Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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10
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2answers
241 views

Common insights into hypothetical complexity of graph problems

I came across two examples of hypothetical hardness of some graph problems. Hypothetical hardness means that refuting some conjecture would imply the NP-completeness of the respective graph problem. ...
1
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0answers
118 views

Generalized path cover problem in DAG

Let $G=(V,E)$ be a directed acyclic graph. Two vertices is transitive if there is a directed path between them. A Path Cover for a Set of Transitive Pairs (PCSTP) is a set of directed paths such that ...
16
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1answer
435 views

What is the complexity of this graph problem?

Given a simple undirected graph $G$, find a subset $A\neq \emptyset$ of vertices, such that for any vertex $x\in A$ at least half of the neighbors of $x$ are also in $A$, and the size of $A$ is ...
5
votes
2answers
180 views

Minimising the root-set of a spanning hyperforest of a hypergraph

I am interested in the complexity of a problem involving spanning hyperforests (a union of hypertrees, which covers all of the vertices) of a $k$-hypergraph. I describe the relevant definitions for ...
1
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0answers
152 views

Are there any NP-complete for continuous mathematics? [closed]

Looking at this wiki page, it seems most NP-complete problems are based on discrete structures, such as graphs. What are some problems that involve real or complex analysis instead of discrete ...
2
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0answers
176 views

Is Non-linear Constrained Optimal Exact Cover NP-Hard?

Playing around I ran into a problem which looks like a Exact Set Covering / Partition Problem, but I am unable to find a reduction to categorize the complexity of the problem, despite it looks NP-Hard....
-5
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2answers
3k views

Minimum-weight feedback edge set in undirected graph - how to find it? Is it NP hard problem?

Let G = (V,E) be an undirected graph. A set F ⊆ E of edges is called a feedback-edge set if every cycle of G has at least one edge in F. Suppose that G is a weighted undirected graph with positive ...
13
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1answer
717 views

Complexity of the problem of words with fewest distinct letters accepted by a finite automaton

Given a finite (deterministic or nondeterministic, I don't think this has much importance) automaton A and a threshold n, does A accept a word containing at most n distinct letters? (By k different ...
3
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1answer
180 views

Matrix Coloring under Vertical and Horizontal Constraints

I'm searching for the correct name of the following NP-complete problem. I would also appreciate answers pointing to problems with similar-looking variations. The input consists of A set of ...
3
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2answers
175 views

Reduction of graph chromatic number to hypergraph 2-colorability

I'm following this paper titled "Coverings and colorings of hypergraphs" by Lovasz 1973, which is referenced in Garey and Johnson's Computers and Intractability, for the Set Splitting Problem. In this ...
0
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2answers
190 views

Geometric max cover

Consider $n$ points and a distance function $d$ that satisfies the triangle inequality. We are also given a number $r$. Each point $p$ defines a set $B_p$ (or a ball) that covers all other points ...
-4
votes
1answer
62 views

Proving NP-complete problem

Suppose the following problem: Given an undirected graph G=(V,E), is it possible to choose a subset V' of vertex set V, such that deleting it removes all triangles (cycles of length 3), where |V'| is ...
6
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1answer
285 views

Maximum Polyhedron Volume in Given $n$ Points

Suppose we are given $n$ points $v_1,v_2,\cdots, v_n\in \mathbb{R}^k$, I want to find $k+1$ points $v_{i_1}, v_{i_2},\cdots,v_{i_{k+1}}$ such that the volume of the convex body spanned by them ...
3
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2answers
210 views

Hardness of exact binomial tail bounds

Chernoff bounds, in their various forms, bound the tails of a Binomial$(n,p)$ random variable $B$. Define the function $F(n,p,t):=P(B>t)$. Naively, computing $F$ requires exponential (in $n$) time. ...
7
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1answer
230 views

What are some problems in $P$ which have lower bounds assuming that $P \neq NP$ or the ETH?

Last year I had watched this talk online about problems in $P$ for which an algorithm which runs in some subclass of $P$, say in subquadratic time, would imply $P = NP$, or violate the ETH (...
2
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1answer
167 views

Solving linear program with 1 quadratic constraint complexity

Consider the following linear program, $$\min y \\ xc_1 \leq c_2 + yz,\\ x = x_1 + \dots + x_n,\\ z \leq x_1 + x_2, \\ z \leq x_2 + x_3, \\ \vdots\\ z \leq x_{n-1} + x_n, \\ x,x_1, \dots, x_n,y,z \...
11
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2answers
8k views

Is the N Queens problem NP-hard?

The N-queen problem is this: Input : N Output : A placement of N "queens" on an NXN chessboard such that no two queens lie on the same row, column or diagonal. Doing a google search on this, I ...
1
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2answers
267 views

Is Asymptotic PTAS $\subseteq$ APX?

The definition of asymptotic polynomial-time approximation scheme (Asymptotic PTAS) is defined as follows: A minimization problem $\Pi$ is Asymptotic PTAS if for all $\epsilon$ there exists an ...
4
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1answer
245 views

String version of even-odd partition problem

Motivated by Hardness proof of EVEN-ODD PARTITION post I came up with a string version. String even-odd partition INPUT: $(x_{1,0},x_{1,1}),\dots,(x_{n,0},x_{n,1})$, i.e., $n$ pairs of strings over ...
24
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6answers
2k views

Are there NP-complete problems with polynomial expected time solutions?

Are there any NP-complete problems for which an algorithm is known that the expected running time is polynomial (for some sensible distribution over the instances)? If not, are there problems for ...
22
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1answer
2k views

Tardos Function Counterexample to Blum's $P\neq NP$ Claim

In this thread, Norbet Blum's attempted $P \neq NP$ proof is succinctly disproved by noting that the Tardos function is a counterexample to Theorem 6. Theorem 6: Let $f \in \mathcal{B}_n$ be any ...
2
votes
2answers
938 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 ...
1
vote
1answer
582 views

Clique cover problem

Consider the following graph problem. We are given a graph $\mathcal{G} = (\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of vertices and $\mathcal{E}$ is the set of edges. For each vertex $...
1
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1answer
214 views

Max weight travel on a graph with deadline

Given a deadline $D>0$ and a complete graph $K_n$ (with loops) in which each edge $e_{ij}$ has a weight $w(e_{ij}) \ge 0$ and a travel time $l(e_{ij}) > 0$. Starting from one of the nodes, we ...
1
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1answer
207 views

Max-weight connected & co-connected subgraph problem

The max-weight connected subgraph problem (MWCS) may be described as follows: given a simple graph $G=(V,E)$ and a weight function $w:V\to\mathbb{R}$, one seeks for a subset $L\subseteq V$ for which ...
3
votes
1answer
263 views

Is deciding whether all satisfying assignments are NAE assignments coNP-complete?

Let the language $L$ consist of the $k$-CNF formulas $\phi$ with the property that any satisfying assignment $x$ of $\phi$ is a Not-All-Equal (NAE) assignment, i.e. every clause of $\phi$ has at least ...
1
vote
1answer
254 views

Does a weighted graph have a path with weight zero?

Given a weighted digraph $G=(V,E)$, where each edge is associated with a weight (could be positive, negative, or zero). We define the weight of a path to be the sum of the weights along this path. ...
1
vote
1answer
202 views

What is the computational complexity of this SAT Variant

Given a 3SAT problem. The question being: 'This Problem has exactly K Solutions'? Now, lets say K=1 (without loss of generality). If the problem has a exactly 1 solution and the answer is True. So, ...
38
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3answers
8k views

Is optimally solving the n×n×n Rubik's Cube NP-hard?

Consider the obvious $n\times n\times n$ generalization of the Rubik's Cube. Is it NP-hard to compute the shortest sequence of moves that solves a given scrambled cube, or is there a polynomial-time ...
-1
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1answer
132 views

A conceptual question regarding hardness proofs by reduction [closed]

If we restrict the input domain of a known NP-hard problem P so that this restricted domain is equal to the input domain of another problem S, then show that we can reduce a solution to P given input ...
1
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1answer
158 views

Is pooling-aware bin packing NP-Hard?

I am unable to prove whether the following problem is NP-Hard. It seems like a bin-packing or a partition problem, without being close enough to either of them (at least I do not see the reduction to ...
6
votes
1answer
247 views

totally-mixed 2SAT with exact cardinality?

Given a 2HornSAT problem, it’s possible in linear time to find the minimum solution to the problem, i.e., a solution that minimizes the number of variables set to 1. Now let us consider the following ...
4
votes
1answer
155 views

Limits of variants of Independent Set?

Independent Set (IS) is the $\mathsf{NP}$-complete decision problem Input: graph $G$ with $v=|V(G)|$, integer $k$ Question: is there an independent set $S \subseteq V(G)$ with at least $k$ vertices? ...
31
votes
2answers
1k views

When does “X is NP-complete” imply “#X is #P-complete”?

Let $X$ denote a (decision) problem in NP and let #$X$ denote its counting version. Under what conditions is it known that "X is NP-complete" $\implies$ "#X is #P-complete"? Of course the existence ...
4
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1answer
1k views

Does P = NP imply NP being a strict subset of PSPACE? [closed]

Does $\textbf{P} = \textbf {NP}$ imply that $\textbf{NP} \subsetneq \textbf{PSPACE}$? Or, for a slightly stronger result, does it imply that $\textbf{NL} = \textbf P$?
15
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1answer
1k views

Is the following problem NP hard?

Consider a collection of sets $F=\{F_1,F_2,\dotsc,F_n\}$ over a base set $U=\{e_1,e_2,\dotsc,e_n\}$ where $|F_i|$ $\ll$ $n$ and $e_i \in F_i$, and let $k$ be a positive integer. The goal is to find ...
16
votes
2answers
518 views

Is intersection of $k \ge 3$ graphic matroids in P?

It is known that intersection of three general matroids is NP-hard (source), which is done via reduction from Hamiltonian cycle. The reduction uses one graphic matroid and two connectivity matroids. ...
5
votes
1answer
839 views

Is approximating Exact Set Cover NP-hard for constant approximation factor? ETH hard?

It is known that Exact Set Cover is an NP-hard problem (Reduction from 3-SAT and 3-Coloring). Also, my minor analysis one can realize that this problem is also ETH-hard, i.e. this cannot be solved in ...
8
votes
1answer
247 views

Vertex isoperimetric number of a graph - NP-hard?

The vertex isoperimetric number of a graph $G=(V,E)$ is $i_V(G) = \min\{\frac{|N(S)|}{|S|} : S \subseteq V, 1\le |S|\le \frac{|V|}{2}\}$. Several academic papers state that the problem of computing ...
1
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0answers
32 views

Sparse coding and matching pursuit algorithms

Is it true that all known sparse coding algorithms which work efficiently in practice don't have convergence proofs and always use an intermediate step of a matching/subspace pursuit algorithm on the ...
6
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3answers
1k views

Complexity of a subset sum variant

Given integers $a_1, \ldots, a_n, b \in \mathbb{N}$. What is the complexity of the following problem $$ \exists x_1, \ldots, x_n \in \mathbb{N} \text{ such that } a_1x_1 + \ldots a_nx_n = b? $$ I ...
13
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1answer
272 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 ...
45
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3answers
4k views

An NP-complete variant of factoring.

Arora and Barak's book presents factoring as the following problem: $\text{FACTORING} = \{\langle L, U, N \rangle \;|\; (\exists \text{ a prime } p \in \{L, \ldots, U\})[p | N]\}$ They add, further ...
25
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1answer
1k views

An edge partitioning problem on cubic graphs

Has the complexity of the following problem been studied? Input: a cubic (or $3$-regular) graph $G=(V,E)$, a natural upper bound $t$ Question: is there a partition of $E$ into $|E|/3$ parts of size $...
14
votes
2answers
550 views

Add a matching to a Hamiltonian path to reduce the max distance between given pairs of vertices

What is the complexity of the following problem? Input: $H$ a Hamiltonian path in $K_n$ $R \subseteq [n]^2$ a subset of pairs of vertices a positive integer $k$ Query: is there a matching $M$ such ...
14
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5answers
1k views

Exact Algorithms for r-Dominating Set on Bounded Treewidth Graphs

Given a graph, $G = (V, E)$, I want to find an optimal $r$-domination for $G$. That is, I want a subset $S$ of $V$ such that all vertices in $G$ are at a distance of at most $r$ from some vertex in $S$...
9
votes
0answers
358 views

What are the best known reductions from SAT to CNF-SAT?

Problems Let SAT denote the following problem: Given a boolean formula, does there exist a satisfying assignment? Let CNF-SAT denote the following problem: Given a ...
13
votes
1answer
620 views

Are there interesting graph classes where the treewidth is hard (easy) to compute?

Treewith is an important graph parameter that indicates how close a graph is from being a tree (although not in a strict topological sense). It is well known that computing the treewidth is NP-hard. ...
13
votes
4answers
546 views

Can we fast generate perfectly uniformly mod 3 or solve NP problem?

To be honest, I don't know that much about how random number are generated (comments are welcome!) but let's assume the following theoretical model: We can get integers uniformly random from $[1,2^n]$ ...
3
votes
1answer
192 views

A maximization problem containing summation and multiplication

Are the following two problems NP-hard? Problem 1 Given $n$ ordered pairs of integers $S=\{(a_i,b_i)\}$, $1\leq i \leq n$, and an integer $k$. Find a subset $A$ of $S$ with $k$ elements, such that $...

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