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

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1answer
70 views

How hard is this constrained $n$-rooks problem?

I asked this over on math.stackexchange.com, then I found out about this forum. Suppose you have an $(n\times n)$-chessboard, together with a constraining function $C : n \times n \to 2$ where ...
1
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1answer
55 views

Set cover problem with constant size subset

Consider a variation of the set cover problem in which the size of the subsets is no larger than a constant $k$. Is this variation still NP-hard?
2
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1answer
31 views

Prove that monotone boolean satisfiability is NP-Complete

I am to prove that monotone boolean formula satisfiability checking when at most k variables are set to 1 is an NP-Complete problem. Proving that it is in NP is easy, but I'm having difficulty ...
1
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1answer
32 views

if $L\in NP\cap Co-NP$ is NP-Hard, then $NP=Co-NP$

I'm looking for a proof to the claim stated in the title: if $L\in NP\cap Co-NP$ is $NP$-Hard, then $NP=Co-NP$. I read the proof from my professor's recitation, but couldn't understand it, and I was ...
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2answers
70 views

Is the Calibron 12 puzzle NP-hard?

So, I was analyzing the Calibron 12 puzzle and to me it looks like a bin-packing problem. Is this puzzle actually a bin-packing problem and thus NP-hard for the perfect solution? Basically, you can ...
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2answers
54 views

Relationship between an NP-hard problems with the subsets of them?

I am writing a paper. I have a problem and I want to prove that it is an NP-hard problem. However, for simplicity, I select a subset from my problem to prove that it is an NP-hard problem. Although I ...
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0answers
25 views

A polynomial reduction from HAMPATH to LONG-PATH [duplicate]

$\text{HAMPATH} = \{(G=(V,E),s',t')| \text{ G has a Hamilton path from s' to t' } \}$ $\text{LONG-PATH} = \{(G,s,t,k) | \text{G has a simple path p from s to t, length(p) $\geq$ k} \}$ I'm trying ...
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2answers
32 views

Finding a 4-clique among $k$ node groups

Given a connected graph $G = (V,E)$, assume that there are partitions $\{p_0, p_1, ..., p_k\}$. Denote the partition set of a vertex $v \in V$ as $p(v)$. The neighborhood of a vertex $v$ is denoted as ...
-2
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1answer
49 views

NP hard: Mixed Q Horn SAT

Prove that Mixed Quantified Horn SAT problem is NP hard by reducing the Q3SAT problem to it. Q3SAT: 3SAT with possibly universally and existentially quantified variables. Mixed Quantified Horn ...
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1answer
61 views

Set of $\mathsf{NP}$-hard languages closed under set inclusion?

As the title says, my question is whether the set of $\mathsf{NP}$-hard languages is closed under set inclusion, i.e. whether for any $\mathsf{NP}$-hard language $L$, all subsets of $L$ are also ...
3
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1answer
79 views

Simplest argument that language decidable in constant time cannot be $\mathsf{NP}$-hard?

My question is specifically about $\emptyset$, but more generally about any language that can be decided in (deterministic or nondeterministic doesn't really make a difference here) constant time. ...
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1answer
41 views

NP hardness of Partition

I'm trying to show that PARTITION is NP-hard. I'm not sure if what I have is correct so I'll write what I have. I tried to reduce it from SUBSET_SUM: $$PART= \{S\subset\mathbb{Z}|\exists C \subset S: ...
2
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1answer
110 views

Are there any PSPACE problems that don't exist in NP-Hard?

The question is in the title, I suppose. I am studying complexity classes, and I understand that NP-Hard is the set of problems that are at least as hard as the hardest problems in NP. Therefore, it ...
3
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1answer
29 views

Minimum weighted arithmetic mean partion?

Assume I have some positive numbers $a_1,\ldots,a_n$ and a number $k \in \mathbb{N}$. I want to partition these numbers into exactly $k$ sets $A_1,\ldots,A_k$ such that the weighted arithmetic mean ...
3
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1answer
30 views

Hardness of approximating hitting set

Consider the hitting set problem with $n$ elements and $m$ sets. I gather from the linked page as well as this that 1) it is NP-hard to approximate the cost of the optimal solution to a ...
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0answers
28 views

Prove that Acyclic Subgraph is NP-Hard by showing Independent Set can be reduced to Acyclic Subgraph

I am trying to prove that the Acyclic Subgraph Problem (AS) is NP-hard by showing that the Independent Set Problem (IS) is polynomially reducible to AS. AS is as follows: Given a directed graph G = ...
3
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1answer
53 views

How can I identify that a restricted variant of Boolean SAT remains hard or not?

While I was studying SAT problem and its different instances, in Algorithms for the Satisfiability (SAT) Problem: A Survey by J. Gu et. al PDF, I came up with this variant (not mentioned there, but I ...
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1answer
103 views

What is the asymptotic runtime of the best known TSP solving algorithm?

I always thought that TSP currently requires time exponential in the number of cities to solve. How, then, has Concorde optimally solved a TSP instance with 85,900 cities?!? Is this a typo? Is ...
1
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1answer
94 views

3-SAT to Max-2-SAT Reduction

I'm trying to find reduction from 3-SAT to Max-2-SAT, so far no luck. Let me first describe it. 3-SAT: Given a CNF formula $\varphi$, where every clause in $\varphi$ has exactly 3 literals in ...
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0answers
21 views

Cyclic definition of NP-completeness [duplicate]

Trying to understand the concept of NP-completeness, I came across this pearl on Wikipedia: From NP-complete: A decision problem L is NP-complete if it is in the set of NP problems and also ...
2
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2answers
64 views

Is summing over all possible $k$-combinations NP-hard?

Say we have a set of numbers $A=\{a_1, a_2, \dots, a_n\}$, and we wish to sum over all possible combinations of $k$ terms to compute $$ \sum_{\substack{C \subseteq \{1,2,\dots,n\} \\ |C|=k}} \prod_{c ...
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0answers
29 views

What is the Unique Games Conjecture? [closed]

What is the unique game conjecture in relatively simple words? What are the consequences of proving it or disproving it? Does it has any relation to game theory? Why is there "game" in the name?
5
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1answer
86 views

How to compute a curious inverse

Let $M$ be a square matrix with entries that are $0$ or $1$ and let $v$ be a vector with values that are also $0$ or $1$. If we are given $M$ and $y = Mv$, we can computer $v$ if $M$ is non-singular. ...
2
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2answers
184 views

Which NPC problems are NP Hard [duplicate]

I have read that TSP and Subset Sum problems are NPC problems which are also NP Hard. There are also problems like Halting Problem which is NP Hard, but not NP Complete And Wikipedia defines this as ...
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4answers
136 views

Can all NP-complete cryptosystems be broken if one is broken?

I was just reading something about NP-hard problems and cryptosystems. I was thinking: Every NP-complete problem can be reduced to another and every NP-complete problem has an equivalent (NP-hard) ...
5
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1answer
185 views

Is building this tournament fixture an NP-Hard / NP-Complete problem?

I'm curious to know if this problem is NP-Hard / NP-Complete, which I believe would mean I'm unlikely to find a polynomial-time algorithm to solve it. I have written a program which randomly ...
0
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1answer
103 views

Relaxed graph coloring, with penalties for assigning adjacent vertices the same color

Consider a set of $N$ nodes. There is a $N\times N$ non-negative valued matrix $D$ where the $(i,j)$th element $d_{ij}$ gives the "positive metric" between node $i$ and $j$, where $i,j\in [N]$. Thus ...
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1answer
25 views

There is equivalence in an NP-hardness proof or not?

I want to show that some problem $P_1$ is NP-hard. I have a problem $P_2$ that is NP-complete. From an instance of $P_2$ I created in polynomial time an instance of the problem $P_1$. My question is: ...
2
votes
1answer
105 views

Bin packing problem or not?

Suppose I have $N$ bins and $M$ items as depicted in the figure below (3 bins and 3 items): Suppose that every bin has unit capacity and the weights of the items depend on the bins used. I want to ...
3
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2answers
106 views

minimizing the summed cardinality of set unions

this optimization problem, I am working on, is kind of making me crazy. ;) Given is a list o of sets (with finite cardinality) of strictly positive integer values ...
3
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0answers
70 views

Is finding all valid nets of a polyhedron NP-hard?

Suppose I wanted to find all valid nets of a polyhedron. Is this kind of problem NP-Hard? My guess is that it is. If you were to increase the "complexity" of the polyhedron (maybe this is the number ...
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1answer
35 views

NP hard relation with NP complete

If any problem P is NP complete then if there is a polynomial time reduction of P to another problem R then what can we say about R.Is it NP-hard or NP complete ? From Theory of computation of ...
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0answers
64 views

Is this problem a knapsack problem?

I have the following problem. Maximize $\sum\limits_{m=1}^M\sum\limits_{n=1}^N x_{mn}$ subject to: $\sum\limits_{\substack{m^\prime=1\\ m^\prime \neq m}}^M\sum\limits_{\substack{n^\prime=1\\ ...
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1answer
160 views

Efficiently pick a largest set of non-intersecting line segments

Given a set of line segments, how do we identify a subset of maximal cardinality where all line segments are pairwise non-intersecting? Brute force we would get $2^n$ sets to check where $n$ is the ...
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1answer
57 views

How do we know that all NP problems reduce to NP-hard problems? [duplicate]

For example, how is it proven that any NP problem can reduce to subset sum, circuit satisfiability, etc.? Or could you link to a proof?
2
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1answer
109 views

3-SAT problem with number of clauses equal to number of variables

Consider the 3-SAT problem where the formula is in conjunctive normal form and we restrict the Boolean formulas such that the number of clauses in the formula is equal to the number of variables. Is ...
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1answer
114 views

Is 0-1 integer linear programming NP-hard when $c^T$ is the all-ones vector?

Karp's 21 NP-complete problems show that 0-1 integer linear programming is NP-hard. That is, an integer linear program with binary variables. If we set the $c^T$ vector of the objective $\text ...
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2answers
147 views

P vs NP: Assuming P = NP

Lets assume $P = NP$. Can we say if every language $L \in P$, then $L \in NPC$? I read $P \subseteq NP$, which means that $L\in NP$. So I know for example, that a language can be $NP \text{ hard}$, ...
3
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1answer
209 views

How to prove the NP-completeness of the ``Exact-3D-Matching`` problem by reducing the ``3-Partition`` problem to it?

Background: The Exact-3D-Matching problem is defined as follows (The definition is from Jeff's lecture note: Lecture 29: NP-Hard Problems. You can also refer to ...
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1answer
52 views

NP-hardness given some reducible language

While reading a passage in an older textbook I came upon this problem which I am having difficulty in justifying whether its true or false. Is this possible? If some problem $A$ is NP-hard, and if ...
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1answer
127 views

Prove the red blue separation problem is NP-complete

Consider the following problem: given a set of $m$ red points and $n$ blue points in the plane, find a minimum length cycle that separates the red points from the blue points. That is, the red points ...
2
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1answer
272 views

Question on SAT reduction

Let Two-Solutions-SAT be the language of Boolean formulas that have exactly two distinct satisfying assignments. Show Two-Solutions-SAT is co-NP-hard. I know how to show that the complement of ...
2
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1answer
71 views

Proving $ \{ \langle D_1, … ,D_K \rangle : \text{ where } D_i \text{ are DFAs and } {\bigcap}_{i=1}^k L(D_i) = \emptyset \} $ is NP-Hard

The question (Prove L is NP-hard) was about proving that the following language is NP-hard: $$ L = \{ \langle D_1, D_2, ... ,D_K \rangle : k \in {N}\text{, the } D_i \text{ are DFAs and } ...
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3answers
221 views

Having trouble proving a language is NP-complete

I'm asked to prove that, if P=NP, that 0*1* is NP-complete, but I'm having trouble going about doing it. I know it's fairly easy to prove it's NP by creating a TM to verify an input (which can be done ...
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1answer
83 views

Prove L is NP-hard

I have no clue how to prove this question. Consider the language $L = \{ \langle D_1, D_2, ... ,D_K \rangle : k \in {N},$ the $D_i$ are DFAs and ${\bigcap}_{i=1}^k L(D_i) = \emptyset \}$ Prove ...
5
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1answer
139 views

Can Euclidean TSP be exactly solved in time better than (sym)metric TSP?

Symmetric/Metric TSP can be solved via the Held-Karp algorithm in $\mathcal O(n^2 2^n)$. See A dynamic programming approach to sequencing problems by Michael Held and Richard M. Karp, 1962. In Exact ...
3
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1answer
68 views

Transforming SAT to Quadratic Programming in polynomial time

I would like to show that Quadratic Programming is NP-hard. I am currently reading a couple of papers which state that QP is NP-Hard and prove it by transforming SAT to QP, however I am finding the ...
2
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1answer
142 views

MIN-2-XOR-SAT and MAX-2-XOR-SAT: are they NP-hard?

What is the complexity of MIN-2-XOR-SAT and MAX_2-XOR-SAT? Are they in P? Are they NP-hard? To formalize this more precisely, let $$\Phi\left(\mathbf x\right)={\huge\wedge}_{i}^{n}C_i,$$ where ...
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1answer
97 views

Is MIN or MAX-True-2-XOR-SAT NP-hard?

Is there a proof or reference that $\left\{\text{MAX},\text{MIN}\right\}\text{-True-2-XOR-SAT}$ is $NP$-hard, or that it (the decision version) is in $P$? Let: $$\Phi\left(\mathbf ...
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2answers
117 views

NP-hardness and FPTAS

I have a problem in understanding how to prove the following question. Let $Q = \langle\max,f,L\rangle$ be an NPO-Problem, where $f$ only supports integers. Define $$L_Q^* =\{(x_0,1^k) : \exists x . ...