Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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8
votes
1answer
224 views

Vertex isoperimetric number of a graph - NP-hard?

The vertex isoperimetric number of a graph $G=(V,E)$ is $i_V(G) = \min\{\frac{|N(S)|}{|S|} : S \subseteq V, 1\le |S|\le \frac{|V|}{2}\}$. Several academic papers state that the problem of computing ...
9
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0answers
303 views

What are the best known reductions from SAT to CNF-SAT?

Problems Let SAT denote the following problem: Given a boolean formula, does there exist a satisfying assignment? Let CNF-SAT denote the following problem: Given a ...
14
votes
2answers
522 views

Add a matching to a Hamiltonian path to reduce the max distance between given pairs of vertices

What is the complexity of the following problem? Input: $H$ a Hamiltonian path in $K_n$ $R \subseteq [n]^2$ a subset of pairs of vertices a positive integer $k$ Query: is there a matching $M$ such ...
13
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1answer
546 views

Are there interesting graph classes where the treewidth is hard (easy) to compute?

Treewith is an important graph parameter that indicates how close a graph is from being a tree (although not in a strict topological sense). It is well known that computing the treewidth is NP-hard. ...
12
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4answers
507 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 $[1,2^n]$ ...
3
votes
2answers
198 views

Hardness of finding roots of a degree $2$ polynomials over $\mathbb{F}_2$

Since every $3$-SAT instance with $n$ variables can be expressed as a degree-$3$ polynomial over $\mathbb{F}_2$ with $n$ unknowns, the NP-hardness of $3$-SAT directly translates to NP-hardness of ...
3
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0answers
79 views

Complexity of Underdetermined Systems [closed]

Given a field $\mathbb{F}$ and a consistent underdetermined system $Ax=b$ over $\mathbb{F},$ $A\in \mathbb{F}^{m \times N}$ and $b \in \mathbb{F}^m,$ finding a vector $z \in \mathbb{F}^N$ such that $...
2
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0answers
120 views

Reduction Unbounded Knapsack < k-Exact Unbounded Knapsack

I'd like to have an explicit reduction among these two problems: (1) Unbounded Knapsack: Given a set of $n$ item types with weight $w_i$ and quality $q_i$ solve: $$maximize \sum_{i=1}^n q_ix_i $$ ...
5
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2answers
361 views

On integer programming

Integer programming is NP-hard. What is the status of integer programming problem that decides between existence of $\leq1$ solution and $>1$ solutions (note $0$ solutions falls in $\leq1$ ...
2
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0answers
149 views

Computational Complexity of cycle double cover

Let $\mathcal{G}$ be the set of all finite simple graphs. Let graph $G\in \mathcal{G}$ and $C_G=\left <C_1,...,C_m \right >$ be a sequence of cycles of $G$ for some $m$. For every edge $e$ of $G$...
14
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1answer
428 views

NP-Complete problems that admit an efficient algorithm under the promise of a unique solution

I was recently reading a very nice paper by Valiant and Vazirani which shows that if $\mathbf{NP \neq RP}$, then there can not be an efficient algorithm to solve SAT even under the promise that it is ...
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 ...
5
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2answers
246 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
202 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 ...
5
votes
1answer
676 views

Is approximating Exact Set Cover NP-hard for constant approximation factor? ETH hard?

It is known that Exact Set Cover is an NP-hard problem (Reduction from 3-SAT and 3-Coloring). Also, my minor analysis one can realize that this problem is also ETH-hard, i.e. this cannot be solved in ...
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
votes
1answer
393 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
110 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" :...
27
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3answers
894 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 ...
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, ...
7
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0answers
214 views

How hard is PromiseFlowFree?

Playing more Flow Free, I think I've realized why I'm so amazingly brilliant at this game: The objective is to connect all pairs while covering the entire board, but in every puzzle there is always ...
1
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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
417 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 $\mathsf{NP}$-...
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
votes
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 ...
5
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0answers
297 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?
8
votes
1answer
225 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 ...
1
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0answers
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 $a_1,\dots,a_{n}\in\Bbb Z$ ...
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
votes
1answer
149 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
votes
1answer
189 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
votes
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 ...
8
votes
1answer
380 views

complexity of a constraint satisfaction promise problem

(This is the "upper end" of my question from over 10 months ago on cs.stackexchange. That question and the "lower end" I asked here over 8 months ago, which I also have a bounty on, are both ...
4
votes
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 ...
5
votes
2answers
208 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 ...
1
vote
1answer
637 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
votes
1answer
155 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://...
4
votes
2answers
439 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, ...
5
votes
1answer
167 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
votes
1answer
130 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 ...
-5
votes
2answers
2k views

Minimum-weight feedback edge set in undirected graph - how to find it? Is it NP hard problem?

Let G = (V,E) be an undirected graph. A set F ⊆ E of edges is called a feedback-edge set if every cycle of G has at least one edge in F. Suppose that G is a weighted undirected graph with positive ...
5
votes
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
votes
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 ...
6
votes
2answers
227 views

Kth best problem that is NP-hard for K polynomial

A Kth best problem is, given some constraint, to find a solution that has the Kth best value compared to all solutions that meet the constraint. One way to write this as a decision problem is to ...
3
votes
1answer
95 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
votes
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
votes
1answer
74 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
votes
1answer
155 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, ...
1
vote
0answers
53 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 ...