Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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6
votes
1answer
368 views

What is the complexity of vertex cover on k-partite graphs?

Given a k-partite graph which is already partitioned into k parts, what is the complexity of finding a vertex cover of minimum size? I guess that it's NP-hard, but couldn't yet prove it or find ...
4
votes
1answer
201 views

Balanced Max-2-SAT NP-Hardness

The Balanced Max-2-SAT is a special case of Max-2-SAT (each clause is a disjunction of exactly 2 literals) in which for every variable $x$, there is a $k$ such that $x$ appears positive exactly $k$ ...
1
vote
0answers
164 views

Is “Binary Interval Tree” NP-hard? [closed]

The input is set of (disjoint) intervals $I$. The output should be the following rooted binary tree. Each leaf node corresponds to an interval from $I$. Each interior node contains an interval which ...
3
votes
0answers
179 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 ...
9
votes
1answer
227 views

Complexity of digraph homomorphism to an oriented cycle

Given a fixed directed graph (digraph) $D$, the $D$-COLORING decision problem asks whether an input digraph $G$ has a homomorphism to $D$. (A homomorphism of $G$ to $D$ is a mapping $f$ of $V(G)$ to $...
16
votes
0answers
645 views

Deeper look at Algorithmica?

Russell Impagliazzo published "A Personal View of Average-Case Complexity" (preprint) back in 1995. He presented five possible worlds we could be living in, depending on how P and NP were related. The ...
18
votes
4answers
1k views

“All-different hypergraph coloring” - known problem?

I am interested in the following problem: Given a set X and subsets X_1, ..., X_n of X, find a coloring of the elements of X with k colors such that the elements in each X_i are all differently ...
1
vote
1answer
268 views

Huffman Tree Depth, Is there any theory?

I'd like to as a variation on this question regarding Huffman tree building. Is there any theory or rule of thumb to calculate the depth of a Huffman tree from the input (or frequency), without ...
1
vote
0answers
141 views

Quantum computer versus Random 3-SAT?

It seems to be commonly believed that quantum computer cannot efficiently solve NP-hard problems. What about the challenging problems in average-case, such as Planted Clique and Random 3-SAT?
-4
votes
1answer
206 views

NP-completeness of one generalized subset sum problem (target sum belongs to interval) [closed]

I need to prove that decision problem: for a given set of positive integers $a_1, ..., a_n$, does it exist a subset that sums up to a value within interval $[\frac{1}{2}\sum a_i; \frac{1}{2}\sum a_i+...
5
votes
1answer
206 views

Graph class with easy chromatic number, but NP-hard coloring

Is there a graph class for which the chromatic number can be computed in polynomial time, but finding an actual $k$-coloring with $k=\chi(G)$ is NP-hard? Without any further restriction the answer ...
1
vote
1answer
325 views

Is there a FNP problem that's NP-hard but not FNP-hard?

For the reductions, choose a class C such that [it's clear what FC means] and FC is not known to be able to solve the satisfaction search problem, and assume that FC indeed can't solve that search ...
3
votes
0answers
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(...
7
votes
1answer
214 views

Complexity of a graph-rewriting problem

I recently came across the following problem which seems to fall in the context of graph rewriting problems: Input: A graph $G=(V,E)$ with maximum degree 3, an edge $e_0 \in E$ and pairs of graphs $(...
8
votes
1answer
267 views

BQP algorithm for two graph bisection problems and its implications on NP $\subseteq$ BQP

I read the paper Ahmed Younes, "A Bounded-error Quantum Polynomial Time Algorithm for Two Graph Bisection Problems", 2015. doi:10.1007/s11128-015-1069-y which is published in Springer's journal ...
10
votes
1answer
787 views

Definition of Planar 3-SAT

What is the standard definition of Planar 3-SAT? I have seen a number of different definitions. What was the original paper that defined it and proved it to be NP-complete?
11
votes
2answers
1k views

On the provability of P versus NP

First of all, my understanding on Gödel's incompleteness theorem (and formal logic in general) is very naive, also is my knowledge on theoretical computer science (meaning only one graduate course ...
1
vote
2answers
1k views

Is longest common subsequence with bounded occurrences NP-complete?

The general longest common subsequence problem (LCS) over a binary alphabet is NP-complete. Does the problem remain NP-complete if each input string has m zeros and n ones, where m and n are constants?...
11
votes
1answer
215 views

maximize MST(G[S]) over all induced subgraphs G[S] in a metric graph

Has this problem been studied before? Given a metric undirected graph G (edge lengths satisfy triangle inequality), find a set S of vertices such that MST(G[S]) is maximized, where MST(G[S]) is the ...
4
votes
1answer
228 views

Find worst case input for a program solving an NP-hard problem

I am trying to find a way to find a worst-case input for a black-box implementation of an algorithm with worst-case exponential runtime. The problem that the program solves (integer linear ...
13
votes
4answers
824 views

Finding the sparsest solution to a system of linear equations

How hard is it to find the sparsest solution to a system of linear equations? More formally, consider the following decision problem: Instance: A system of linear equations with integer coefficients ...
5
votes
2answers
190 views

Popular average-case complexity assumptions

Except for planted clique and random 3SAT, what are the other commonly used average-case complexity assumptions?
4
votes
2answers
235 views

Partition graph into 2 or more claw-free subgraphs

Is it NP-hard to partition the vertex set of graph G into k subsets so that they induce k claw-free subgraphs of G?
14
votes
0answers
553 views

Does solving matrix multiplication in quadratic time imply that SETH is false?

I have a little conjecture that if you could perform matrix multiplication (or solve 3-clique) in $O(n^2 \log(n))$ time, then you could solve CNF-SAT in $O(2^{(1-\epsilon)n})$ time. In other words, ...
2
votes
0answers
124 views

Empty sudoku and NP-completeness [closed]

My question is straightforward: Is an empty sudoku grid (not partially completed) still NP-complete?
4
votes
1answer
271 views

Bipartite Graphs - Maximum subset of one partition with at most n neighbours - NP-hard?

Given: A bipartite graph G=(U,V,E) Integers n and k. Decision Problem: Is there a subset of U of size k that has at most n neighbours? I am trying to figure out whether this problem is NP-hard (...
1
vote
1answer
291 views

Is this problem #P-hard and why?

Problem: In a directed graph $G=(V,E)$, each edge $e\in E$ is associated with a weight $w_e$ which is geometrically distributed with a parameter $p$, i.e. $P(w_e=i)=p(1-p)^{i-1}, i\geq 1$. $s,t$ are ...
1
vote
0answers
64 views

Multicuts (or multiway cuts) for 3 Terminals of Minimum Capacity

For each fixed $k \geq 3$, the following well-known problem is NP-complete. k-Multicut (aka "Multiway Cut", "k-Way Cut", "Multicut") Input: $(N,l)$ where $N=(V,E,T,c)$ is an undirected $k$-terminal ...
6
votes
1answer
290 views

NP-hardness of coloring uniform hypergraphs

Since a $2$-uniform hypergraphs are just graphs. The problem of deciding if $2$-uniform is $k$-colorable for $k=1,2$ is easy, and NP-hard for $k \geq 3$ colors. This is well know and I have seen ...
7
votes
0answers
1k views

Reduction from Vertex Cover to Max-Cut? [closed]

I am referring to Computational Complexity by Arora and Barak for my course. In the section on NP-completeness reductions, the book has a diagram that is represents how one NP-complete problem ...
0
votes
0answers
33 views

On approx-preserving P- and A-reducibilities

Let $X$ and $Y$ be two NPO problems. Let $(f,g)$ be a reduction between $X$ and $Y$, in particular, assume that $(f,g)$ is both P-reduction and A-reduction, i.e., there exist two poly-time ...
16
votes
2answers
2k views

Linear diophantine equation in non-negative integers

There's only very little information I can find on the NP-complete problem of solving linear diophantine equation in non-negative integers. That is to say, is there a solution in non-negative $x_1,x_2,...
3
votes
1answer
197 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 ...
2
votes
1answer
166 views

When can a convex function induce submodularity?

Say I have a real valued convex function $f$ on the hypercube $[-1,1]^n$. Let $f'$ be the induced function on the discrete hypercube $\{-1,1\}^n$. Now I want to find a vertex on $\{-1,1\}^n$ on which ...
22
votes
1answer
650 views

Is it still open to determine the complexity of computing the treewidth of planar graphs?

For a constant $k \in \mathbb{N}$, one can determine in linear time, given an input graph $G$, whether its treewidth is $\leq k$. However, when both $k$ and $G$ are given as input, the problem is NP-...
8
votes
1answer
277 views

NP-hardness of a Set Cover specialization

Is the following problem NP-hard? Given a set of $N$ real numbers (targets) $x_1,\dotsc,x_N$ and a "trident" defined by two distances $a$, $b$ from the center of the trident, what is the minimum ...
6
votes
0answers
582 views

Is this permutation-sum problem NP-complete?

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,...
3
votes
0answers
56 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,...
4
votes
2answers
355 views

Graph coloring/partitioning problem

I'm interested in the complexity of the following problem: Problem $P$: Given an undirected planar graph $G=(V,E)$ and a weight function $w: E \rightarrow \mathbb{Z}$ (so weights can be negative, ...
11
votes
4answers
768 views

Does NP-completeness/hardness have to be constructive?

Is there any $L\in {\bf NP}$ with the following properties: It is known that $L\in {\bf P}$ implies ${\bf P}={\bf NP}$. There is no (known) polynomial time Turing reduction of $SAT$ (or some other ${\...
2
votes
0answers
210 views

Convex hull of polytopes

Consider a set of polytopes $P_i : i=1,2,...,k$ each of which has a structure as $P_i:= \{(x_{i1},x_{i2},..., x_{in})\; |\; x_{ij} \in [a_{ij}, b_{ij}] \subseteq [0,1]\}\;\; \text{for all}\;\; j=1,......
-1
votes
1answer
303 views

What is the simplest known solver for a np-complete problem?

Lets define the simpler of two terms as the one with shortest description length on the untyped λ-calculus. Trying to find the simplest solver for a np-complete problem, I've got this: ...
3
votes
2answers
175 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 $D'$ ...
54
votes
11answers
2k 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, ...
0
votes
0answers
154 views

NP-hardness of minimizing sum of complicated objective function

In our research, we faced the following problem optimization problem: Input: a list of $k$ pairs of positive integers $(n_1,d_1), \ldots, (n_k,d_k)$; an integer $m$. Output: $P$, a partition of the ...
4
votes
1answer
604 views

If BQP contains NP, does this mean that P=NP?

There is a question raised by Scott Aaronson in one of his papers [1]: "Could we show that if NP ⊆ BQP, then the polynomial hierarchy collapses?". Assuming the answer is yes, and it is also know that ...
4
votes
0answers
137 views

What is the current “state-of-the-art” solver for quadratic knapsack problems?

New to this forum, so please let me know if my question format is incorrect. For linear KP with $n$ items and $c$ capacity, dynamic programming can find exact solutions in $\mathcal{O}(nc)$. I have ...
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 ...
4
votes
1answer
181 views

Is SAT with two “opposite” solutions NP-hard?

Here is a variant of the SAT problem in which a satisfying assignment must have additional properties. Input: A 3-CNF formula $f$ with variables $x_{1\dots k}$. Output: For an assignment $S$ of $x_{1\...
5
votes
0answers
119 views

Completing a matrix (over the reals) to be singular

Consider the following problem: you are given a matrix (say, with rational entries) some of whose entries are actually left blank; can these blanks be filled in with real numbers so that the resulting ...