Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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List of strongly NP-hard problems with numerical data

I am looking for strongly NP-hard problems for a reduction. So far I have found the following problems: 3-partition problem bin-packing problem Numerical 3-dimensional matching TSP Any NP-complete ...
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2answers
3k views

Hamiltonicity of k-regular graphs

It is known that it is NP-complete to test whether a Hamiltonian cycle exists in a 3-regular graph, even if it is planar (Garey, Johnson, and Tarjan, SIAM J. Comput. 1976) or bipartite (Akiyama, ...
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2answers
513 views

Testing whether letters can be scheduled to achieve a word in a regular language

I fix a regular language $L$ on an alphabet $\Sigma$, and I consider the following problem that I call letter scheduling for $L$. Informally, the input gives me $n$ letters and an interval for each ...
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3answers
662 views

Is it hard to find optimal addition chains?

An addition chain is a sequence of positive integers $(x_1, x_2, \dots, x_n)$ where $x_1 = 1$ and each index $i\ge 2$, we have $x_i = x_j + x_k$ for some indices $1\le j,k < i$. The length of the ...
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2answers
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Are there known NP-complete problems, neither NP-hard in the strong sense nor having pseudopolynomial algorithm?

In their paper (p. 503) Garey and Johnson remark: ... there could exist an NP-complete problem which is neither NP-complete in the strong sense nor solvable by a pseudo-polynomial time algorithm ......
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3answers
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Complexity of edge coloring in planar graphs

3-edge coloring of cubic graphs is $NP$-complete. Four Color Theorem is equivalent to "Every cubic planar bridgeless graphs is 3-edge colorable". What is the complexity of 3-edge coloring of cubic ...
13
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2answers
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H-free partition

This is a question inspired by the H-free cut problem. Given a graph, a partition of its vertex set $V$ into $r$ parts $V_1, V_2, \ldots, V_r$ is $H$-free if $G[V_i]$ does not induce a copy of $H$ for ...
13
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1answer
741 views

Intermediate $\mathsf{NP}$-complete problems?

Partition problem is weakly NP-complete since it has polynomial (pseudo-polynomial) time algorithm if input integers are bounded by some polynomial. However, 3-Partition is strongly NP-complete ...
12
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2answers
980 views

Reducing P vs. NP to SAT

The following question uses ideas from cryptography applied to complexity theory. That said, it is a purely complexity-theoretic question, and no crypto knowledge whatsoever is required to answer it. ...
12
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3answers
869 views

Edge-partitioning cubic graphs into claws and paths

Again an edge-partitioning problem whose complexity I'm curious about, motivated by a previous question of mine. Input: a cubic graph $G=(V,E)$ Question: is there a partition of $E$ into $E_1, E_2, \...
12
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1answer
944 views

NP-hardness of a special case of Number Partitioning

Consider the following problem, Given a set of $n = k m$ positive numbers $\{ a_1, \dots, a_n \}$ in which $k \ge 3$ is a constant, we want to partition the set into $m$ subsets of size $k$ so that ...
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2answers
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Complexity of finding 2 vertex-disjoint $(|V|/2)$-cycles in cubic graphs?

I posted this on mathoverflow but with no luck: Finding a connected 2-factor is $NP$-complete since it is equivalent to the Hamiltonian cycle problem. I'm interested in the complexity of finding two ...
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3answers
474 views

Graph classes in which CLIQUE is known to be NP-hard?

Given a graph $G$ and a positive integer $k$, the CLIQUE problem asks if $G$ contains a clique (complete subgraph) on at least $k$ vertices. This problem is long known to be NP-complete --- in fact, ...
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1answer
592 views

Hardness of node partitioning under shortest path constraint

Given a direct graph $G=(V,E)$. $\forall (i,j) \in E$, there is a weight $w(i,j) \in R$ (negative weight is possible). A label $l(i)$ is associated with each node $i \in V$. How to assign $k$ (or less)...
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2answers
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Is the half-filled magic square problem NP-complete?

Here is the problem: We have a square with some numbers from 1..N in some cells. It's needed to determine if it can be completed to a magic square. Examples: ...
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2answers
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Any fast algorithm for minimum cost feedback arc set problem?

In a directed graph, $G=(V,E)$, $F\subset E$, if $G\setminus F$ is a DAG(directed acyclic graph), $F$ is called a feedback arc set. If each edge is associated with a weight $w$, the minimum cost ...
11
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1answer
283 views

Minimum weight subforest of given cardinality

This question was motivated by a question asked on stackoverflow. Suppose you are given a rooted tree $T$ (i.e. there is a root and nodes have children etc) on $n$ nodes (labelled $1, 2, \dots, n$). ...
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 ...
11
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1answer
544 views

What is the complexity of (possibly succinct) Nurikabe?

Nurikabe is a constraint-based grid-filling puzzle, loosely similar to Minesweeper/Nonograms; numbers are placed on a grid which is to be filled with on/off values for each cell, with each number ...
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3answers
691 views

Could there be an extremely large hidden subset of Polynomially solvable problems within NP-Complete problems?

Suppose P != NP. We know that we can make easy instances of 3-SAT at any time. We can also generate what we believe to be hard instances (because our algorithms can't solve them quickly). Is there ...
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1answer
336 views

Is there a FNP problem that's NP-hard but not FNP-hard?

For the reductions, choose a class C such that [it's clear what FC means] and FC is not known to be able to solve the satisfaction search problem, and assume that FC indeed can't solve that search ...
22
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1answer
653 views

Does NP-hardness imply P-hardness?

If a problem is NP-hard (using polynomial time reductions), does that imply that it is P-hard (using log space or NC reductions)? It seems intuitive that if it is as hard as any problem in NP that it ...
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 ...
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10answers
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Problems that are easy on unweighted graphs, but hard for weighted graphs

Many algorithmic graph problems can be solved in polynomial time both on unweighted and weighted graphs. Some examples are shortest path, min spanning tree, longest path (in directed acyclic graphs), ...
21
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1answer
886 views

Is finding Logspace reductions harder than P reductions?

Motivated by Shor's answer related to different notions of NP-completeness, I am looking for a problem that is NP-complete under P reductions but not known to be NP-complete under Logspace reductions (...
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2answers
646 views

H-free cut problem

Suppose you are given a connected, simple, undirected graph H. The H-free cut problem is defined as follows: Given a simple, undirected graph G, is there a cut (partition of vertices into two ...
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2answers
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What is the following variation on Set Cover known as?

What is the following variation on set cover known as? Given a set S, a collection C of subsets of S and a positive integer K, do there exist K sets in C such that every pair of elements of S lies in ...
13
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4answers
522 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 $[1,2^n]$ ...
13
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3answers
896 views

Transitive feedback arc set (TFAS): NP-complete?

Some time ago, I posted a reference request for graph problems where we want to find a 2-partition of the edges where both sets fulfill a property not related to their cardinality. I was trying to ...
11
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1answer
392 views

Computation of max H-free sets

In a graph, an independent set is a vertex subset which doesn't contain an edge as an induced subgraph. The problem of finding largest independent sets in a graph is a fundamental algorithmic question,...
10
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1answer
634 views

Probability of generating a desired permutation by random swaps

I'm interested in the following problem. We're given as input a "target permutation" $\sigma\in S_n$, as well as an ordered list of indices $i_1,\ldots,i_m\in [n-1]$. Then, starting with the list $L=...
10
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0answers
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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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0answers
163 views

What relations are there between a problem hardness and the hardness of verifying a witness?

I had some hard times trying to formulate the question, so I'll start with some examples: Suppose you are given a Dominating Set instance, $<G,k>$. Now suppose I give you a set of vertices $D$ ...
6
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1answer
212 views

Graph class with easy chromatic number, but NP-hard coloring

Is there a graph class for which the chromatic number can be computed in polynomial time, but finding an actual $k$-coloring with $k=\chi(G)$ is NP-hard? Without any further restriction the answer ...
6
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1answer
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Is integer factorization an NP-complete problem? [duplicate]

Possible Duplicate: What are the consequences of factoring being NP-complete? What notable reference works have covered this?
5
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3answers
441 views

Partitioning a segmented stick

Problem : We are given a stick partitioned into n - equal parts. Each of these parts has a weight, let's say x. Number of times x appears as weight of some part is guaranteed to be even. For ...
4
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1answer
146 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? ...
4
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1answer
419 views

Is This Scheduling Problem NP-Hard?

The scheduling problem (arising from distributed computing) is defined as a decision problem as follows: Instance: A trace is comprised of $n$ processes histories (denoted $p_0, p_1, \ldots, p_{n-...
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0answers
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Showing hardness of maximizing stochastic objective function over graph

Consider a graph $G = (V, E)$ with $n$ vertices and $m$ edges. Each vertex $v_i$ can take positive value $a_i$ with probability $p_i$ and value $0$ with probability $1-p_i$. The challenge is to ...
2
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1answer
157 views

Computational complexity of finding a (1D deterministic) CA that achieves a desired spacetime history “patch” after $t$ iterations

Question description: Consider the problem of finding a minimum $n$-color $k$-state one-dimensional cellular automata (minimizing $k$ for some fixed value of $n$ or vice versa), with communication ...
17
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1answer
916 views

What is the complexity of this edge coloring problem?

Recently, I have encountered the following variant of edge coloring. Given a connected undirected graph, find a coloring of the edges that uses the maximum number of colors while also satisfying ...
13
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2answers
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Lexicographically minimal topological sort of a labeled DAG

Consider the problem where we are given as input a directed acyclic graph $G = (V, E)$, a labeling function $\lambda$ from $V$ to some set $L$ with a total order $<_L$ (e.g., the integers), and ...
7
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1answer
922 views

Is it NP-hard to _play_ minesweeper perfectly?

This paper shows that it is NP-hard "to determine if there is some pattern of mines in the blank squares that give rise to the numbers seen." If there is a way to "lead a perfect player into" such ...
5
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1answer
430 views

Comparing $\mathbf{NP}$ and $\mathbf{E}$

We know that $\mathbf{NP} = \mathbf{NTIME}(n^{O(1)})$ and $\mathbf{E} = \mathbf{DTIME}(2^{O(n)})$. The complexity zoo states that $\mathbf{E}$ does not equal $\mathbf{NP}$, and cites the following ...
5
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1answer
340 views

A reachability problem

Let $P$ be a length-preserving (i.e. $|P(x)|=|x|$) polynomial-time computable program. I. Given two strings $x$ and $y$, we want to decide if $y$ can be obtained by repeated applications of $P$ ...
4
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1answer
179 views

Example of a function problem which is $\mathrm{FP}^{\mathrm{NP}}[wit, log]$-hard?

The usage of an $\mathrm{NP}$-oracles which delivers a witness has been proposed for example in [Buss1995]. I would like to see an example of an $\mathrm{FP}^{\mathrm{NP}}[wit, log]$-hard problem. Can ...
4
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1answer
405 views

Proof that the graph optimization problem is NP-hard

I'm trying to prove that the following optimization problem is NP-hard: Given a graph $G=(V,E)$, non-negative vertex weight functions $w(v)$ and $s(v)$, and a non-negative edge weight function $t(u,v)...
4
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3answers
492 views

bin packing with overlapping objects

I have $N$ bins with capacity $M$ and $k$ objects with size $s_i$. The goal is to pack these objects in the bins. Until now it is similar to the bin-packing problem. But the twist is that each object ...
3
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1answer
204 views

Is the following optimization problem (a variant to a previous problem) NP-hard?

This problem is a following up question on this one. The only difference is the newly added constraint in the bold font. Set S, which is an non-empty finite subset of $\{ (i,j) : i, j \in N \land i \...
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2answers
475 views

Is the following optimization problem NP-hard?

Set S, which is an non-empty finite subset of $\{ (i,j) : i, j \in N \land i \neq j \}$, is given. E.g. $S=\{(1,3), (2,3), (1,4), (2,4), (3,1), (3,4)\}$ . For each element $(i,j)$, we have weight $w_{...