Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

541 questions
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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 ...
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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 ...
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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 ...
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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 $\min \mathbb{E}_\mu [f(x)]$ ...
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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 $\mathsf{NP}$-...
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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 ...
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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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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?
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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 ...
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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 $a_1,\dots,a_{n}\in\Bbb Z$ ...
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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 (...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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-complete when parameterized by the ...
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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://...
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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, ...
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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 ...
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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 ...
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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 ...