Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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1answer
277 views

Best parameterized algorithm for maximum clique

I have seen the basic algorithm for the maximum clique problem parameterized by the maximum degree at an algorithms course. However, I struggle to find anything better. Searching for things like "...
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1answer
76 views

Is the counting version of 1-in-3 Sat #P-complete?

In the paper "Hard Tiling Problems with Simple Tiles", Moore and Robson prove that Cubic Planar Positive 1-in-3 Sat in NP-complete by a reduction from Positive 1-in-3 Sat. Cubic Planar Positive 1-in-...
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1answer
78 views

Determining if a word of specific length exists that is not accepted by a NFA

It is known that the problem of determining if an NFA accepts every word is PSPACE-COMPLETE, meaning it is also NP-Hard, but is this weaker version of the problem still NP-hard? Given an NFA and a ...
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0answers
75 views

NP-Hard Knapsack Instances

Consider the classic Knapsack optimization problem (KP): Given $p_1, \dots, p_n, w_1, \dots, w_n, B\in\mathbb N$, compute a solution $I\subseteq \{1,\dots,n\}$, such that $\sum_{i\in I} w_i \leq B$ ...
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5answers
3k views

Implications of unprovability of $P\neq NP$

I was reading "Is P Versus NP Formally Independent?" but I got puzzled. It is widely believed in complexity theory that $\mathsf{P} \neq \mathsf{NP}$. My question is about what if this is not ...
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0answers
45 views

Complexity of multi-objective optimization problems

How can we define and prove the worst-case complexity of multi-objective optimization problems (MOOP)? It is easy to see that, if one of the objectives is an NP-Hard optimization problem, then the ...
1
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0answers
92 views

how to model community detection into np-hard problem

My problem is that given a fully connected graph G={V, E} and a seed set $S=\{s_1, s_2,\dots,s_n\}$, the target is to find clusters under some constrains that the average edge weight within each ...
1
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1answer
71 views

The decision procedure of theory of closed real field is in NP-hard?

The decision procedure of theory of closed real field refers to https://en.wikipedia.org/wiki/Decidability_of_first-order_theories_of_the_real_numbers
4
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1answer
115 views

Hardness of finding if a vertex lies on a simple directed path between two vertices

Given a directed graph $G = (V, E)$ and three vertices $u, v, w \in V$. Is it NP-Hard to find whether there is a simple path from $u$ to $v$ passing through $w$? I found a couple of hardness ...
6
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2answers
143 views

$NP$-Completeness of $\epsilon$-balanced graph partitioning for fixed $\epsilon$

Consider this graph partitioning problem: Let $G = (V, E)$ be a simple undirected graph and $0 \leq \epsilon \leq 1, M \geq 0$ be constants. Are there disjoint subsets $V_1, V_2$ with $V = V_1 \cup ...
9
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2answers
112 views

Complexity of finding an edge set yielding specified vertex degrees

I'm trying to figure out if the following two problems are known in general to be in P or NP-complete: Q1: Given a graph $G=(V,E)$ and integers $d_i,\,1\leq\,i\leq|V|$, does there exist a subset $E'...
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16answers
6k views

Everyday encounters with NP-complete problems

Mark Dominus collected a few examples of polynomial-time reductions from various NP-hard problems to “regular expression” matching. Envisioning polynomial-time verifications isn't an enormous leap. ...
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0answers
122 views

NP-Hardness of 4-cycle packing problem in complete bipartite digraph?

A directed complete bipartite graph is a bipartite graph where there is exactly one directed edge between any two vertices from its two different parts. In other words, it's an orientation of a ...
10
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1answer
692 views

Relationship between two graph optimization problems

Let $Q$ be a polynomial time computable graph property of simple, undirected graphs. Consider the following two optimization problems on any input graph: P1. Find a largest induced subgraph of the ...
2
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1answer
136 views

What Is the Complexity of This Two-to-One Matching Problem?

Given a graph $G=(V,E)$ and a function $c:V\mapsto\{1,2\}$. The function $c(\cdot)$ divides the vertices into two disjoint sets $V_1$ and $V_2$, where for all $v_1\in V_1$, we have $c(v_1)=1$ and for ...
2
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0answers
72 views

How hard is it to approximate distance of linear code

I'm trying to figure out what is the current knowledge about how hard it is, given a generating matrix of a linear code over a field $F_{q}$, approximate it's distance. I of course found that ...
15
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1answer
601 views

Can one efficiently uniformly sample a neighbor of a vertex in the graph of a polytope?

I have a polytope $P$ defined by $\{ x : Ax \leq b, x \geq 0\}$ . Question: Given a vertex $v$ of $P$, is there a polynomial time algorithm to uniformly sample from the neighbors of $v$ in the graph ...
44
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4answers
9k views

Is finding the minimum regular expression an NP-complete problem?

I am thinking of the following problem: I want to find a regular expression that matches a particular set of strings (for ex. valid email addresses) and doesn't match others (invalid email addresses). ...
-4
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1answer
100 views

Example of decidable NP-hard problem that is not NP-complete [closed]

I am looking for an example of a decision problem which fulfills the following conditions: 1. It is decidable 2. It is NP-hard 3. It is not NP-complete All my search attempts yielded examples that ...
10
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0answers
296 views

Is unary $\Pi_2$-SUBSETSUM coNP-complete?

Consider the following problem: for given integers $a_1, \ldots, a_{2n}$ and $A$ that are given in unary representation define is it true that for every $S \subseteq \{1, ..., 2n \}$ such that $|...
7
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0answers
129 views

Optimal bee swarm plots: NP-hard?

Bee swarm plots are a way of visualizing one-dimensional data sets, similar to box plots. The idea is that if there's not too many points (e.g. <300) we can just plot them along the $x$-axis with ...
1
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1answer
174 views

What languages can be reduced to a NP-complete problem in polynomial time

NP-complete: Language is NP-complete, when it is in NP and every problem in NP is reducible to it in polynomial time. But what languages are reducible to a NP-complete problem (for example SAT) in ...
7
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1answer
202 views

NP-hardness of a planar SAT variant

Background: An instance of 3-SAT is called monotone if each clause consists only of positive literals or only of negative literals. Given an instance $\phi$ of 3-SAT, we consider the bipartite graph ...
2
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1answer
90 views

Diameter vs. Tractability

I would like to find examples of (unweighted) optimization problems $\Pi$ that satisfy 1) or 2): 1) $\Pi$ is NP-hard but polytime solvable in the class of graphs with diameter $\le k$, for all $k\ge ...
8
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3answers
421 views

A partition problem in which some numbers may be cut

In the standard partition problem, we are given some numbers whose sum is $2s$ and have to decide whether they can be partitioned into two subset whose sum is $s$. It is known to be NP-hard. However,...
2
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0answers
52 views

Complexity of Block Design?

What is known about the complexity of creating Block Designs (https://en.wikipedia.org/wiki/Block_design)? I've found one paper that creates approximately solutions using Metaheuristics that claims ...
22
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8answers
3k views

NP-hard problems on paths

everybody knows there exist many decision problems which are NP-hard on general graphs, but I'm interested in problems that are even NP-hard when the underlying graph is a path. So, can you help me to ...
2
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0answers
90 views

How hard is it to determine ex(n,G)?

Define the extremal Turán function $ex(n,G)$ of a graph $G$ as the most edges a graph on $n$ vertices can have without having a subgraph that is isomorphic to $G$. This function is known ...
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1answer
157 views

Find the shortest s-t trail(edge disjoint path) in a graph with negative weight edges

A walk in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. A trail is a walk in which all edges are distinct. Note that a trial may visit a vertex multiple times ...
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0answers
101 views

Network Reliability Problem

Network reliability, in which we are given an undirected graph $G$ with a failure probability $p_e$ for each edge and we are asked to calculate the probability that the network becomes disconnected ...
3
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2answers
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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0answers
62 views

Generalization of k-Coloring: maximizing the number of vertices with no neighbours of same color

One can consider the following generalization of the $k$-Coloring problem: Let be given a graph $G$ and an two integers $k$ and $p$. A vertex $v$ of $G$ is properly colored if $v$ has no neighbour ...
2
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0answers
54 views

Is minimal cover under symmetric 3-deduction NP-complete?

Forgive me if this problem is known by another name, I do not know any references for it. Symmetric deduction. An equation $e \in E$ is a subset of variables $V$ such that knowing $|e| - 1$ of the ...
3
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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 ...
8
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1answer
203 views

A partition problem with order constraints

In the OrderedPartition problem, the input is two sequences of $n$ positive integers, $(a_i)_{i\in [n]}$ and $(b_i)_{i\in [n]}$. The output is a partition of the ...
60
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12answers
3k views

Parameterized complexity from P to NP-hard and back again

I'm looking for examples of problems parametrized by a number $k \in \mathbb{N}$, where the problem's hardness is non-monotonic in $k$. Most problems (in my experience) have a single phase transition, ...
13
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0answers
208 views

Which monotone DNFs are evasive?

A Boolean function $\phi$ on variables $X$ is evasive if every decision tree for $\phi$ has height $|X|$. In other words, for any strategy that picks variables of $X$ and asks for their value, an ...
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1answer
175 views

Can we map this problem to subset-sum?

Let there be $n$ set of ordered pairs $s_1=\{(c_1,f_1),(c_1,f_2) ...(c_1,f_m)\}$, $s_2=\{(c_2,f_1),(c_2,f_2) ...(c_2,f_m)\}$, $s_3=\{(c_3,f_1),(c_3,f_2) ...(c_3,f_m)\}$, .... $s_n=\{(c_n,f_1)(c_n,f_2) ...
47
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20answers
7k views

NP-hard problems on trees

Several optimization problems that are known to be NP-hard on general graphs are trivially solvable in polynomial time (some even in linear time) when the input graph is a tree. Examples include ...
8
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1answer
334 views

Planar Exact Cover by even-size sets

Major edit on June 6, 2019: Replaced the target problem with a simpler (but equivalent) one. Is the following problem NP-complete? Planar Exact Cover by even-size sets Input: A set $U$, a ...
5
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1answer
134 views

Complexity of Finding Largest Set of Intersecting Convex Polytopes

I have a set of $n$ convex polytopes, and I wish to find the largest subset of those polytopes that shares at least one point in common. I think that this problem should be NP-hard, but I am ...
14
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1answer
410 views

Sampling a uniformly random satisfying assignment

Problem: Given $\phi : \{0,1\}^n \to \{0,1\}$ represented by a boolean circuit, generate a uniformly random $x \in \{0,1\}^n$ such that $\phi(x)=1$ (or output $\perp$ if no such $x$ exists). Clearly ...
4
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3answers
274 views

Variation on partial Set Cover with penalties

I'm looking at the following problem, which I'm hoping is perhaps a known variant of the Set Cover problem: Input: We're given a universe $U$ and two disjoint subsets $A$ and $B$ such that $A \cup B=...
5
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0answers
71 views

Complexity of a specific class of definite integrals

INTRODUCTION: From the answer to this question I learned that deciding whether a definite integral is $0$ or not can be NP-complete, as the following integral representation of the Number Partition ...
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6answers
2k views

Is there a natural problem on the naturals that is NP-complete?

Any natural number can be regarded as a bit sequence, so inputting a natural number is the same as inputting a 0-1 sequence, so NP-complete problems with natural inputs obviously exist. But are there ...
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0answers
32 views

Scheduling with Separation Constraint

There are $N$ types of jobs. For each $i$, we have to schedule $T/D_i$ jobs of type $i$ in $T$ timeslots. We know that $\sum_{i=1}^N 1/(D_i+1) = 1$. For each type $i$, the distance between two ...
10
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1answer
302 views

How hard is deciding the existence of Red-Blue perfect matching?

Two-colorable perfect matching problem is to decide whether a graph has coloring with two colors such that each node has exactly one neighbor the same color as itself. The problem was proven to be NP-...
1
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0answers
81 views

Minimum cut with nonlinear objective function

Let $G$ be an undirected graph. The classic minimum (cardinality) cut problem asks for a cut $C\subseteq E(G)$, such that $|C|$ is minimum. Let us generalize it the following way: let $f$ be a ...
7
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0answers
141 views

NP-hardness proof of selecting the ''fittest'' topological sequence of a DAG

Given a directed acyclic graph (DAG) with $n$ vertexes $V=\{v_1, v_2,...,v_n\}$ and a given permutation of those $n$ vertexes $P=[p_1, p_2,..., p_n]$ that $\forall i, p_i\in V$. Note that $P$ could ...
2
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0answers
82 views

Finding 3SUM witness when promised a solution

Suppose we have a 3SUM instance given with the promise that there exists at least one solution. Is the trivial $O(n^2)$ (modulo logarithmic improvements) solution still the best algorithm or is there ...