected graph with two distinguished nodes $u\neq v$. How hard is it to find an $u-v$ path, such that its length is as close to the average $u-v$ path length as possible? Formally, for ...
-2 votes
1 answer
618 views

Solving 3-SAT in O(n^6)?

There's an algorithm (published on GitHub) which is claimed to solve any 3-SAT formulation in polynomial time with a complexity of max O(n^6). I would usually brush claims like this away, but having ...
2 votes
0 answers
91 views

Is there a natural problem in P but no polynomial time algorithm is known?

Most of the time, a problem is shown to be in P by the construction of a polynomial time algorithm. I wonder if there is a natural problem that was proved in P but no explicit algorithm was known? I'm ...
0 votes
1 answer
215 views

list of 3-CNF formula that can be solved in polynomial time

Suppose i want to program a 3-SAT solver. I want my solver to first check whether a formula is in the list of 3-CNF that currently known can be solved in polynomial time before resorting to brute ...
0 votes
1 answer
114 views

Can one find any solution to this matrix problem in polynomial time?

I am given an M * N (M > 1, N > 1) matrix with all the numbers blackened but their row and column sums are visible. For example, I am given this 3 * 3 matrix. And one of the possible matrix ...
10 votes
3 answers
1k views

Choosing random permutations in "strict" polynomial time

This question compares "strict" polynomial time, as opposed to "expected" polynomial time. Let $S = \{1,2,…,n\}$, and let $O$ be an ordering on elements of $S$ (the number of ...
10 votes
5 answers
440 views

Problems in quasi-polynomial time that could conceivably be in P (without causing collapses or violating widely held beliefs)

What are some examples of problems with quasi-polynomial time ($QP$) algorithms that could conceivably be in $P$. In other words, they are in $QP$ for no obvious reason other than no one has found a ...
3 votes
1 answer
169 views

Efficient transformation into CNF preserving entailment

Suppose you have two propositional formulas $\varphi$ and $\psi$, not necessarily in CNF. I want to convert them to 3CNF efficiently (hence introducing auxiliary variables) in such a way that $\varphi ...
5 votes
0 answers
107 views

Joint Scheduling Problem with Variables Arrival Times

The scheduling problem $1\bigl|r_j\bigl|\Sigma\,U_j$ is strongly NP-hard but when preemption is allowed the scheduling problem $1\bigl|pmtn,r_j\bigl|\Sigma\,U_j$ becomes strongly polynomial. I have a ...
2 votes
0 answers
69 views

Is there a poly-time algorithm to compute the drawing of a simple graph (need not be planar) in a 2D-plane such that any two edges cross at most once?

Does there exists a ploynomial time algorithm to embed a simple graph(need not be planar) in a plane satisfying the following conditions? No edge touches vertices other than its end vertices. At any ...
-1 votes
1 answer
66 views

Interpolation to find polynomial multivariate derivative

This question came when reading a paper here about affine projections of polynomials. The publisher claims in Proposition 22 that Let $f(\mathbf{x}) \in F[\mathbf{x}]$ be an $n$-variate polynomial of ...
1 vote
0 answers
45 views

How to deal with the time to minimize a function in a given interval?

I'm writing a paper in which I designed an algorithm running in $O(n^2m)\cdot T(f)$ to solve my problem, where $n,m$ is the size of input and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a function, and $T(...
1 vote
1 answer
167 views

3-SAT runtime if an optimal order to eliminate possible solutions is known

As a mental exercise I have been playing around with the 3-SAT problem, but I am having difficulty proving or disproving the usefulness of a current idea that I am playing around with. My current ...
1 vote
1 answer
192 views

Young Diagrams and distinguishing between two distributions

Introduction: The reference for everything is this paper. The Robinson–Schensted–Knuth (RSK) algorithm is a well-known combinatorial algorithm with diverse applications throughout mathematics, ...
1 vote
0 answers
164 views

Is polynomial-time the same in all classical computational models?

There are many models of computability, all giving the same notion of 'computable function'. To pick a few examples: Turing machines (with variants: one-ended, two-ended, multiple tapes...) RAM ...
16 votes
0 answers
498 views

a geometric variant of k-medians. NP-hard or in P?

The following problem is a special case of k-medians. Is it NP-hard? Is it in P? Input: $n$ points $(x_1,y_1), (x_2,y_2), \ldots, (x_n, y_n)$ with each $y_i \ge 0$, and an integer $k$. Output: a set ...
2 votes
0 answers
213 views

Simultaneous evidence for $L\neq NL$ and $P\neq NP$

We believe $L\neq NL$ and $P\neq NP$. Is there any evidence which simultaneously imply $L\neq NL$ and $P\neq NP$?
4 votes
1 answer
325 views

Complexity of linear programming

It is known that Linear Programming (LP) is P-complete. I am interested in approximation algorithms for LP. There are numerous inapproximability results for NP optimization problems, e.g. it is NP-...
0 votes
0 answers
120 views

$\mathsf{ACC}^0$ and $\mathsf{TC}^0$ with $\mathsf{Cuniform}$-$\oplus\mathsf{L}$ or $\mathsf{Cuniform}$-$\mathsf{NC}^1$ oracle?

$\mathsf{TC}^0$ is a small class with $\oplus\mathsf{L}$ containing it. Following inclusions are known: $$\mathsf{Cuniform}\mbox{ -}\mathsf{ACC}^0\subseteq\mathsf{Cuniform}\mbox{ -}\mathsf{TC}^0\...
13 votes
0 answers
361 views

NP-hardness for one-dimensional facility location problem with entrance fee for each customer [closed]

We have $n$ customers, $(x_1, \dots, x_n)$, sorted on the read line. For convenience, we also use $x_i$ to denote its coordinate on the line. We need to locate $m$ facilities on the real line. We note ...
5 votes
2 answers
267 views

Can we recover integers $a_i$ from the sum $a_0 + a_1e+a_2e^2+\cdots+a_ne^n$?

Since $e$ is transcendental, the function $f:\mathbb Z_{\geq 0}^{n+1}\to \mathbb R$ is injective, $$ f(\underset{\text{Integers}\ \geq\ 0}{\underbrace{a_0,a_1,\ldots, a_n}}) = a_0 + a_1e+a_2e^2+\cdots ...
1 vote
1 answer
147 views

Complexity of a satisfiability problem

I would like the know the complexity of a specific satisfiability problem. I have a feeling it could be solved in polynomial time, but I am not sure about it. The problem is described below. Given $n$ ...
25 votes
7 answers
3k views

Is there a natural problem in quasi-polynomial time, but not in polynomial time?

László Babai recently proved that the Graph Isomorphism problem is in quasipolynomial time. See also his talk at University of Chicago, note from the talks by Jeremy Kun GLL post 1, GLL post 2, GLL ...
3 votes
0 answers
95 views

Are there enumerations of machines for all languages in 𝑃 such that there exists a simulator that can efficiently run all the machines enumerated?

From Kozen INDEXINGS OF SUBRECURSIVE CLASSES: "the class of polynomial time computable functions is often indexed by Turing machines with polynomial time counters.... The collection of all (...
-4 votes
1 answer
150 views

Is there an algorithm for 3x3 sudokus without backtracking? [closed]

From what I can see on Wikipedia and the Internet at large, all sudoku solving algorithms (including human ones) employ some kind of EXPTIME backtracking search for some sudokus. Are there any SAT ...
-2 votes
1 answer
1k views

Sequential vs Distributed algo question

If a certain graph problem in the $\textbf{sequential}$ setting is proven to have "no" better constant-factor approximation algorithm than say a 2-approx. algorithm in polynomial time, then does this ...
1 vote
1 answer
109 views

Induction on all polynomial runtimes?

Has there ever been a proof technique to show that a language isn't in $\mathrm{P}$, by showing inductively there isn't any $k$ for which the language is in $\mathrm{TIME}(n^k)$? e.g.: $L\notin \...
10 votes
0 answers
225 views

On Courcelle's question about Monadic second-order logic with cardinality predicates

I have found the following question at openproblemgarden.org: The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the ...
9 votes
1 answer
495 views

On theoretical aproaches for solving $\mathsf{SAT}$ in special cases

In what cases $\mathsf{SAT}$ can be solved in polynomial time? I know two cases: $2$-$\mathsf{SAT}$ and Horn-$\mathsf{SAT}$. Question 1: Is there a reference with algorithms for solving $\mathsf{...
10 votes
2 answers
485 views

"Relatives" of the shortest path problem

Consider a connected undirected graph with non-negative edge weights, and two distinguished vertices $s,t$. Below are some path problems that are all of the following form: find an $s-t$ path, such ...
0 votes
0 answers
86 views

When can convex optimization be considered to be exactly solvable?

If one is trying to find the global minima of a convex function using gradient descent then one will get a run-time which is a function of $\epsilon >0$ where $\epsilon$ measures the accuracy of ...
2 votes
0 answers
113 views

How to prove a general convex set is nonempty or empty in polynomial time?

The general convex set should be represented by a set of (generalized) inequalities $f_{i}(x)\leq 0 $ with $ f_{i}(x) $ being convex in $ x $. I know ellipsoid method and interior method, but I do ...
9 votes
3 answers
906 views

Non-Orthogonal Vectors Problem

Consider the following problems: Orthogonal Vectors Problem Input: A set $S$ of $n$ Boolean vectors each of length $d$. Question: Do there exist distinct vectors $v_1$ and $v_2 \in S$ ...
2 votes
0 answers
73 views

Complexity of comparing extended integer power towers

Inspired by this stackexchange question, is it an open problem to compare two power towers of positive integers if we additionally allow numbers lower in the tower to themselves be represented by ...