Questions related to NP-hardness and NP-completeness.

learn more… | top users | synonyms (1)

2
votes
0answers
134 views

Is the nonnegativeness of a polynomial hard for $\mathsf{NP}_\mathbb{R}$?

It is clear that the following problem is in $\mathsf{NP}_\mathbb{R}$. Input: a list $P$ of triplets $(a,s,t)$ where $s$ and $t$ are nonnegative integers. Output: is there an $x\in \mathbb{R}$ such ...
3
votes
0answers
78 views

NP-hardness of a quadratic programming problem

Motivated by the mean-variance optimization, I came up with the following question: Given $n$ integers $a_1, \cdots, a_n$; $n$ lower bounds $0<\ell_1, \cdots, \ell_n<1$ $n$ upper bounds ...
-1
votes
0answers
86 views

If a problem is not in NP, does it have to be NP-hard [closed]

Is every non-NP problem NP-hard? In other words, does every problem in NP reduce (in polynomial time) to every problem outside of NP? Equivalently, does 3SAT (or your favorite other NP-complete ...
1
vote
1answer
131 views

NP-Hardness for an optimization problem

I want to prove that the following optimization problem is NP-Hard. max $\prod_{i = 1}^{N} \frac{\left[\sum_{j =1}^M x_j \mathcal{R}_{ij}\right]^2}{ \sum_{j=1}^M x_j}$ subject to $x_j \in ...
6
votes
0answers
78 views

Edge-partitioning into rainbow triangles

I'm wondering if the following problem is NP-hard. Input: $G = (V,E)$ a simple graph, and a coloring $f : E \to \{1,2,3\}$ of the edges ($f$ does not verify any specific property). ...
25
votes
1answer
585 views

Recognizing sequences with all permutations of $\{1, \ldots, n\}$ as subsequences

For any $n > 0$, I say that a sequence $s$ of integers in $\{1, \ldots, n\}$ is $n$-complete if, for every permutation $\mathbf{p}$ of $\{1, \ldots, n\}$, written as a sequence of pairwise distinct ...
7
votes
1answer
152 views

What do we know about checking real-stability of multivariate complex polynomials?

Given a polynomial $p : \mathbb{C}^n \rightarrow \mathbb{C}$ it is to be called "real-stable" if (1) all its coefficients are real and (2) if it has no roots such that all the coordinates of the root ...
4
votes
1answer
120 views

Example of a function problem which is $\mathrm{FP}^{\mathrm{NP}}[wit, log]$-hard?

The usage of an $\mathrm{NP}$-oracles which delivers a witness has been proposed for example in [Buss1995]. I would like to see an example of an $\mathrm{FP}^{\mathrm{NP}}[wit, log]$-hard problem. Can ...
10
votes
0answers
229 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 ...
1
vote
0answers
123 views

Finding an equivalent NP-complete instance for this game-theory problem

I apologize if this question is not a good fit for CSTheory. I'm a PhD student who has just started out and I'm working on a game-theory problem in one of my classes. Although my professor hasn't ...
2
votes
0answers
106 views

Complexity of Knapsack-type problem with applications to computational workflows

Consider the following problem: Let there be a set A of $n$ items $A=\{z_1, ..., z_n\}$, and let $W$ be a strictly positive integer. Each item $z_i$ has a value $v_i$ and a weight $w_i$. Finding a ...
2
votes
0answers
86 views

Is minimizing sum of distances hard?

The Problem Given a set of $n$ points $S = \{v_1, v_2, \cdots, v_n\} \subset \Re^d$, find a unit vector $s \in \Re^d$ such that $s$ minimizes $$ \sum_{i=1}^{n}\sqrt{\|v_i\|^2 - \langle v_i, s ...
0
votes
0answers
68 views

Is there a reduction from a 0-1 knapsack problem to the unbounded problem?

As we know, an unbounded knapsack problem could be described as: $\max \sum_{i=1}^nc_1x_i$ s.t. $\sum_{i=1}^na_ix_i\le b$ $x_i\ge0,x_i\in\mathbb Z,i=1,\cdots,n$ And for an 0-1 knapsack problem, we ...
5
votes
1answer
287 views

NP-hardness of minimum distance over a code

There happens to be this NP-complete question, Minimum-Distance-Over-$\mathbb{F}_{2^m}$ Given $w \in \mathbb{Z}^+$ and a $r \times n$ matrix $H$ over $\mathbb{F}_{2^m}$, is there a $x \in ...
-2
votes
1answer
99 views

Proving NP-hardness of scheduling problem (total weighted completion time)

Consider the problem $P \mid \mid \sum w_j C_j$. I want to prove that this problem is (strongly) NP-hard by reducing from $3$-Partition, but I am not really sure how to do this. Just to be precise, ...
11
votes
2answers
1k views

To which complexity class does this language belong?

I was thinking about which class this language belongs: $L =\{ \langle G,k \rangle \mid G $ is a graph, $k$ is a natural number and $k$ is the chromatic number of $G\}$ I thought of $L$ as (1) " ...
1
vote
0answers
52 views

Hardness of Minimizing Submodular Functions with Cardinality Constraints

I am new to submodular functions and I am reading the introductions to submodular functions and applications ( https://www.ima.umn.edu/optimization/seminar/queyranne.pdf ). In this introduction, it ...
1
vote
1answer
194 views

$k$-clique in $k$-partite graph

Is the decision whether a $k$-clique exists in a $k$-partite graph NP-hard? I have found only a very limited number of references on this problem, and they seem to be concerned with heuristics to ...
4
votes
1answer
100 views

Generalized geography on solid grid graphs

A post here For which families of graphs is Generalized Geography in $P$? mentioned that generalized geography on solid grid graphs is open. Is the question still open? A quick search on Google shows ...
0
votes
1answer
237 views

NP-complete problems with optimal approximation in poly-time

I'm looking for examples of hard optimization problems, for which we have an optimal approximation (not that this is not the same as $PTAS$, as we require a completely tight approximation, and not ...
4
votes
1answer
358 views

The complexity of a multi-objective shortest path problem

I have the following shortest path problem. Consider a directed graph with $n$ levels. Each level has $m$ nodes. Each node at level $i$ is connected to all nodes at level $i+1$. Let us also make a ...
0
votes
0answers
63 views

Finding exact value with a quotients of products of random values

Sorry for the haphazard title: really not sure what this should be called Suppose we have a set of $z$ random values $S = r_1, \dots, r_z$ drawn from $\mathbb{Z}_N$ (where $N$ is some large prime). ...
3
votes
3answers
379 views

Minimum vertex cover on k-regular graphs, for fixed k>2 NP-hard proof?

Good Evening! I am aware that the minimum (cardinality) vertex cover problem on cubic graphs ($3$-regular) graphs is $NP$-hard. Say positive integer $k>2$ is fixed. Has there been any ...
2
votes
3answers
163 views

How hard is it to find a “well-distributed” subset of models of a propositional formula?

We consider the propositional language $\mathcal{L}_{\mathit{PS}}$ defined over a finite alphabet $\mathit{PS}$ and the usual logical connectives. An interpretation is an assignment $\mathit{PS} ...
13
votes
1answer
325 views

“Snake” reconfiguration problem

While writing a small post on the complexity of the videogames Nibbler and Snake; I found that they both can be modeled as reconfiguration problems on planar graphs; and it seems unlikely that such ...
7
votes
1answer
357 views

Does $NP$-hardness of $c$-approximation (for some $c>1$) imply $APX$-hardness?

Assume that for a given minimization problem with only integer solutions, it is $NP$-hard to decide if the optimal solution is 5 or 6. I.e., a polynomial-time algorithm with an approximation ratio ...
7
votes
0answers
122 views

Maximum weight “fair” matching

I'm interested in a variant of the maximum weight matching in a graph, which I call "Maximum Fair Matching". Assume that the graph is full (i.e. $E=V\times V$), has even number of vertices, and that ...
2
votes
2answers
180 views

Balls & Bins: A punishment and reward game

Consider a game where one has a set of bins $(b_1, b_2, ...)$, and each bin has an associated initial count of balls $(c_1, c_2, ...)$. The rules of the game are as follows: (1) Once a bin has a ...
2
votes
0answers
76 views

Calculating the ground state of an Ising model with $\sigma_i = (0,1)$ spin state assignments (do Barahona & Istrail's NP-hardness results hold?)

In a typical Ising model, one has possible spin assignments of $\sigma_i = \pm 1$. However, one can also imagine a $q = 2$ Potts model generalization with spin assignments $\sigma_i = (0,1)$. Is ...
9
votes
0answers
80 views

Reconstructing labeled poset from linear extensions

Let $(P, <, \mu)$ be a labeled poset, that is, a partial order $(P, <)$ with a labeling function $\mu$ that maps the elements of $P$ to labels in an alphabet $\Sigma$. A label list (or word) is ...
9
votes
0answers
119 views

Advances towards proving the Held-Karp conjecture for TSP

I've only began my research into the Held-Karp conjecture and I was wondering about recent progress in proving the conjecture. The Held-Karp relaxation is conjectured to have an integrality gap of ...
0
votes
0answers
64 views

Trying to find polynomial-time algorithms for knapsack-like problems

Consider two related problems: You have n cannisters that must go into m trucks that can each carry k cannisters. You require that no truck becomes overloaded, and for each cannister, there is a ...
2
votes
0answers
74 views

Which complexity information of Ising model is more important?

In 1982, Barahona proved that finding the ground state of an Ising model is NP-hard. Later, in 2000, Istrail proved that it is NP-complete. When I look up the citations of these two papers using ...
1
vote
0answers
45 views

Bounding the cost of an approximation algorithm when subtraction involve [closed]

Given an algorithm with approximation ratio $\alpha$, and another algorithm with approximation ratio $\beta=n^\epsilon$, and a solution to a problem with cost $c$. What is the standard way to bound ...
4
votes
2answers
135 views

Variation on partial Set Cover with penalties

I'm looking at the following problem, which I'm hoping is perhaps a known variant of the Set Cover problem: Input: We're given a universe $U$ and two disjoint subsets $A$ and $B$ such that $A \cup ...
0
votes
0answers
75 views

Minimum Weight Ordering of nodes on a directed graph

I'm bumping my head against the wall trying to prove this problem is NP-complete (it might not be) Let $G = (V,E)$ be a directed graph with weights $w:E \to \mathbb{R_{\geq 0}}$ on the edges. The ...
0
votes
0answers
57 views

Minimum vector sets span spaces cover problem

Instance: a set of vectors $V=\{\mathbf{v}_1,\mathbf{v}_2,...,\mathbf{v}_n\}$ and $m$ vector sets $V_1,V_2,...,V_m$, each of which contains multiple vectors ($V_i$ may not be a subset of $V$). In our ...
4
votes
1answer
125 views

Inapproximability of $(\alpha, \beta)$ bi-criteria approximation

An $(\alpha, \beta)$ bi-criteria approximation algorithm for $k$-center is defined as an algorithm that returns a solution whose value is $\beta \cdot OPT$ ($OPT$ being the optimal solution for the ...
10
votes
1answer
355 views

Probability of generating a desired permutation by random swaps

I'm interested in the following problem. We're given as input a "target permutation" $\sigma\in S_n$, as well as an ordered list of indices $i_1,\ldots,i_m\in [n-1]$. Then, starting with the list ...
6
votes
0answers
213 views

NP-hardness of tasks graph assignment to two heterogenous servers

I have a problem with determinig if the following assignment problem is NP-hard. Any comments and suggestions would be appreciated. Problem definition Given is a directed acyclic graph $G=(V,E)$ ...
1
vote
1answer
105 views

Node-weighted steiner problem with few terminals

Consider the node-weighted steiner problem: Input: a graph $G=(V,E)$, a set $T\subseteq V$ of terminals, a weight function $w: V\setminus T \to \mathbb{R}_+$. Output: a minimum weight ...
1
vote
0answers
156 views

Definition of Planar 3-SAT

What is the standard definition of Planar 3-SAT? I have seen a number of different definitions. What was the original paper that defined it and proved it to be NP-complete?
0
votes
1answer
143 views

Hardness of an extended maximum set packing problem

(Edited) The maximum set packing problem when the sets are all of equal size, say $k$, is known to be NP-hard for $k \ge 3$. The requirement in this problem is that the sets in the solution will be ...
2
votes
1answer
164 views

Complexity class of a subset problem similar to subset sum

What could be the complexity class of the following problem: Given a positive integer $K$, a positive integer $d$ (say $2$) and a set $S$ of all non-negative integers less than $K$ find a $S' ...
12
votes
2answers
1k views

Hamiltonian cycle on graphs without small cycles

While answering this question on cstheory, I (informally) proved on the fly the following theorem: Theorem: For any fixed $l \geq 3$ the Hamiltonian cycle probem remains NP-complete even if ...
6
votes
0answers
87 views

Hard extendability problems

In extendability problem, we are given part of the solution and we want to decide whether we can extend it to a complete solution. Some extendability problems are efficiently solvable while other ...
5
votes
0answers
188 views

NP-completeness of the Dominating set problem for planar graphs of maximum degree 3?

I am trying to learn about some techniques that are used for proving the NP-completeness of domination related problems. One of the problems that is known to be NP-complete is the domination number of ...
3
votes
1answer
316 views

Hardness of 3-dimensional matching variant

I am thinking about a variant of 3-dimensional matching. In normal 3d matching, we have three sets of vertices $X$, $Y$ and $Z$, and a set of edges $E \subseteq X \times Y \times Z$. We want to choose ...
2
votes
1answer
125 views

NP-hardness of a bilinear program?

Given $a_i, b_i, c_i, d_{ij}\in [0,1]$ for $i,j\in [n]$ and $i\neq j$ such that $\sum_{i\in [n]} a_i=1$ and $d_{ij}=d_{ji}$. I have the following bilinear program: max $\sum_{i=1}^n (x_i-a_i)y_i$ ...
2
votes
0answers
126 views

Graph partition with objective over intra-partition weights

I have a problem in which I need to find an optimal graph cut that maximizes an objective over weights not on the cut. I have looked at the literature but have not been able to find any similar ...