Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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Is mathematical proof itself NP-hard?

At the 8:00 mark of this video, he claims that proving things is itself an NP problem. I'm looking for more insight into this. Could someone help explain this concept to me and also provide a link to ...
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1answer
48 views

Minimizing the gaps with incremental capacity

There are a single job, a machine and a set of $n$ slots. The machine has a capacity that increments by $\zeta(t)$ every slot $t=1,2,\ldots,n$. Initially (before the first slot), the machine has 0 ...
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83 views

On the paper “Quantum Computing Hamiltonian cycles”

The paper Quantum Computing Hamiltonian cycles claims: An algorithm for quantum computing Hamiltonian cycles of simple, cubic, bipartite graphs is discussed. It is shown that it is possible to evolve ...
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1answer
111 views

Recent progress on the next-to-shortest-path problem for directed graphs?

In the paper "Computing strictly-second shortest paths" (1997), Lalgudi and Papaefthymiou consider the following problem: Let $G$ be a directed graph with edge-weighting $w$. Let $u,v$ be vertices in ...
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1answer
78 views

Choosing one number from each set so that the sum of squares of each distinct number counts is minimized

Problem is as follows: We are given $K$ subsets of $\{1,2,...,n\}$. We need to pick one number from each of these subsets such that $\sum_{i=1}^n p_i^2$ is minimized where $p_i$ is the number of times ...
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1answer
176 views

Conjecture about ASP reductions between NP-complete problems

$ASP$-complete reductions, introduced by Ueda and Nagao, relate the hardness of computational problems in $FNP$. Basically, $ASP$-reduction is a polynomial time reduction between instances and a ...
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36 views

Would a machine learning algorithm benefit from an “optimization oracle”?

I'm trying to understand the behavior of machine learning algorithms where the loss function is non-convex and the problem of training the ML on a specific data set is computationally hard. Now let'...
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1answer
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Polynomial-time reducibility of Primality and 3-SAT

Is 3-SAT $\leq_{p}$ Primality? And/or is Primality $\leq_{p}$ 3-SAT? I think the answer is no and yes, respectively, but I'm not sure. Any help would be appreciated. Thank you.
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Topological sorting of a DAG where special vertices have to come in even groups

Consider the following problem. The input is a directed acyclic graph (DAG) $G = (V, E)$, and a subset $V' \subseteq V$ of vertices, which we call special vertices. The question is to determine ...
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74 views

NP-Hard or PTIME?

I am working on my research problem that essentially boils down to the following question. Consider an $N \times N$ matrix. There is a man at given a starting point $(x,y)$. In each unit of time, the ...
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1answer
192 views

Is $\{0,1\}$-Vector bin packing NP-Hard when vectors have constant dimension?

The paper https://cs.brown.edu/people/seny/pubs/vbponline.pdf discusses $\{0,1\}$-Vector Bin packing in the online setting and give lower bounds. However, they do not mention anything about the ...
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59 views

Is it possible to approximate the solution of NP-Hard problems in polynomial time using linear programming? [closed]

Suppose we have a NP-Hard problem such as the k-col, which is meant to determine if a graph may be colored using at most ...
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1answer
85 views

NP hard proving: separate graph into a set of the same size disjoint parts by maximizing the shared neighbours of each part

Given a graph $G=\{V,E\}$ where $V$ denotes the nodes and $E$ denotes edges. The size of the node $|V| = nk$. The target is to separte the graph into $n$ disjoint parts $P=\{V_i\}_{i=1}^n$ and the ...
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1answer
70 views

Interval partitioning with restrictions: NP-complete or efficiently solvable?

The interval partitioning problem can be solved efficiently using a greedy algorithm. However, adding restrictions on the interval assignment to the problem results in a problem that appears harder. ...
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Uniformly sampling or counting connected graph partitions with any number of blocks

Question: Is it possible to uniformly sample in polynomial time from the set of all connected partitions of a graph? Or is there a JVV type argument that proves this to be NP-hard? To clarify: By a ...
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1answer
67 views

3-dimensional matching variant

In the normal version of the matching problem, we are given a set of vertices $X$, $Y$, and $Z$, each of size $n$, and a set of edges $E\subseteq X\times Y\times Z$. We need to find a matching $M\...
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1answer
110 views

What is the complexity of this weighted b-edge matching problem?

I'm wondering about the complexity of the following variant of the Generalized Weighted b-edge Matching problem: Input: An undirected multigraph $G = (V, E)$ without loops, an edge partition $(E_1,...
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108 views

Order of quantifiers in the definition of NP-completeness: does the reduction allow arbitrary polynomials? [closed]

Arora and Barak define NP-completeness as the following: "We say that a language $A \subseteq \{0, 1\}^∗$ is polynomial-time Karp reducible to a language $B \subseteq \{0, 1\}^∗$ denoted by $A \leq_p ...
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Complexity of a scheduling problem with a fixed left bound of jobs

Consider the following scheduling problem. We have a finite set of jobs $\mathcal{J}= \{j_1, j_2, ..., j_n\}$ to do. Every job $j_i$ has its own value $c_i$ (the amount of money we are paid for ...
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5answers
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NP-complete decision problems on deterministic automata

Do you know any NP-complete decision problems on deterministic automata? Most NP-complete problems that come to my mind are either (see, or here) graph theoretical, or involve some string rewriting or ...
7
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1answer
275 views

Complexity of constructing minimum depth decision trees

I am interested in the computational complexity of Problem 1: Given a finite, non-empty set $J$, given $A, B \subseteq \{0,1\}^J$ such that $A \cap B = \emptyset$, and given $n \in \mathbb{N}$, does ...
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1answer
102 views

is this selection problem np-hard? [closed]

Give $n$ clusters $C=\{C_i\}_{i=1}^n$ where each cluster consists of a set of similar points, i.e., $C_i=\{c_j\}_{j=1}^{|C_i|}$. The similarty between two points $c_i$ and $c_j$ is denoted as $w(c_i,...
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3answers
436 views

Best parameterized algorithm for maximum clique

I have seen the basic algorithm for the maximum clique problem parameterized by the maximum degree at an algorithms course. However, I struggle to find anything better. Searching for things like "...
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1answer
98 views

Is the counting version of 1-in-3 Sat #P-complete?

In the paper "Hard Tiling Problems with Simple Tiles", Moore and Robson prove that Cubic Planar Positive 1-in-3 Sat in NP-complete by a reduction from Positive 1-in-3 Sat. Cubic Planar Positive 1-in-...
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NP-Hard Knapsack Instances

Consider the classic Knapsack optimization problem (KP): Given $p_1, \dots, p_n, w_1, \dots, w_n, B\in\mathbb N$, compute a solution $I\subseteq \{1,\dots,n\}$, such that $\sum_{i\in I} w_i \leq B$ ...
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1answer
92 views

Determining if a word of specific length exists that is not accepted by a NFA

It is known that the problem of determining if an NFA accepts every word is PSPACE-COMPLETE, meaning it is also NP-Hard, but is this weaker version of the problem still NP-hard? Given an NFA and a ...
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60 views

Complexity of multi-objective optimization problems

How can we define and prove the worst-case complexity of multi-objective optimization problems (MOOP)? It is easy to see that, if one of the objectives is an NP-Hard optimization problem, then the ...
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1answer
134 views

The decision procedure of theory of closed real field is in NP-hard?

The decision procedure of theory of closed real field refers to https://en.wikipedia.org/wiki/Decidability_of_first-order_theories_of_the_real_numbers
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1answer
124 views

Hardness of finding if a vertex lies on a simple directed path between two vertices

Given a directed graph $G = (V, E)$ and three vertices $u, v, w \in V$. Is it NP-Hard to find whether there is a simple path from $u$ to $v$ passing through $w$? I found a couple of hardness ...
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2answers
115 views

Complexity of finding an edge set yielding specified vertex degrees

I'm trying to figure out if the following two problems are known in general to be in P or NP-complete: Q1: Given a graph $G=(V,E)$ and integers $d_i,\,1\leq\,i\leq|V|$, does there exist a subset $E'\...
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2answers
145 views

$NP$-Completeness of $\epsilon$-balanced graph partitioning for fixed $\epsilon$

Consider this graph partitioning problem: Let $G = (V, E)$ be a simple undirected graph and $0 \leq \epsilon \leq 1, M \geq 0$ be constants. Are there disjoint subsets $V_1, V_2$ with $V = V_1 \cup ...
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137 views

NP-Hardness of 4-cycle packing problem in complete bipartite digraph?

A directed complete bipartite graph is a bipartite graph where there is exactly one directed edge between any two vertices from its two different parts. In other words, it's an orientation of a ...
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1answer
713 views

Relationship between two graph optimization problems

Let $Q$ be a polynomial time computable graph property of simple, undirected graphs. Consider the following two optimization problems on any input graph: P1. Find a largest induced subgraph of the ...
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How hard is it to approximate distance of linear code

I'm trying to figure out what is the current knowledge about how hard it is, given a generating matrix of a linear code over a field $F_{q}$, approximate it's distance. I of course found that ...
2
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1answer
142 views

What Is the Complexity of This Two-to-One Matching Problem?

Given a graph $G=(V,E)$ and a function $c:V\mapsto\{1,2\}$. The function $c(\cdot)$ divides the vertices into two disjoint sets $V_1$ and $V_2$, where for all $v_1\in V_1$, we have $c(v_1)=1$ and for ...
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1answer
108 views

Example of decidable NP-hard problem that is not NP-complete [closed]

I am looking for an example of a decision problem which fulfills the following conditions: 1. It is decidable 2. It is NP-hard 3. It is not NP-complete All my search attempts yielded examples that ...
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1answer
222 views

NP-hardness of a planar SAT variant

Background: An instance of 3-SAT is called monotone if each clause consists only of positive literals or only of negative literals. Given an instance $\phi$ of 3-SAT, we consider the bipartite graph $...
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1answer
355 views

What languages can be reduced to a NP-complete problem in polynomial time

NP-complete: Language is NP-complete, when it is in NP and every problem in NP is reducible to it in polynomial time. But what languages are reducible to a NP-complete problem (for example SAT) in ...
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136 views

Optimal bee swarm plots: NP-hard?

Bee swarm plots are a way of visualizing one-dimensional data sets, similar to box plots. The idea is that if there's not too many points (e.g. <300) we can just plot them along the $x$-axis with ...
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1answer
95 views

Diameter vs. Tractability

I would like to find examples of (unweighted) optimization problems $\Pi$ that satisfy 1) or 2): 1) $\Pi$ is NP-hard but polytime solvable in the class of graphs with diameter $\le k$, for all $k\ge ...
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55 views

Complexity of Block Design?

What is known about the complexity of creating Block Designs (https://en.wikipedia.org/wiki/Block_design)? I've found one paper that creates approximately solutions using Metaheuristics that claims ...
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0answers
91 views

How hard is it to determine ex(n,G)?

Define the extremal Turán function $ex(n,G)$ of a graph $G$ as the most edges a graph on $n$ vertices can have without having a subgraph that is isomorphic to $G$. This function is known ...
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1answer
198 views

Find the shortest s-t trail(edge disjoint path) in a graph with negative weight edges

A walk in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. A trail is a walk in which all edges are distinct. Note that a trial may visit a vertex multiple times ...
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Network Reliability Problem

Network reliability, in which we are given an undirected graph $G$ with a failure probability $p_e$ for each edge and we are asked to calculate the probability that the network becomes disconnected ...
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Is unary $\Pi_2$-SUBSETSUM coNP-complete?

Consider the following problem: for given integers $a_1, \ldots, a_{2n}$ and $A$ that are given in unary representation define is it true that for every $S \subseteq \{1, ..., 2n \}$ such that $|...
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64 views

Generalization of k-Coloring: maximizing the number of vertices with no neighbours of same color

One can consider the following generalization of the $k$-Coloring problem: Let be given a graph $G$ and an two integers $k$ and $p$. A vertex $v$ of $G$ is properly colored if $v$ has no neighbour ...
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55 views

Is minimal cover under symmetric 3-deduction NP-complete?

Forgive me if this problem is known by another name, I do not know any references for it. Symmetric deduction. An equation $e \in E$ is a subset of variables $V$ such that knowing $|e| - 1$ of the ...
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1answer
122 views

Complexity of existence of simple polygonalization with prescribed area?

This is a followup on my previous question. Fekete proved the NP-completeness of deciding the existence of simple polygonalization with minimum (or maximum) enclosed area (simple polygonalization is ...
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222 views

Which monotone DNFs are evasive?

A Boolean function $\phi$ on variables $X$ is evasive if every decision tree for $\phi$ has height $|X|$. In other words, for any strategy that picks variables of $X$ and asks for their value, an ...
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215 views

A partition problem with order constraints

In the OrderedPartition problem, the input is two sequences of $n$ positive integers, $(a_i)_{i\in [n]}$ and $(b_i)_{i\in [n]}$. The output is a partition of the ...

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