# Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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• 1,049
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### Shortest path with affine updates and fixed dimension

One may look at the shortest path problem on a weighted directed graph with weights on $\mathbb{Q}$ as the problem of minimizing a rational value $x$ which is updated at each edge of the graph with ...
• 1,049
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### A Travelling Salesman variant where the next distance depends on distance travelled so far

The travelling salesman problem can be seen as a problem of selecting a permutation on $\{1,\ldots,n\}$ of minimun length, where the length of a permutation $\sigma$ is determined by pairwise ...
• 2,262
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### On The Complexity of Block-Interchange Distance for Binary Strings

The block-interchange distance problem is defined as finding the minimal number of subsequences swaps to apply to an input string to turn it into a desired string. It is a well studied tractable ...
361 views

### Is sorting NP-complete?

SORTING problem. Input: A poset which corresponds to a partially sorted list of different numbers. Output: Number of pairwise comparisons needed (in the worst case) to get a completely sorted array. ...
• 14k
217 views

### Is Optimal Swap Sorting NP-Hard?

Given an array of integers with duplicates, find the minimum number of swaps to sort the array. According to this question, the problem is NP-Complete but the reference given proves NP-Completeness ...
32 views

### Compute a feasible schedule for scheduling on identical parallel machines?

I am considering the offline version of identical parallel machine scheduling with arrival time and deadlines while allowing preemption and no assumption regarding the agreeability of the arrival and ...
• 101
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### Complexity of finding a room-partitioning

In a paper by Jack Edmonds and Laura Sanità (link: https://www.sciencedirect.com/science/article/pii/S1571065310001605) the following intriguing result is given: Given a triangulation $T$ of $3n$ ...
• 1,336
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### Set Cover with Multiple covers

I am interested in whether a set cover instance that covers all elements $q$ times may have the property that every sufficiently small subset of this set cover will not cover the elements even once. ...
• 412
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### Is this problem on unambiguous DNFs hard?

Call a DNF formula $\varphi = \bigvee_{i=1}^n C_i$ unambiguous if for every $i\neq j$, $C_i \land C_j$ is unsatisfiable. In other words, the disjunct $C_i$ contains some literal $l$ and $C_j$ contains ...
• 1,429
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### maximum independent set in graphs with small number of edges

For the classic maximum independent set problem, a hardness of approximation result of $n^{1-\varepsilon}$ is known by [Hastad, 1996] assuming $\textsf{NP} \not \subseteq \textsf{ZPP}$, where $n$ is ...
• 412
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### What's the complexity of the "decision version" of counting the paths in a graph?

I learned that "counting the simple paths in a graph(whether directed or not)" is #P-Complete. I'm wondering what the complexity is for its decision version. Here are two types I'm ...
96 views

### Complexity of Identifying SAT Problems with a Unique Solution from Satisfiable Instances

I am curious about the computational complexity involved in identifying SAT problems that have only one solution from a set of satisfiable SAT instances. input and output: input: A satisfiable cnf ...
• 316
1 vote
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### Hardness for find the clause for statisfiable 3-SAT problems

The 3-SAT problems are known to be NP-complete so the decision problems are believed to be non efficiently solvable unless P=NP. Yet, there are cases where the satisfiability can be answered such as ...
• 111
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### What’s the complexity of this decision problem with bit shifting?

I’ve been wondering about the computational complexity of a problem that involves bit shifting. Let me define some notation before I present the problem. If $\langle{b}\rangle$ is a bitstring ...
• 96
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### Hardness of 3-Partition with Small Target Value

In the 3-partition problem, we are given a set of positive integers $a_1,\ldots,a_n$ and a target value $T$; the goal is to decide if there is a partition of the numbers to triplets such that the sum ...
• 412
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### The hardness of active learning with fixed budget

I have been looking for theoretical papers studying this question of the hardness of PAC active learning algorithms. I found a few papers studying the problem from a fixed perspective (particular ...
• 65
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### Is this variant of Facilities location problem a NP-hard problem?

Given a set of locations $P=\{p_1,p_2,\dots\}$ and a set of facilities $F=\{f_1,f_2,\dots\}，|F|\ge k$ on a plane. We want to partition the facilities into $k$ disjoint subsets (each subset has at ...
• 21
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### Certifying the promise in hard promise problems

Do we happen to know of any promise problems where the problem is both conditionally hard (say, NP-hard) while simultaneously being able to certify that the instance satisfies the given promise? For ...
• 987
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### Could we build PSPACE-based cryptography - more secure post-quantum?

It seems not safe to exclude possibility of e.g. some next generation quantum computers being able to attack NP problems (e.g. 2WQC) - so maybe it is worth to start thinking of shifting the ...
• 325
1 vote
92 views

### Generalization of the Hamiltonian path problem on Grid Graphs

Fix a cost to each of these actions: move up, move down, move left, move right. I.e. fix some function $f: \{\text{move up, move down, move left, move right}\} \to \mathbb N$. Define the following ...
• 31
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### Is this edge-partitioning NP-Hard?

Let $G = (V,E)$ be an undirected graph with $m = |E|$ edges (assume that $m = 3t$ for some $t \in \mathbb{N}$). Problem: Partition $E$ to $q = \frac{m}{3}$ sets $S_1,S_2,\ldots, S_q \subseteq E$ sets ...
• 412
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### Is $\mathsf{NP}\subseteq\mathsf{NSPACE}(n)$?

It is well-known that $\mathsf{P}\neq\mathsf{SPACE}(n)$, either for $\mathsf{SPACE}=\mathsf{DSPACE}$ or $\mathsf{NSPACE}$, and it is conjectured that both $\mathsf{P}\not\subseteq\mathsf{DSPACE}(n)$ ...
• 131
289 views

### Complexity of 1-or-3-in-3-SAT (odd-3-SAT)

Consider a 3-CNF formula $\Phi$, i.e., a conjunction of clauses of 3 literals. I call odd-SAT (or 1-or-3-in-3-SAT) the problem of checking whether there is an assignment of the variables such that ...
• 9,547
35 views

### Hardness of finding minimal subsets that will change the maximum of a univariate polynomial

Given a univariate polynomial of the form $p(x) = \prod_{0 \leq i \leq N}{(x*a_i + b_i)}$ when all of the $a_i$ and $b_i$ are numbers in the range [-1,1] and $i$ goes from $0$ to $N$ (we are given all ...
1 vote
80 views

### Hardness of maximization of a univariate polynomial (as a function of its degree)

Given a univariate polynomial of the form $p(x) = \prod_{i}{(x*a_i + b_i)}$ when all of the $a_i$ and $b_i$ are numbers in the range [-1,1] and $i$ goes from $0$ to $N$. What is the complexity of ...
69 views

### Enumerating all set covers with sets of size at most two

I am working on enumerating all the set covers (need not be minimal). A branching algorithm runs in $O^*(1.2353^{|U|+|S|})$ time that branches on all the sets of size at least three. As the branching ...
94 views

### Prove that Vertex Cover is NP-Complete by reducing MaxCut to Vertex Cover

This is not the most straight forward reduction available on the internet since most people start from the fact that vertex cover is NP-complete and reduce a given vertex cover instance to MaxCut ...
1 vote
105 views

### Complexity of n-rooks completion

I am motivated by the post, Complexity of n-queens-completion. I am interested in completion problem of non-attacking rooks on a chessboard. Input: Given a chessboard of size $n*n$ with $n-k$ rooks ...
158 views

### Complexity of numerical 3-dimensional matching where the three multisets are identical

Consider the following problem: Input: one multiset $S = \{s_1, \ldots, s_n\}$ of positive integers written in unary Output: can we build $n$ triples that sum to the same value, using each element of ...
• 9,547
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### clique problem in graphs with bounded degree

Is the problem of finding a clique of size $d$ in a graph of maximum degree $d$ NP-complete ($d$ part of the input)?
213 views

### Which 1-player games are EXPTIME-complete? Also, are there any known games that are EXPSPACE-complete?

I noticed a lot of 1-player games have been shown to be NP-Hard, like Pac-Man, The World's Hardest Game, Tetris, etc. For PSPACE-Complete, I noticed that Wikipedia listed these 1-player games: It is ...
• 33
223 views

### Complexity of solving a higher-order degree polynomial equation? P-problem or NP-problem or neither?

I am a mathematician and I am very new to theoretical computer science. The definition of P/NP problem I found in wiki is that: P is the set of decision problems solvable in polynomial time by a ...
304 views

### Complexity of permanent verification

Consider the problem of permanent verification: $\bullet \$ Given a $n\times n$ matrix $A$ with entries in $\{0,1\}$, and given $k\ge 0$, does $Per(A)=k$? Question: Is it known to be NP-hard? Should ...
• 812
1 vote
106 views

### Is Maximum Independent Set polynomial-time solvable in $(p,1)$-colorable graphs for general $p$?

It is well-known that Maximum Independent Set (MIS) in bipartite graphs is polynomial-time solvable. What happens if we generalize the input graphs by replacing the vertices in one partite with ...
• 421
72 views

### Is this Knapsack/Subset Sum Variant NP-Hard?

The problem: Let $A_1 = \{a^1_1,\ldots,a^1_n\}, A_2 = \{a^2_1,\ldots,a^2_n\}, \ldots, A_k = \{a^k_1,\ldots,a^k_n\} \subset \mathbb{N}$ be $k$ sets of $n$ integers, and let $U,L \in \mathbb{N}$ be ...
• 412