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Questions tagged [np-hard]

decision problems that are at least as hard as NP-complete problems

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1answer
26 views

Proving NP hardness about graph creation problem with triangle number

I have graph creation problem. Given a set of nodes of graph, and node constraints such as given every node's number of neighbors (degree). I am also provided with the total number of triangles in ...
2
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1answer
62 views

Max flow and Matching problem

Where can i find a list of problems reducible to max flow and matching problems. I need such examples to learn and practice .
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0answers
20 views

Karp hardness of two vertex sets in a digraph

Given a digraph $G(V,A)$ and a number $k$, we want to find two vertex subsets $S,T\subseteq V$ such that: $|S|+|T|=k$ For every $v\notin S\cup T$, $v$ has no arcs coming to $S$, and no arcs coming ...
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17 views

Finding if a given problem is a Np-Hard problem - recruitment problem

I have to prove that the following Recruiting problem a NPC-problem. Input: n candidates and m positions and a matrix A $\in {Q^{n\times n}}$. Each entry $A_{ij}$ with i < j tells how much gets ...
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2answers
84 views

How to generate an instance for an NP_hard proof, where each element has two inputs?

I want to prove the NP-hardness of an scheduling problem. The problem seems to be NP-hard in the ordinary sense, so I am trying with the Partition Problem, precisely the Equal Cardinality Partition (...
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0answers
22 views

Testing if a 4-regular graph can be decomposed into two edge-disjoint Hamiltonian cycles

Given a 4-regular graph, is it NP-complete to test whether it can be decomposed into two edge-disjoint Hamiltonian cycles?
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1answer
33 views

Graph Edit Distance

Source: K. Riesen, Structural Pattern Recognition with Graph Edit Distance, Advances in Computer Vision and Pattern Recognition. Link: https://www.springer.com/cda/content/document/...
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1answer
39 views

Reducing subset sum to even subset sum

I'm trying to learn reduction. I have this problem called "even subset sum" that's very similar to subset sum. It's the same problem as as subset sum except that the only numbers allowed are even ...
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1answer
28 views

Karp reduction from NP-hard problem to unknown problem

If I know that problem $A$ is NP-hard, but know nothing of problem $B$ and I know that the following Karp reduction is true: $$A \to B \, .$$ Is it correct to conclude that $B$ must also be NP-hard?
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Relation between Undecidable problems and NP-Hard

I drew these pictures to check whether I comprehended the ideas of P, NP, NP Complete and NP Hard correctly. And then, I realized that it is not certain where undecidable problems should be placed. ...
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0answers
13 views

Regex engine: A polynomial algorithm for backreferences?

During my search for a way to integrate efficiently backreferences (using backtracking) into a text-directed regex engine, I found a paper entitled Extending finite automata to efficiently match Perl-...
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1answer
60 views

Karp hardness of minimum number of arc reversals needed to turn a graph into an Eulerian one

What is the complexity of this problem EULERIAN ARC REVERSAL Input: a directed graph $G(V,A)$ and an integer $k$ Output: YES if $k$ arc reversals are enough to transform $G$ into an ...
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0answers
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Proof that MAX-2-SAT is NP-hard [duplicate]

According to Wikipedia, while the 2SAT problem is polynomial, its maximization variant MAX2SAT is NP-hard. But, they do not provide a reference for this claim. Is this obvious? If not, where can I ...
2
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1answer
24 views

NP-Hard on Complete Graphs

I have a problem (A) on undirected graphs that I wish to show is NP-Hard. I can show that there is a reduction from a well known NP-Hard problem (B) to A by constructing an instance of A with a ...
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1answer
52 views

Gerrymandering Problem: Variant on Set Partitioning

I was recently helping a friend with homework from a dynamic programming class, and this was the question: Given a set of n precincts P1 ,... Pn , each containing m votes, with <...
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1answer
30 views

Variant of an approximation algorithm for vertex cover

Here is an approximation algorithm that finds vertex cover of a graph. ...
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1answer
106 views

How can I show that a problem is not $NP$

Consider the following image: The problem is: can we cover the bigger rectangle with small rectangles such that no two rectangles overlap and no gap opens up? Prove that this problem is $NP-Hard$. I ...
0
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1answer
59 views

Finding minimum number of edges such that when adding into the graph, the graph is a 2-connected graph

Given is a undirected and 1-connected graph G=(V,E). Between every two node b=(u,v) in graph G there is a cost c_b to build another edge(Regardless of whether (u,v) ∈E or not). We use a C={c_b |c_b∈Z^+...
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2answers
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If P = NP, can all NP problems be solved within time $O(n^k)$ for fixed $k$?

I came across this question while studying for an exam: T/F: Suppose we can show for some fixed $k$, an NP-complete problem P has a time $O(n^k)$ algorithm. Then every problem in NP has a $O(n^k)$ ...
1
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1answer
25 views

Complexity of two cycles which differ by $1$ in length

Given an undirected graph $G(V,E)$, our problem asks whether $G$ contains $2$ simple cycles which differ by $1$ in length. What is the complexity of this problem?
2
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1answer
59 views

Is SAT known to be non-context-free or even non-regular?

We have seen various languages proven to be outside of REG and CFL by corresponding pumping lemmas. Has something similar been done for SAT?
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1answer
32 views

Computational complexity of maximizing sum of rational functions

I have a optimization problem: $$\max_z\ \sum_{i=1}^n \frac{W_i}{D_i - z_i} \quad \text{s.t.}\ \sum_{i=1}^n z_i \leq k, z_i \in [0,k],$$ where each $W_i$, $D_i$ are constants and $z_i$ are integer ...
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3answers
473 views

What will happen to NP-Hard problems if P=NP

I was going through the lecture of prof. Erik Demaine and he said that a problem X is NP-Hard if it is at-least as hard as every problem Y that belongs to NP class. He further said that if we can ...
1
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1answer
50 views

Karp hardness of directed monochromatic triangle problem

Monochromatic problem is a classic NP-complete problem. Does the complexity stay NP-complete if we use directed graph? DIRECTED MONOCHROMATIC TRIANGLE problem: Input: A digraph $G(V,A)$ ...
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1answer
58 views

Is this an $NP$-complete problem: Product-2-Partition

I want to prove the NP-hardness of my problem P in scheduling. I was trying with Partition, 3-Partition and Subset product, But neither was successful. Now, I can reduce a problem, say PRODUCT-2-...
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0answers
10 views

Karp hardness of a supportive clique in digraphs

In a directed graph (digraph) $G(V,A)$, a set of vertices $C\subseteq V$ is called a supportive clique if (1) for every $u\neq v$ in $C$, either $(u,v)\in A$ or $(v,u)\in A$ and (2) for every $u\in C$,...
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1answer
21 views

Karp hardness of a diameter-decreasing planted clique

A diameter-decreasing planted clique in an undirected graph $G(V,E)$ is a set of vertices $\mathcal{C}\subseteq V$ such that if we add all the missing edges between any pair of vertices in $\mathcal{C}...
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1answer
46 views

For each given set choosing either it or its complement such that their union exactly has a given size

Given an integer $k$ and $n$ sets $A_1,\ldots,A_n$, denote $U=A_1\cup A_2\cup\cdots\cup A_n$, $A_i^0=A_i$ and $A_i^1=U\backslash A_i$. The problem asks whether there exists $(b_1,\ldots, b_n)\in\{0,1\}...
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1answer
48 views

Karp hardness of a clique with given number of outer-incident edges

A clique in an undirected graph $G(V,E)$ is a subset of vertices $\mathcal{C}\subseteq V$ every pair of which is adjacent $u,v\in \mathcal{C}\implies uv\in E(G)$. Given a clique $\mathcal{C}\subseteq ...
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0answers
41 views

Karp hardness of a module in a graph

DEFINITION: A set of vertices $A\subseteq V$ in a graph $G(V,E)$ is called a module if it satisfies the following property: For every $v\in V\setminus A$, either $A\subseteq N(v)$ or $A\cap N(v)=\...
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1answer
38 views

On the proof of NP-Hardness of the Cardinality Constrained Quadratic Knapsack Problem

in Polyhedral Study of the Cardinality Constrained Knapsack Problem the authors prove that the Cardinality Constrained Knapsack Problem is NP-Hard by reducing PARTITION to it. Besides, it's easy to ...
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2answers
66 views

Karp hardness of testing for homomorphisms to a fixed non-bipartite graph

Description: Suppose that we are given a fixed non-bipartite graph $H$. A graph $G$ is loopless surjectively homomorphic to $H$ if there exists a loopless surjective homomorphism $\varphi:V(G)\to V(H)$...
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1answer
38 views

Karp hardness of a vector system allocation

Given an undirected graph $G(V,E)$, a vector system is a set $S$ of ordered pair (intuitively called vector) $(u,v)$ (we shall call $u$ the initial point, and $v$ the terminal point) that satisfies ...
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1answer
32 views

Karp hardness of a guarding set in digraph

Our problem is to decide whether a digraph has a guarding set of size at most $k$. Definitions are following. A digraph $G(V,A)$ has $V(G)$ as its vertex set and $A(G)$ as its arc set. A guarding set ...
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2answers
48 views

Karp hardness of digraph coloring without monochromatic dicycle

Problem statement: Input: a digraph $G(V, A)$ and a natural number $k$ Output: YES if it is possible to color all vertices of $V(G)$ by $k$ colors such that no directed cycle is monochromatic, ...
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1answer
35 views

Karp hardness of an equidistant set in digraph

Following the success of the undirected version: Karp hardness of an equidistant vertex set Inspired by the success of this long ago question: NP-hardness of problem with indices and subsets We ...
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0answers
44 views

Karp hardness of a simply equidistant vertex set

Following the success of the previous question: Karp hardness of an equidistant vertex set I continue to propse yet another computational problem. This time, we modify the notion of an equidistant ...
2
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1answer
63 views

Karp hardness of an equidistant vertex set

What is the hardness of the following problem? Input: An undirected graph $G(V, E)$ and a natural number $k$ Output: YES if $G$ has an equidistant vertex set of size $k$, otherwise NO $\...
1
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1answer
26 views

Is Rectilinear Steiner Tree still NP-complete when points have integral coordinates?

Garey proved that the Rectilinear Steiner Tree problem is (strongly) NP-hard. I wonder if it is still true when we retrict the points to have integral coordinates and lie on a square of side lenght n^...
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0answers
28 views

Finding the “most modular” subset of graph vertices, i.e. that minimize disagreement inside and outside

Let $G = (V, E)$ be a graph. I want to find the subset of vertices of $G$ that minimizes a certain modularity cost. In our setting, the modularity cost of a subset $X$ is defined as the number of ...
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0answers
27 views

Edge-midpoints cover with radius 1

This is in a series of posts. Previous quetion: Vertex cover with covering radius 2 Other series: Karp hardness of searching for a matching split In this problem, our cover for a given undirected ...
4
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1answer
50 views

Vertex cover with covering radius 2

Our problem is: Given an undirected graph, does it have a vertex cover consisting of $k$ vertices? A vertex included in this vertex cover variant will cover every edge incident to it and every edge ...
3
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1answer
67 views

Karp hardness of searching for a matching split

UPDATE: In 2 days, if no more convincing answer is posted, then bounty of 50 rep. will go to xskxzr. Due to lack of connectedness and a clean & clear cut, the bounty is still open for 2 days. (UTC ...
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1answer
69 views

Maximum induced tree

Given an undirected graph $G(V, E)$, how hard is it to decide whether it has an induced tree consisting of $k$ vertices, where $k$ is also given in the input?
3
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1answer
36 views

Karp hardness of searching for a matching erosion

First, read the previous question: Karp hardness of searching for a matching cut As mentioned in the supposed-to-be-comment answer in that question, without the requirement of cardinality $k$, the ...
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3answers
79 views

Defining a graph decision problem not in NP

I have been doing some research online looking for graph problems that are decidable but not in NP. I have found the concept of succinct graphs, which if I understand properly, consist of making the ...
4
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2answers
87 views

Karp hardness of searching for a matching cut

Follow-up question in the series: Karp hardness of searching for a matching erosion Karp hardness of searching for a matching split Maximum Matching Cut problem Input: An undirected graph $G(...
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1answer
23 views

Hardness of $2$ edge-disjoint spanning trees decomposition

The question is clear from the title. What is the complexity of the following decision problem: Input: An undirected graph $G(V, E)$ Output: $\mathrm{YES}$ if $G$ can be decomposed into two ...
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2answers
161 views

Does deep learning infer P = NP?

The question comes from the following scenario, assume we have the traveler problem which is NP (the one where a traveler wants to visit all countries with the lowest cost(by summing up all flights)) ...
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3answers
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Collection of meta-reductions in theory of $\mathrm{NP}$-completeness

I want to start a wiki post about meta-result of meta-reductions in the theory of $\mathrm{NP}$-completeness. This can be regarded as a reference request post. Any links are appreciated. At least, ...