Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

Filter by
Sorted by
Tagged with
4
votes
0answers
32 views

Neighborly properties in a bipartite graph

Input: Let $G$ be a connected, bipartite graph with parts $A$ and $B$, each of size $n$. For $S\subseteq A$, let $N(S)\subseteq B$ be the set of neighbors of $S$ in $B$. Similarly, for $T\subseteq B$, ...
8
votes
1answer
129 views

Recent progress on the next-to-shortest-path problem for directed graphs?

In the paper "Computing strictly-second shortest paths" (1997), Lalgudi and Papaefthymiou consider the following problem: Let $G$ be a directed graph with edge-weighting $w$. Let $u,v$ be vertices in ...
-1
votes
0answers
30 views

Reduce a decision problem to an optimization problem

In this paper, the author shows that finding an edge coloring that minimizes the total sum on a multi-tree is NP-hard. In theorem 3.1, the author does so by reducing from 3SAT. I find his proof quite ...
0
votes
1answer
67 views

Minimizing the gaps with incremental capacity

There are a single job, a machine and a set of $n$ slots. The machine has a capacity that increments by $\zeta(t)$ every slot $t=1,2,\ldots,n$. Initially (before the first slot), the machine has 0 ...
-3
votes
0answers
69 views

Is mathematical proof itself NP-hard?

At the 8:00 mark of this video, he claims that proving things is itself an NP problem. I'm looking for more insight into this. Could someone help explain this concept to me and also provide a link to ...
1
vote
1answer
85 views

On the paper “Quantum Computing Hamiltonian cycles”

The paper Quantum Computing Hamiltonian cycles claims: An algorithm for quantum computing Hamiltonian cycles of simple, cubic, bipartite graphs is discussed. It is shown that it is possible to evolve ...
9
votes
2answers
3k views

What are good approximation algorithms for the subset sum problem so far?

By "good", I mean either the algorithm provides a relatively tight bound or it has a relatively fast running time. Any reference is welcome.
39
votes
3answers
3k views

Is the integer factorization problem harder than RSA factorization: $n = pq$?

This is a cross-post from math.stackexchange. Let FACT denote the integer factoring problem: given $n \in \mathbb{N},$ find primes $p_i \in \mathbb{N},$ and integers $e_i \in \mathbb{N},$ such that $...
7
votes
2answers
585 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 $\sum\limits_{v\in V}|succ(v)|\times|pred(v)|$ ...
5
votes
2answers
145 views

Topological sorting of a DAG where special vertices have to come in even groups

Consider the following problem. The input is a directed acyclic graph (DAG) $G = (V, E)$, and a subset $V' \subseteq V$ of vertices, which we call special vertices. The question is to determine ...
3
votes
3answers
439 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 "...
1
vote
1answer
78 views

Choosing one number from each set so that the sum of squares of each distinct number counts is minimized

Problem is as follows: We are given $K$ subsets of $\{1,2,...,n\}$. We need to pick one number from each of these subsets such that $\sum_{i=1}^n p_i^2$ is minimized where $p_i$ is the number of times ...
-1
votes
1answer
176 views

Conjecture about ASP reductions between NP-complete problems

$ASP$-complete reductions, introduced by Ueda and Nagao, relate the hardness of computational problems in $FNP$. Basically, $ASP$-reduction is a polynomial time reduction between instances and a ...
1
vote
1answer
192 views

Is $\{0,1\}$-Vector bin packing NP-Hard when vectors have constant dimension?

The paper https://cs.brown.edu/people/seny/pubs/vbponline.pdf discusses $\{0,1\}$-Vector Bin packing in the online setting and give lower bounds. However, they do not mention anything about the ...
-5
votes
1answer
22 views

Polynomial-time reducibility of Primality and 3-SAT

Is 3-SAT $\leq_{p}$ Primality? And/or is Primality $\leq_{p}$ 3-SAT? I think the answer is no and yes, respectively, but I'm not sure. Any help would be appreciated. Thank you.
0
votes
0answers
36 views

Would a machine learning algorithm benefit from an “optimization oracle”?

I'm trying to understand the behavior of machine learning algorithms where the loss function is non-convex and the problem of training the ML on a specific data set is computationally hard. Now let'...
20
votes
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 ...
3
votes
1answer
116 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 ...
13
votes
2answers
3k views

Lexicographically minimal topological sort of a labeled DAG

Consider the problem where we are given as input a directed acyclic graph $G = (V, E)$, a labeling function $\lambda$ from $V$ to some set $L$ with a total order $<_L$ (e.g., the integers), and ...
1
vote
1answer
80 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 ...
4
votes
0answers
181 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 ...
20
votes
2answers
1k views

Is feedback vertex set problem is solvable in polynomial time for 3-degree bounded graphs?

Feedback Vertex Set is NP-complete for general graphs. It is known to be NP-complete for degree-8 bounded graphs due to a reduction from vertex cover. The Wikipedia article says that it is poly-time ...
0
votes
0answers
81 views

NP-Hard or PTIME?

I am working on my research problem that essentially boils down to the following question. Consider an $N \times N$ matrix. There is a man at given a starting point $(x,y)$. In each unit of time, the ...
6
votes
0answers
142 views

An optimal subspace projection problem

Suppose we have a $k$-dimensional subspace $V$ in $\mathbb{R}^n$ given by a basis $\{v_1,\cdots,v_k\in \mathbb{R}^n\}$, find an index set $I\subset [n]$ with $|I|=m$ where $k\le m\le n$, such that $$\...
-1
votes
1answer
59 views
3
votes
1answer
70 views

Interval partitioning with restrictions: NP-complete or efficiently solvable?

The interval partitioning problem can be solved efficiently using a greedy algorithm. However, adding restrictions on the interval assignment to the problem results in a problem that appears harder. ...
2
votes
1answer
85 views

NP hard proving: separate graph into a set of the same size disjoint parts by maximizing the shared neighbours of each part

Given a graph $G=\{V,E\}$ where $V$ denotes the nodes and $E$ denotes edges. The size of the node $|V| = nk$. The target is to separte the graph into $n$ disjoint parts $P=\{V_i\}_{i=1}^n$ and the ...
3
votes
0answers
65 views

Uniformly sampling or counting connected graph partitions with any number of blocks

Question: Is it possible to uniformly sample in polynomial time from the set of all connected partitions of a graph? Or is there a JVV type argument that proves this to be NP-hard? To clarify: By a ...
25
votes
1answer
625 views

Is there a problem that is easy for cubic graphs but hard for graphs with maximum degree 3?

Cubic graphs are graphs where every vertex has degree 3. They have been extensively studied and I'm aware that several NP-hard problems remain NP-hard even restricted to subclasses of cubic graphs, ...
2
votes
1answer
67 views

3-dimensional matching variant

In the normal version of the matching problem, we are given a set of vertices $X$, $Y$, and $Z$, each of size $n$, and a set of edges $E\subseteq X\times Y\times Z$. We need to find a matching $M\...
3
votes
1answer
110 views

What is the complexity of this weighted b-edge matching problem?

I'm wondering about the complexity of the following variant of the Generalized Weighted b-edge Matching problem: Input: An undirected multigraph $G = (V, E)$ without loops, an edge partition $(E_1,...
7
votes
5answers
394 views

NP-complete decision problems on deterministic automata

Do you know any NP-complete decision problems on deterministic automata? Most NP-complete problems that come to my mind are either (see, or here) graph theoretical, or involve some string rewriting or ...
0
votes
2answers
108 views

Order of quantifiers in the definition of NP-completeness: does the reduction allow arbitrary polynomials? [closed]

Arora and Barak define NP-completeness as the following: "We say that a language $A \subseteq \{0, 1\}^∗$ is polynomial-time Karp reducible to a language $B \subseteq \{0, 1\}^∗$ denoted by $A \leq_p ...
2
votes
0answers
95 views

Complexity of a scheduling problem with a fixed left bound of jobs

Consider the following scheduling problem. We have a finite set of jobs $\mathcal{J}= \{j_1, j_2, ..., j_n\}$ to do. Every job $j_i$ has its own value $c_i$ (the amount of money we are paid for ...
7
votes
1answer
276 views

Complexity of constructing minimum depth decision trees

I am interested in the computational complexity of Problem 1: Given a finite, non-empty set $J$, given $A, B \subseteq \{0,1\}^J$ such that $A \cap B = \emptyset$, and given $n \in \mathbb{N}$, does ...
-3
votes
1answer
102 views

is this selection problem np-hard? [closed]

Give $n$ clusters $C=\{C_i\}_{i=1}^n$ where each cluster consists of a set of similar points, i.e., $C_i=\{c_j\}_{j=1}^{|C_i|}$. The similarty between two points $c_i$ and $c_j$ is denoted as $w(c_i,...
0
votes
1answer
98 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-...
6
votes
1answer
92 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 ...
0
votes
0answers
85 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 $\sum_{i\in I} w_i \leq B$ ...
24
votes
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 ...
0
votes
0answers
60 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
vote
1answer
134 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
votes
1answer
124 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
votes
2answers
145 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
votes
2answers
115 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'\...
37
votes
16answers
7k 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. ...
12
votes
0answers
137 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
votes
1answer
713 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
votes
1answer
142 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
votes
0answers
76 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 ...

1
2 3 4 5
12