Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

Filter by
Sorted by
Tagged with
1
vote
0answers
35 views

Is the following special case of multiway number partitioning NP-hard?

The following problem is a decision problem of multiway number partitioning (wikipedia) (Note that $k$ is also a part of an input in the following problem, while $k$ is a fixed number in wikipedia ...
1
vote
0answers
37 views

What type of Problems (Class and types) Can Be Solved Effectively with Quantum Computers?

The Question: I'm trying to understand the type of problems that Quantum Computers are/will be good at solving and if there is a special class to categorizes these types of problems (e.g. Do we ...
1
vote
1answer
167 views

How to show that Color Tiles is NP-Complete

Color Tiles is a puzzle game where color tiles are laid out in a rectangular grid. A tile is visible to an empty grid cells if there is a clear line-of-sight in one of the 4 cardinal directions (You ...
1
vote
1answer
66 views

Set cover with rewards

I am dealing with the following problem: Given a universe $U$, let $\mathcal{S}$ be a family of subsets of $U$. Each subset $S\in\mathcal{S}$ is associated with a non-negative reward, and each element ...
2
votes
0answers
77 views

NP-hardness of Euclidean k-Median for k = 2

In the Euclidean $k$-median problem, we are given a set $C$ of clients in $\mathbb{R}^d$. The task is to open a set $F \subset \mathbb{R}^d$ of $k$ facilities such that the cost function $\Phi(F) = \...
4
votes
0answers
90 views

Analogue of Chow-Liu tree for $L_1$

Say $\Omega$ is a finite set and $f$ a probability mass function (pmf) over $\Omega^d$. Now let $T$ be a spanning tree on the set $V=\{1,2,\ldots,d\}$, and consider a collection of one- and two- ...
1
vote
1answer
110 views

Hardness of computing the dimension of an integral polytope?

Given a set of linear inequalities $Ax \leq b$ let $P = \text{conv}\{x \in \{0,1\}^n \mid A x \leq b \}$ be the convex hull of all binary vectors that satisfy the given inequalities. I am interested ...
1
vote
0answers
56 views

A reduction from the maximum $k$-closure problem to the clique problem

Fix a partially ordered set $(P, \le)$ with $N$ elements and real weights $w(p)$ for each $p \in P$. A subset $S \subset P$ is called closed if for any $x, y$ with $y \in S$ and $x \le y$ we also ...
0
votes
0answers
55 views

Subset Sum Problem exact Algorithm (hypothetical)

Let assume there is way to apply a divide & conquer approach to the classical subset sum solver of Horowitz and Sahni. And for this, we design a decomposition function, when applied to the ...
0
votes
0answers
35 views

Bounded-Frequency Minimum Set Cover Problem

Consider the special case of the minimum set cover problem where each element of the universe occurs in at most 3 sets. Can this problem be solved in polynomial time? Is there a nontrivial upper ...
1
vote
1answer
54 views

Maximize the absolute value of connected nodes after $k$ modifications

Given a graph $G=\{V,E\}$, each node $i$ has a value $v_i$. Given budget $k$, we have $k$ chance to add 1 or minus 1 for a node's value, for example, $v'_i=v_i+1$ or $v'_i=v_i-1$. In particular, $v'_i$...
2
votes
1answer
99 views

Can this special case of Node Weighted Steiner Tree be solved in polynomial time?

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 subset $S \...
1
vote
1answer
198 views

Can this NP-hardness proof for Super Mario Brothers (and other games) be simplified?

In "Classic Nintendo Games are (Computationally) Hard", a generalized framework based on reducibility of 3-SAT for proving NP-hardness of classic Nintendo games is presented, and several ...
2
votes
2answers
116 views

Status of certain problems in knot theory

I found it somewhat difficult to understand the status of certain problems from knot theory. Is it correct to say that it's been neither proved nor disproved that any of the following problems are NP-...
-1
votes
1answer
87 views

Is finding the shortest consistent term to fill a missing line in a truth table still NP-hard?

I understand the logic minimization problem is NP-hard when given the onset, since the last step is equivalent to set cover optimization. If instead we are given a partial truth table, and we just ...
1
vote
0answers
124 views

For what parameters is minimum distance hard?

It has been shown by Vardy that minimum distance of a code is NP-hard (see Alexander Vardy, “The Intractability of Computing the Minimum Distance of a Code,” IEEE Trans. Inf. Thy., Vol. 43 pp. 1757--...
1
vote
1answer
138 views

Is QMA known to contain Co-NP?

Is QMA known to contain Co-NP? If not, would Co-NP being contained in QMA have any implications for other complexity classes. (e.g. Causing the polynomial heirachy to collapse.)
0
votes
1answer
168 views

Lexicographic Boolean satisfiability

Maximal Satisfying Assignment (Lexicographic Boolean satisfiability / LexMaxSAT), the problem of finding the lexicographically maximum x_1, . . . , x_n ∈ {0, 1}^n that satisfies Boolean formula φ, or ...
0
votes
0answers
50 views

Is p-dimensional matching with (p−1)n edges NP-hard? What about 3n edges?

Let $p≥3$ an integer. I am wondering whether or not the following problems are NP-hard or not (and if they are, I am looking for a convincing argument, or even better a detailed proof): Let $V_1,\...
4
votes
0answers
127 views

An NP-hard Hidden Subgroup Problem

I've encountered a model which can be thought of as a version of the Hidden Subgroup Problem (https://en.wikipedia.org/wiki/Hidden_subgroup_problem), but that doesn't quite meet the standard problem's ...
2
votes
1answer
154 views

Coloring intersection graph of squares

It is known that the coloring intersection graph of axis-parallel rectangles is NP-Hard. What about squares and more specific case "unit squares"? Thanks.
1
vote
0answers
40 views

Modifying sets to minimize the distance among each pair of the mean value of sets

Given $n$ points, each point $x_i$ has a value $v_i \in \mathbb{R}^{d}$, and there are $m$ point sets $\{S_1,\dots, S_m\}$ that each point set consists of some points. The size of point sets can be ...
7
votes
1answer
282 views

Hardness of maximizing $x^TAy$ with $\{-1,1\}$ entries

My question concerns the NP-hardness of the following discrete optimization problem: Given a matrix $A \in \{ \pm 1 \}^{m\times n}$, $$\begin{array}{ll} \underset{x \in \{ \pm 1 \}^m ,\, y \in \{ \pm ...
-2
votes
1answer
62 views

Is time-slot matchmaking NP-hard?

I have the following problem, which I intuitively expect to be NP-hard but cannot see how to write a reduction for: Givens: There is a fixed set of time slots, e.g. {9-10, 10-11, 11-12, 12-1, etc.}. ...
5
votes
1answer
231 views

$\rho OPT + k$ approximation for bin packing (Unpublished result of David P. Williamson)

I am currently stuck on Exercise 5.12 in this book, which is apparently an unpublished result of David P. Williamson according to the book notes. The problem asks to use randomized rounding and first ...
1
vote
0answers
147 views

Triangle detection hardness in regular graphs

Consider a tripartite graph over $n^{1-\epsilon}$ vertices each in sets $I, J, K$. Suppose we impose a constraint that every vertex has degree $n^\epsilon/c$ for some constant $\epsilon > 0$ and ...
3
votes
0answers
138 views

Computational complexity of minimum distance of rate $\frac{1}{2}$ codes

We know from Berlekamp, McEliece and Van Tilborg [On the inherent intractability of certain coding problems, IEEE Trans. Information Theory, 24 (1978)] that computing minimum distance of a (binary) ...
3
votes
1answer
343 views

How hard is this combinatorial optimisation problem?

Suppose we have multiple intervals $R_1,R_2,...,R_i$ of non-negative integers. These intervals may overlap and we use $R_h(\mathrm{median})$ to denote the median integer in the $h$-th interval $R_h$, ...
8
votes
1answer
153 views

Finding vertex separator such that the induced subgraph has minimal number of edges

My problem is related to edge and vertex cuts with a little twist. Given a graph $G$ and two vertexes $u$ and $v$. I want to find a set of vertexes $S \subset V$ that disconnects $u$ and $v$ such that ...
2
votes
1answer
72 views

Covering a binary relation as a union of rectangles

Given finite sets $X$ and $Y$ and a subset $R\subset X\times Y$, I want to express $R$ as a union $R=\bigcup_{i=1}^n X_i\times Y_i$ with $n$ as small as possible. Here, each $X_i\subset X$ and $Y_i\...
2
votes
2answers
362 views

Proving not NP-complete by non-existence of gadget

Suppose we suspect a problem to be polynomial time solvable, but we are unable to prove this. So, we attempt to prove that the problem cannot be NP-hard. Known proofs in this direction show that if ...
4
votes
1answer
219 views

Complexity of relaxed edge colouring

A (proper) $k$-edge colouring of a graph $G(V,E)$ is a function $f:E\to\{1,2,\dots,k\}$ such that adjacent vertices are mapped to different colours; that is, $f(e)\neq f(e')$ if $e$ and $e'$ are ...
1
vote
0answers
86 views

Is there a "common" name for this type of combinatorial optimization problem?

I'm trying to find papers that discuss approaches (in particular, any Deep Learning or Deep Reinforcement Learning techniques) that could be used used to solve the problem described in the next ...
4
votes
0answers
88 views

Inverting Kronecker product on vectors is in P?

Problem: Given a vector V of positive integers, find two vectors v1 and v2 such that the Kronecker product of v1 and v2 is equal to p(V) (where p(V) is a suitable permutation of V). Example: Input: V={...
1
vote
2answers
200 views

Two valued variant of subset sum problem

I'm interested in the complexity of the following problem: Given a multiset $S$ containing only two positive integers $a$ and $b$, find a $k$-partition of $S$ that maximizes the sum of part with ...
6
votes
2answers
396 views

Is that edge orientation optimization problem NP-hard?

Is the following optimization problem NP-hard? Problem. For a given undirected graph $G=(V,E)$, find an orientation of the edges that minimizes the objective value $\sum_\limits{u\in V} ~\left( d_{...
5
votes
1answer
258 views

Is this edge orientation optimization problem NP-hard?

Is the following optimization problem NP-hard? Problem. For a given undirected graph $G=(V,E)$, find an orientation of the edges that minimizes the objective value $\sum_\limits{v\in V} ~d_{out}(v)\...
2
votes
0answers
183 views

Is this node permutation optimization NP-Hard?

Let $G=(V,E)$ be an undirected graph and let $\pi$ be a permutation of the vertices in $V$. For a node $v\in V$, we denote by $\text{succ}_{\pi}(v)$ the set of neighbors of $v$ that occur after $v$ in ...
2
votes
0answers
193 views

Is the Fermat-Weber problem $\mathsf{NP}$-hard?

Given a set of $n$ points in a Euclidean space, the Fermat-Weber problem asks to find a center that minimizes the sum of distances of points to that center. There are iterative algorithms known for ...
1
vote
0answers
49 views

Does a decision oracle imply an algorithm for $\mathbb{NP}$ - hard problems with several parameters?

If we consider the decision version of the classical graph coloring problem, then we have some graph $G$ and some integer $k$ and we want to color $G$ with at most $k$ colors. It is well known that, ...
5
votes
0answers
171 views

Is this problem in P? Given a bipartite graph, find a minimum cardinality set of edges which intersect every vertex cover

This problem came up in my study of digraphs: Given a connected bipartite graph $G = (A \cup B, E)$, a vertex cover is a set $S$ of vertices such that every edge has some endpoint in $S$. Note that $A$...
-2
votes
1answer
78 views

Set cover with small subsets [closed]

Has the variant of the Set Cover problem where each set is of size at most $d$, for some given $d$, been studied? Is it polynomial-time solvable for $d=2$ and NP-hard for $d=3$ like SAT?
-4
votes
1answer
105 views

Partition vertices of graph into two sets such that there are at least $k$ edges between sets [closed]

I have to show that for every integer $k$, the problem whether the vertices of input graph can be partitioned into two sets such that there are a least $k$ edges between the sets can be solved in ...
12
votes
0answers
347 views

NP complete problem help

I'm currently trying to find a reduction to this problem: Given a set S of n points (in the plane) in general position, is there a set of at least k triangles (formed using only points in S as ...
0
votes
2answers
120 views

Knowing if there are two solutions to the subset sum problem

I was wondering if there are any results that say how hard it is to answer the question are there TWO subsets that sum to a fixed value? In other words, the subset sum problem but asking if there are ...
13
votes
0answers
335 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 ...
0
votes
0answers
83 views

Is the traveling salesman problem still NP-hard if all edges need to be covered as well?

If we formulate the travelling salesman problem with an added edge-covering constraint as follows, is it still NP-hard? Given a graph G with non-negative edge weights, is there a circular walk in G ...
0
votes
1answer
61 views

How hard is deciding the existence of a polygonization with prescribed perimeter?

Polygonization problem of a set of points in the Euclidean plane (2D lattice) is to find a simple polygon that passes through all points. Deciding the existence of a polygonization with minimum (or ...
6
votes
1answer
227 views

Why does Dinur's proof of the PCP theorem fail to work for unique games?

What is the critical step where things go wrong if one attempts to use Dinur's proof the PCP theorem to prove the unique games conjecture by starting from a unique label cover instance and doing gap ...
7
votes
1answer
320 views

Is solving the following system of boolean equations NP-hard?

I reduced a problem I'm currently working on to the following system of boolean equations: $$ X_i \iff \begin{cases} \bigvee_{B \in A_i} \bigwedge_{k \in B} X_k \\ true \\ false \end{cases} $$ Where $|...

1
2 3 4 5
13