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Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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2answers
214 views

NP-hard problems on the class of caterpillars

My question is whether there exist an NP-hard problem that has only a caterpillar as input. By saying only caterpillar as input, I wanted to emphasize that no function (eg: weights on vertices or ...
20
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4answers
1k views

Positive topological ordering, take 3

Suppose we have an n by n matrix. Is it possible to reorder its rows and columns such that we get an upper-triangular matrix? This question is motivated by this problem: Positive topological ordering ...
2
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0answers
161 views

Best polynomial-time approximation factor for NP-optimization problems

Let us say that a function $f(n)$ is the best approximation factor for an NP-optimization problem, if both of the following hold: There exist a polynomial-time algorithm $A,$ and an integer $n_0$, ...
3
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1answer
94 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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2answers
86 views

Is it possible to have a sorting algorithm that computes faster than QuickSort? [closed]

Given an unsorted array, QuickSort has to touch each source element it is trying to sort multiple times before it declares an array as sorted. (notice how many times the 2 is touched [circled in red ...
2
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1answer
290 views

What are some example problems for integer programming that are *not binary*

I'm interested in NP-hard problems that have a "nice" integer-programming formulation (quadratic or linear, with quadratic or linear constraints) that is not binary. Of course it is always possible ...
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1answer
143 views

Subset Sum Problem and hard looking instances that are not really hard

I have been working in a subset sum solver (some new approach) and while working on the time complexity analysis I found what I describe below. Maybe this could explain why some "hard looking" ...
128
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28answers
19k views

Problems Between P and NPC

Factoring and graph isomorphism are problems in NP that are not known to be in P nor to be NP-Complete. What are some other (sufficiently different) natural problems that share this property? ...
2
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1answer
140 views

A least sized partition of a set under a distance metric

What is the worst case complexity of an algorithm to find a least partition of a set under a distance metric, described as follows: Input: A set $S=\{s_1,\ldots,s_n\}$, where the elements $s_i$ are ...
9
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1answer
145 views

Best known asymptotic PCP sizes / 3-SAT

What are the best known asymptotic upper bounds on sizes of probabilistically checkable proofs? Ideally, I am looking for a contemporary survey on this broad question, but if there is none, I am ...
1
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1answer
113 views

A dominate vector subset sum problem

Let $k$ be some constants (e.g. one can take $k=2$ for simplexity), for any $u,v\in \mathbb{R}$, we say $u$ dominate $v$ if $\forall 1\le i\le k,~ u[i]\ge v[i]$, write it as $u\succ v$. Consider the ...
9
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0answers
129 views

Complexity of fractional SAT

Let $(a, k)$-SAT be $k$-SAT with the promise that if there is there is a satisfying assignment, then there is such an assignment that satisfies at least $a$ literals of every clause. Can 3-SAT with $...
2
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0answers
103 views

Counting the maximum number of paths of length $n$ that differ in at least $k$ edges

What is known about the complexity of solving (or approximately solving) the following problem? INPUT: Graph $G=(V,E)$ and constants $L$ and $K$. OUTPUT: The maximum size of any set $S$ of simple ...
10
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2answers
684 views

Set optimization problem - is it np-complete?

Set $S=\{e_1,\cdots,e_n\}$ is given. For each element $e_i$, we have weight $w_i>0$ and cost $c_i>0$. The goal is findIng the subset $M$ of size $k$ that maximize the following objective ...
3
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0answers
57 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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0answers
57 views

Reasoning about NP hardness of optimization problems with closed form functions as input

(This may not be a research level question per se. I can delete this question if the community thinks this way too) I am trying to understand how to reason about hardness of optimization problems ...
232
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11answers
116k views

Is Norbert Blum's 2017 proof that $P \ne NP$ correct?

Norbert Blum recently posted a 38-page proof that $P \ne NP$. Is it correct? Also on topic: where else (on the internet) is its correctness being discussed? Note: the focus of this question text has ...
1
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1answer
268 views

Is the following optimization problem (another variant to a previous problem) NP-hard?

This problem is a following up question on this one. The only difference is the addition of $3^{rd}$ constraint "$\sum_{i} x_{ij} \le M$", where M is a constant number. This constraint essentially ...
3
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2answers
217 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 ...
2
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1answer
130 views

Is partitioning a multiset into two multisets with equal averages NP-complete?

The subset sum problem of partitioning a multiset of integers into two multisets with equal sums is NP-complete. Is the seemingly related problem of partitioning a multiset of integers into two ...
4
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0answers
114 views

A variant of hitting set: finding a matching to hit all edge sets

In general, the hitting set problem is given a family $\cal S$ of sub-sets, $\{S_1, \cdots, S_h\}$, and a universal set $U = \bigcup_{i\in [1,h]} S_i$. It asks for a minimum set $H \subseteq U$ ...
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1answer
75 views

Restricted Universe Exact Cover

Apologies for a simple question - I am a beginning graduate student in TCS. Consider the following $\mathrm{ExactCover}$ problem: Given a collection $\mathcal{S}$ of subsets of a universe set $U$ and ...
3
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0answers
207 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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1answer
89 views

NP hard? Maximize the average for a subset of numbers

Given a set of real numbers, choose a subset to maximize the average subject to the average not exceeding a given threshold. Is it NP hard? I think so, but I cannot come out with a proof. Thanks a ...
15
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1answer
1k views

Password hashing using NP complete problems

Commonly used password hashing algorithms work like this today: Salt the password and feed it into a KDF. For example, using PBKDF2-HMAC-SHA1, the password hashing process is ...
22
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2answers
2k views

Are there any hard instances of 3-SAT when the clauses can only use literals that are “nearby” each other?

Let the variables be $x_1 , x_2 , x_3 ... x_n$. The distance between two variables is defined as $d(x_a , x_b) = |a-b|$. The distance between two literals is the distance between the corresponding ...
9
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1answer
266 views

Maximum weight “fair” matching

I'm interested in a variant of the maximum weight matching in a graph, which I call "Maximum Fair Matching". Assume that the graph is full (i.e. $E=V\times V$), has even number of vertices, and that ...
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1answer
61 views

How to make any graph 2-degenerate?

I have to show a PPT(polynomial time reduction) from 'Colorful graph Motif' to '2-Degenerate Steiner Tree'. As input graph should be 2-degenerate, but here is normal graph G (that is, basically an ...
5
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3answers
8k views

what is the real difference between traveling salesman problem (TSP) and vehicle routing problem (VRP)?

Both problems are well-known NP-hard problems with great similarities. In fact, I do not see the real difference between these two problems. It seems relatively easy to model TSR in the form of VRP ...
5
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1answer
512 views

Sparsification Lemma for k-SAT and Exponential Time Hypothesis

According to R. Impagliazzo, R. Paturi and F. Zane, 2001 an instance of $k$-SAT is called sparse if $m = O(n)$ where $m$ denotes the number of clauses and $n$ the number of variables. The ...
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0answers
88 views

Directed NP Hard Problem on DAG

There are problems that are NP-Hard on undirected graphs(maximum weight independent set and graph coloring) but are polynomial time solvable on trees. Tree decomposition is a good tool to talk about ...
14
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1answer
435 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 ...
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1answer
53 views

NP-Complete graph problems where a special vertex is given as input?

I am currently working on a graph theory problem where the instance includes a graph and a special vertex in the graph. I am trying to prove the NP-completeness of the problem as well as explore ...
8
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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 ...
21
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4answers
827 views

Problems that are counter-intuitively solvable in practice?

Recently, I went through the painful fun experience of informally explaining the concept of computational complexity to a young talented self-taught programmer, who never took a formal course in ...
6
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1answer
111 views

NP completeness of classes of spanning trees

I am teaching a complexity course, and I want to give some examples of similar looking problems such that one is in P, and the other is NP complete. This made me think of the following problem: does ...
4
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1answer
188 views

How hard is it to determine the chromatic number of a unit distance graph?

For example, is it NP-complete to decide whether a unit distance graph is 3-colorable?
1
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1answer
87 views

Minimum number of columns making each row different

I'm curious whether this problem is NP-hard: suppose you are given an arbitrary $m\times n$ 0-1 matrix (each element is either 0 or 1, for the simplicity of the problem), and any pair of rows (i.e. a ...
1
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0answers
95 views

Best way to represent NP-Hardness result for decision problem with two decision parameters

What is the best way to represent a NP-Hardness result for a decision problem with two decision parameters? Suppose we have a problem $P$ which asks to minimize two parameters $x$ and $y$ and we show ...
0
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1answer
75 views

Constrained Topological Sorting with bounded number of chains

In general, constrained topological sorting is NP-hard. Now we add another constraint to it, such that take any k+1 nodes and there will be at least one pair ...
11
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2answers
241 views

Determining what can be achieved by a permutation of elements of a noncommutative group

Fix a finite group $G$. I am interested in the following decision problem: the input is some elements of $G$ with a partial order on them, and the question is whether there is a permutation of the ...
6
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2answers
696 views

Edge and vertex fault tolerance in graphs

Suppose we are given two graphs $G$ and $H$, where $H$ is a subgraph of $G$. What is the maximum number $k$ such that if any $k$ edges are removed from $G$, $H$ still remains a subgraph of $G$? What ...
1
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2answers
177 views

Check the match of the maximum of each subset

Given a number of vectors with $n$ elements, i.e., $S=(a_1, \cdots, a_n)$, $T_j=(b_1^j, \cdots, b_n^j)$ for $j=1,\cdots, m$ where each $a_i$ or $b^i_j$ is a natural number. Question: determine ...
3
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0answers
81 views

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 ...
3
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0answers
89 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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0answers
85 views

Is NP-complete the existence of paths of a given length in a directed graph? [closed]

Given a directed graph G= (V,E), a pair of vertices s and t, a natural number K encoded in binary, whether the problem to decide there exists a path (not necessarily simple) from s to t of length K is ...
4
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0answers
90 views

min weight k-closure on DAG

The problem Given a (connected) DAG $G(V,E)$ where each node is assigned an (non-negative) integer weight an integer k where $0\leq k\leq|V|$ Find a induced subgraph $H$ of $G$ consisting of $k$ ...
5
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0answers
79 views

Complexity of bounded degree full contraction

This paper defines the problem $\mathrm{B{\scriptsize OUNDED} \ D{\scriptsize EGREE}\ C{\scriptsize ONTRACTION}}$ as follows: Instance: A graph $G$ and two integers $d$ and $k$. Question: Is there a ...
0
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1answer
62 views

About using smoothness of the Hessian for getting to approximate criticality of a non-convex objective

Is there any algorithm which shows that under the assumption of Lipschitz smoothness of the Hessian of a non-convex function one can get to its critical point faster?
-4
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1answer
436 views

Proving NP-hardness of scheduling problem (total weighted completion time)

Consider the problem $P \mid \mid \sum w_j C_j$. I want to prove that this problem is (strongly) NP-hard by reducing from $3$-Partition, but I am not really sure how to do this. Just to be precise, ...