Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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2
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0answers
52 views

Examples of “Sandpile” TSP Instances

This question is closely related to this MO question. I would like to know, whether any (planar Euclidean) TSP instances are known, that exhibit avalanche effects similar to those ecountered in ...
26
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3answers
880 views

Did “Where the really hard problems are” hold up? What are current ideas on the subject?

I found this paper to be very interesting. To summarize: it discusses why in practice you rarely find a worst-case instance of a NP-complete problem. The idea in the article is that instances usually ...
5
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2answers
243 views

ETH-Hardness of $Gap\text-MAX\text-3SAT_{c}$

The PCP theorem can be stated like this : There is a polynomial time reduction from SAT to $Gap\text-MAX\text-3SAT_{c}$ i.e. there is a reduction that maps an instance $\phi$ of SAT to an instance $...
2
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0answers
200 views

Graph optimization problem with multiple objectives/constraints

Let's assume that we have a directed acyclic graph $G = (V, E)$, non-negative vertex weight functions $w_a(v)$ and $w_b(v)$, and a non-negative edge weight function $t(u,v)$. We can divide vertices in ...
3
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0answers
90 views

Some questions about the Ryan O'Donnel and Yuan Zhou's paper “Approximability and proof complexity”

My question is particularly about the set-up in section $8$ (``Analysis of the KV Max-Cut instances") of the paper, https://arxiv.org/pdf/1211.1958.pdf. What they call the Khot-Vishnoi UG instance ...
3
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0answers
108 views

Can the Lasserre relaxation be defined over the reals?

If one wants to say minimize a function $f : \{-1,1\}^n \rightarrow \mathbb{R}$ on its domain then a degree$-d$ Lasserre relaxation of it would be to solve the problem of $\min \mathbb{E}_\mu [f(x)]$ ...
4
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1answer
388 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)...
3
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0answers
109 views

SOS and the small set expansion property

For what graphs do we know that their small set expansion property has a low degree SOS proof? Is this known to be true for say the complete graphs? A terminology issue about what is ``low degree" :...
2
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0answers
64 views

Is there a relationship between the probabilistic interepretation of Sherali-Adams SDP hierarchy and the Lasserre SDP hierarchy?

Firstly note this paper http://ttic.uchicago.edu/~madhurt/Papers/reductions.pdf where a Lasserre SDP is being setup for the independent set probblem at the bottom of page 4 where the author says says, ...
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0answers
27 views

How does one know what is not in a certain class of pseudo-distributions?

We consider working in the function space $\mathbb{R}^{\{ -1,1\}^n}$ where the expectation inner-product makes the juntas form a $2^n$ dimensional orthonormal basis. Now say one has found a degree $...
15
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1answer
415 views

Validity of exponentiation in a polynomial time reduction

I asked this question 10 days ago on cs.stackexchange here but I didn'y have any answer. In a very famous paper (in the networking community), Wang & Crowcroft present some $\mathsf{NP}$-...
2
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391 views

Post Correspondence Problem “binary” variant

Bounded Post Correspondence Problem is defined as follows: given list of pairs of words $ (x_1,y_1), \ldots, (x_n, y_n) $ and $K$ find sequence of indexes $i_1, \ldots, i_k$, $k \leq K$ so that $x_{...
3
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0answers
123 views

Given a matrix $A$ maximize the number of positive elements in $Ax$ under specific constraints for $x$

Let $A = [a_{ij}]$ be a symmetric matrix with nonnegative values and $k << n/2$ a given constant. We want to rearrange the columns of the matrix such that the number of rows with the following ...
5
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1answer
188 views

Finding a positive point for a collection of polynomials

I am wondering about the complexity of the following problem: Given $k$ polynomials $p_1(x_1, \ldots, x_n)$, $p_2(x_1, \ldots, x_n)$, $\ldots$, $p_k(x_1, \ldots, x_n)$ over the $n$ real ...
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182 views

At what parameters is following $NP$-hard?

Problem Instances at given $\alpha>0$. $(1)$ Given $a_1,\dots,a_{n^\alpha}\in\Bbb Z$ with $|a_i|\in(2^{n-1},2^n-1)$ is there a subset of that sums to $0$? $(2)$ Given $a_1,\dots,a_{n}\in\Bbb Z$ ...
8
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1answer
224 views

Consequences of a distillation algorithm for PSPACE

The following notion of a distillation algorithm comes from "On Problems Without Polynomial Kernels". Let a language $L$ be given. A distillation algorithm for $L$ takes a given list of input ...
5
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0answers
278 views

On the shortest vector problem (is it $NP$-complete?)

Ajtai has shown that shortest vector problem is $NP$-hard by using randomized reduction from subset sum. Has this been derandomized?
9
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0answers
304 views

Triangle arrangement problem

Suppose you are given an undirected graph $G$, with each vertex representing an equilateral triangle with sides of unit length. Does there exist an arrangement of these triangles in two dimensions (...
2
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1answer
147 views

Maximum stable matching/allocation

I checked some papers on two-side stable allocation/matching (marriage, worker/company, doctor/hospital), but has not found any literature on the following problem. Can someone point out if I missed ...
4
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1answer
187 views

Is there any relationship of hardness between the two problems?

Assuming F(x,y,D) is a function, and we can evaluate it in polynomial time with input x, y and D. Consider the problem P1: With D as input, computes $(x^*,y^*)=argmax_{(x,y)}F(x,y|D)$ where x and y ...
4
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0answers
168 views

NP-completeness of a specific topological sorting problem

Consider $(V, E)$ be a DAG, and $p_1, \dots, p_n$ be its topological sorting (i.e. such permutation $p$ of $V$ that $\forall(x, y) \in E.\ p^{-1}(x) < p^{-1}(y)$). Let's call the goodness of $p$ a ...
5
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2answers
205 views

Hardness of $k$-Plex

Definition. Given an undirected graph $G = (V,E)$, a $k$-plex is a subgraph $G'$ of $G$ such that each vertex in $G'$ is connected to at least $s - k$ other vertices in $G'$, where $s$ is the # of ...
4
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1answer
99 views

From Lasserre maps to pseudo-distributions

Let me define a ``Lasserre map of degree $d$" as a linear map $L : \mathbb{R}_n[x] \rightarrow \mathbb{R}$ i.e a real valued linear map on polynomials over $n$ variables with real coeffients. This is ...
21
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2answers
928 views

$NP$-completeness of recognizing the difference of two permutations

Shor stated, in his comment to anonymous moose's answer to this question Can you identify the sum of two permutations in polynomial time?, that it is $NP$-complete to identify the difference of two ...
1
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1answer
620 views

Positive 1-in-3 SAT FPT or Fixed Parameter Intractable

There are a number of satisfiability problems that are difficult to solve even in the fixed parameter sense. For example, Weighted q-CNF Satisfiability is W[1]-complete when parameterized by the ...
7
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1answer
154 views

What are the hardness results known for CSP over $\mathbb{F}_q$?

I found two related papers, There is a UGC hardness result here, https://www.cs.cmu.edu/~venkatg/pubs/papers/qaryCSP.pdf A kind of a stronger result might be found in these two other papers, http://...
3
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2answers
412 views

Does the problem “partition a vertex-weighted graph into $k$ balanced connected parts” have a standard name?

Consider the following problem: Given an integer $k$ and a vertex-weighted graph $G=(V,E)$, find a partition of $V$ into $V_1,\ldots,V_k$ such that each subgraph induced by $V_i$ is connected, ...
12
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4answers
1k views

Directed NP-hard problems on DAGs

Tree width measures how close a graph is to a tree. Several NP-hard problems are tractable on graphs with bounded tree width. If a problem remains NP-hard on trees then tree width cannot save us. This ...
10
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1answer
401 views

What are the complexities of the following SAT subsets ?

Assume $P \neq NP$ Let use the following notation ${}^ia$ for tetration (ie. ${}^ia = \underbrace{a^{a^{\cdot^{\cdot^{\cdot^{a}}}}}}_{i \mbox{ times}}$). |x| is the size of the instance x. Let L be ...
5
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1answer
164 views

Is sparse embedding of a NP-complete problem in a polynomial problem NP-complete?

Consider the following problem P: Input is a finite graph G. If the number of vertices in G is 2^2^i for some integer i, then output a minimum vertex cover of G; otherwise output empty set. Can I say ...
-1
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1answer
127 views

Given oracle for Max-3SAT compute clauses that cannot be satisfied

We know that Max-3SAT is NP-hard to compute exactly (and also hard to approximate to a particular constant multiplicative factor). However, suppose you are given an oracle for Max-3SAT that tells you ...
15
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3answers
1k views

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 ...
5
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2answers
187 views

Solution/Hardness of the following (integer) budgeted problem?

I have no idea how to solve the following INTEGER problem or prove its hardness. Thanks for any help/comment/open discussion! Assume there are $N$ startups. For each startup $i$, you can invest $x_i\...
9
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1answer
193 views

What is known about the hardness of the chromatic index for restricted graph classes?

There is a nice paper from 1991 that contains three diagrams about different graphclass-families showing what is known about the hardness of determining the chromatic index for them. Are there any ...
3
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1answer
93 views

What is the recognition complexity of k-uniform k-partite hypergraphs? [duplicate]

We can easily recognize bipartite graphs, but I surprisingly couldn't find anything on the recognition complexity of 3-uniform tripartite hypergraphs, though I'm sure this has been studied. It's also ...
8
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1answer
253 views

NP-hardness on Cayley graphs

What is known about complexity of NP-hard problems on Cayley graphs? Suppose that the graph is given explicitly as the multiplication table of the group and the list of generators. So the input ...
2
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1answer
73 views

Constant Width Max Sum Product Multi-objective Shortest path problem

This question is a follow-up on the question I asked three days ago here. For convenience I restate it here. I am given a graph. Each edge is labelled by a vector of numbers, called weights. They ...
4
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1answer
154 views

Max Sum Product Multi-objective Shortest path problem

Is anything known about the following problem: I am given a graph. Each edge is labelled by a vector of numbers, called weights. They are numbers between 0 and 1. A path is first assigned a vector, ...
29
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2answers
2k views

Can you identify the sum of two permutations in polynomial time?

There were two questions asked recently on cs.se which were either related to or had a special case equivalent to the following question: Suppose you have a sequence $a_1, a_2, \ldots a_n$ of $n$ ...
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0answers
52 views

About increasing the objective values of certificates for Max-Clique SDP

Say you write a round-k Lasserre (or any other hierarchy!) SDP relaxation of the Max-Clique problem. Lets say one now finds (or knows how to sample with high probability) a graph $G_1$ of size $n$ and ...
2
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0answers
84 views

Max common sub forest on $k$ graphs

Not sure how to phrase this really, but here goes. Suppose you are given $k$ simple graphs, each having exactly $m$ edges. The edges in each graph are labeled from 1 to $m$. The problem is to find ...
8
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0answers
171 views

NP-hardness of approximation for unconstrained submodular maximization

The problem of unconstrained submodular maximization can be phrased as follows: Given a non-negative submodular function $f$ on a domain $D$ find a set $S \subseteq D$ maximizing $f(S)$. Here a ...
10
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6answers
2k views

Do many-one reductions and Turing reductions define the same class NPC

I wonder if NPC classes defined by many-one reductions and Turing reductions are equal. Edit: Another question, are Turing reductions only collapsing C and co-C classes for some C or is there a class ...
3
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1answer
257 views

Variant of Subset Sum Problem with Changing Bound

Given a sequence of decreasing integers, i.e., $a_1 \geq a_2 \geq \cdots \geq a_T $ and a positive real $k\geq 1$, find a subset $S$ such that $$\max_{S\subseteq \{1,\ldots,T\}} \sum_{i\in S} a_i$$ $$...
1
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1answer
215 views

Packing sets to maximize overlap

For a set of sets $A$, let $\cup A := \cup_{S \in A} S$. Consider the following problem: Input: a list of $m$ weights $w = (w_1, \ldots, w_m)$, a list of $n$ distinct subsets $T = (S_1, \ldots, ...
3
votes
2answers
3k views

Using decision version of TSP to solve optimization version

Given an oracle for solving the decision version of TSP, how would I use this to solve the optimization version of TSP. This is not a homework assignment, but of general interest. I have been trying ...
2
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0answers
90 views

About representability of optimization versions of NP-complete questions as polynomial optimization over the hypercube

Naively, in my very limited awareness, it feels that the Max-CUT is a very "special" NP-Hard problem because for a graph with edge-set $E$, it can be written as the question of trying to maximize a $n$...
0
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0answers
59 views

About complexity of recovering or learning Bayesian networks

Are there complexity theoretic results about recoverability or learnability of the marginals (of the source vertices) and the conditionals (along each of the edges) of a Bayesian network from having ...
3
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2answers
177 views

Finding a minimal context free grammar that recognizes a finite set of strings of bounded length

Problem: Given a finite set of strings $\{x_1, x_2, ..., x_n\}$ of length $\ell$ or less from some finite alphabet $\Sigma=\{a_1, a_2, ..., a_k\}$, find the minimal context free grammar that ...
6
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1answer
585 views

On reducing the hardness of CNF-SAT to k-Clique

CNF-SAT refers to the following problem: Given a boolean formula $\phi$ in conjunctive normal form, does there exist an assignment to the variables that satisfies $\phi$. There are several ...