Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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Exercises of polynomial and turing reductions

I'm following a graduated course in theoretical computer sicence. A good part the theory we see in this course has to do with polynomial and Turing reductions of NP problem (to prove NP-completeness), ...
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449 views

Is this vertex ordering optimization NP-Hard?

Could you help me to prove that the following problem is NP-hard? Problem. Given an undirected graph $G=(V,E)$, find an ordering of the nodes such that $\sum\limits_{v\in V}|succ(v)|\times|pred(v)|$ ...
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2answers
224 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 ...
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1answer
2k views

The significance of NP-Hard Problems in Cryptography

I didn't refer any literature but thought this was ideal to get views from people here.. Assuming that P=NP is proved would cryptography(only provable security) be impossible? Since the adversary can ...
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2answers
172 views

Reduction of graph chromatic number to hypergraph 2-colorability

I'm following this paper titled "Coverings and colorings of hypergraphs" by Lovasz 1973, which is referenced in Garey and Johnson's Computers and Intractability, for the Set Splitting Problem. In this ...
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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
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 ...
3
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1answer
859 views

Is it NP-Complete to determine if a quadratic program (QP) has multiple solutions?

If this is known, can someone point me to a proof? Edit: A QP is essentially a LP with a quadratic objective. That is, it looks like: minimize $\frac{1}{2} x^T Q x + c^T x$ s.t. $Ax \leq b$ It's ...
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259 views

Is deciding whether all satisfying assignments are NAE assignments coNP-complete?

Let the language $L$ consist of the $k$-CNF formulas $\phi$ with the property that any satisfying assignment $x$ of $\phi$ is a Not-All-Equal (NAE) assignment, i.e. every clause of $\phi$ has at least ...
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1answer
826 views

The complexity of a multi-objective shortest path problem

I have the following shortest path problem. Consider a directed graph with $n$ levels. Each level has $m$ nodes. Each node at level $i$ is connected to all nodes at level $i+1$. Let us also make a ...
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1answer
231 views

Maximize the expected number of “losers” - Is it NP-hard?

I am trying to find a reduction for a problem that seems NP-hard: Let me start from a toy example. Consider 3 elements, $a$, $b$, and $c$. You want to choose two pairs out of the three pairs and ...
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1answer
413 views

Do you know this problem of deciding whether a given 0-1 matrix contains k pairwise disjoint “column-paths”?

Problem: $X$ Instance: A $m\times n$ 0-1 Matrix $A$, $k \in \mathbb{N}$. Question: Does $A$ contain $k$ pairwise disjoint "column-paths"? A column-path starts in the first column, ends in the ...
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334 views

Complexity of the Hamiltonian Subcycle problem

The problem is as follows: Given a graph $G$, find a (vertex) disjoint set of cycles $C$ on $G$ such that every vertex is visited by a cycle exactly once. My question is then: what is the ...
3
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1answer
99 views

how to achieve a topological sort of an given sequence with minimum swaps

For example, given the constraints {$a<b,c<d$} and a sequence $[b,a,c,d]$. we just need swap $a$ with $b$ to get an topological sort, I want to ask how to find the sort solutions with minimum ...
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1answer
273 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 $$\max_{S\subseteq \{1,\ldots,T\}} \sum_{i\in S} a_i$$ $$...
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2answers
188 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 ...
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1answer
301 views

reduction of maximum independet set to minimum distance of code

Is there a reference for direct reduction of computing maximum independent set of a suitably constructed graph to computing minimum distance of a linear code when the code is specified by its parity ...
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1answer
203 views

How hard is recognizing a permutation that is a square for the shift product?

This is a continuation of my attempts to generate simple combinatorial computational problems that turn out to be computationally hard (NP-complete). In this pursuit, I came up with a permutation ...
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1answer
114 views

Is there an algorithm that's “like” cross-validation for approximation algorithms of NP-hard problems?

I normally do machine learning work, and when I'm evaluating an algorithm on a data set, I always use cross-validation to determine how effective the algorithm is. Is there a similar method for ...
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1answer
421 views

Is this multiprocessor scheduling problem with overlaps NP-Hard?

The problem statement is: "Given a set $J$ of jobs where job $J_i$ has length $L_i$ and a number of processors $m$, jobs have inter-overlapping (For example, if job $J_i$ and $J_k$ are assigned to the ...
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1answer
397 views

Approximate bound/algorithm for “product of sums maximization” problem

I am looking for some approximate algorithm with upper/lower bound for the following problem: Given a set of positive integers $\{a_1, a_2, \dots, a_n\}$, partition $\{1, 2, \dots, n\}$ into disjoint ...
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1answer
277 views

Is the problem “MIN-SET-PARTITION” an NP-hard problem?

Given a family of sets $F = S_1, ..., S_n$ of elements from a universe $U$, find the minimal integer $k$ for which there is a partition of $F$ of size $k$, such that every two sets in the same ...
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167 views

What is the best approximation and exact algorithm for vertex cover on cubic graphs?

"Best" = best performing in terms of run-time for exact algorithm and approximation ratio for an approximation algorithm.
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1answer
175 views

Matrix Coloring under Vertical and Horizontal Constraints

I'm searching for the correct name of the following NP-complete problem. I would also appreciate answers pointing to problems with similar-looking variations. The input consists of A set of ...
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206 views

Hardness of exact binomial tail bounds

Chernoff bounds, in their various forms, bound the tails of a Binomial$(n,p)$ random variable $B$. Define the function $F(n,p,t):=P(B>t)$. Naively, computing $F$ requires exponential (in $n$) time. ...
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1answer
192 views

A maximization problem containing summation and multiplication

Are the following two problems NP-hard? Problem 1 Given $n$ ordered pairs of integers $S=\{(a_i,b_i)\}$, $1\leq i \leq n$, and an integer $k$. Find a subset $A$ of $S$ with $k$ elements, such that $...
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1answer
169 views

bounded outdegree bipartite spanners

Given an undirected graph $G = (V,E)$ and an integer $k > 0$, our objective is to find a subgraph $G' = (V ,E')$ where $E' \subseteq E$ such that $G'$ has the three following properties : $G'$ ...
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1answer
87 views

Complexity of existence of simple polygonalization with prescribed area?

This is a followup on my previous question. Fekete proved the NP-completeness of deciding the existence of simple polygonalization with minimum (or maximum) enclosed area (simple polygonalization is ...
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245 views

Finding a set which dominates the Minimum Dominating Set

Given an unweighted, undirected graph, a dominating set $S$ is a set of nodes such that every node is in $S$ or adjacent to a node in $S$. The dominating set problem is NP-hard, but I am considering ...
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1answer
97 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 ...
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188 views

Is this covering problem NP-hard?

Given a rectangular region $R$ and a set $D$ of $n$ disks such that the union of all disks in $D$ cover the entire rectangular region $R$, the objective is to find the minimum cardinality set $D'$ ...
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1answer
1k views

Hardness of 3-dimensional matching variant

I am thinking about a variant of 3-dimensional matching. In normal 3d matching, we have three sets of vertices $X$, $Y$ and $Z$, and a set of edges $E \subseteq X \times Y \times Z$. We want to choose ...
3
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1answer
286 views

NP-hardness of a bilinear program?

Given $a_i, b_i, c_i, d_{ij}\in [0,1]$ for $i,j\in [n]$ and $i\neq j$ such that $\sum_{i\in [n]} a_i=1$ and $d_{ij}=d_{ji}$. I have the following bilinear program: max $\sum_{i=1}^n (x_i-a_i)y_i$ ...
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161 views

Take a NEXP-complete problem and then have the input in unary. Why is this not NP-complete?

It is known that if any unary language is NP-complete, then P=NP. Suppose we take a NEXP-complete language with input $x$ in binary and witness $y\in\{0,1\}^{2^{poly(|x|)}}$ such that the verifying ...
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70 views

Rearranging angles of a convex polyline to make it closed

Let {$\alpha_1, \alpha_2, ... ,\alpha_n$} be a string of n positive reals summing up to 2$\pi$. We inductively construct the following 2D polyline, denoting with $R[\alpha]$ the clockwise rotation by ...
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59 views

Hardness of ancilla free quantum circuit extraction from circuit with ancillas

Is there any known result regarding the hardness of the following problem: Given a quantum circuit with ancillae implementing a unitary, find a quantum circuit that does not use any ancillae that ...
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208 views

What is consequence of $PH\subseteq NSPACE((\log n)^2)$?

What is consequences of $PH\subseteq NSPACE((\log n)^2)$? We don't even know PH is equals to L or not. I am wondering what will be happened when $PH\subseteq NSPACE((\log n)^2)$?
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PTIME or NP-Hardness of stochastic objective function

I will begin by linking a previous post where I asked a general question for a stochastic setting which I describe below. It turns out that my "proof" for a restricted case had a mistake and there is ...
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90 views

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 ...
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141 views

Random self reducibility and NP

I was reading the Wikipedia page Random self-reducibility and it states: If an NP-complete problem is non-adaptively random self-reducible the polynomial hierarchy collapses to $\Sigma_3$. I am ...
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82 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 $...
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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 ...
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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 $\min \mathbb{E}_\mu [f(x)]$ ...
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116 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" :...
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124 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 ...
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184 views

Min dominating set software

I need a fast min-dominating-set code for some complexity lower bounds research I am doing. I could transform to SAT and use an off the shelf SAT solver; but I was hoping min-dom-set had something ...
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126 views

Complexity of minimizing monotone arithmetical formulas

Let's say that I have a multi-variate arithmetical expression $A(x_1, \ldots, x_n)$ that uses addition and multiplication operations and is also in a very simple form of sums of products, e.g., $A_1(...
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61 views

Is this permutation-sum problem NP-complete? [duplicate]

A new, tighter tardiness bound has been found for global Earliest-Deadline-First scheduling of jobs on symmetric multiprocessors. But this bound seems to be particularly hard to compute. In particular,...
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242 views

NP-hardness of a quadratic programming problem

Motivated by the mean-variance optimization, I came up with the following question: Given $n$ integers $a_1, \cdots, a_n$; $n$ lower bounds $0<\ell_1, \cdots, \ell_n<1$ $n$ upper bounds $0<...
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814 views

What's the hardest problem with a non-trivial exact algorithm?

Exact algorithms for NP-complete problems are sometimes feasible, if the input is small enough. I’ve also came across some algorithms which are not practical even for very small inputs, and their ...