Questions related to NP-hardness and NP-completeness.

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Is sparse embedding of a NP-complete problem in a polynomial problem NP-complete?

Consider the following problem P: Input is a finite graph G. If the number of vertices in G is 2^2^i for some integer i, then output a minimum vertex cover of G; otherwise output empty set. Can I say ...
-2
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1answer
59 views

Given oracle for Max-3SAT compute clauses that cannot be satisfied

We know that Max-3SAT is NP-hard to compute exactly (and also hard to approximate to a particular constant multiplicative factor). However, suppose you are given an oracle for Max-3SAT that tells you ...
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1answer
51 views
+50

Minimum-weight feedback edge set in undirected graph - how to find it? Is it NP hard problem?

Let G = (V,E) be an undirected graph. A set F ⊆ E of edges is called a feedback-edge set if every cycle of G has at least one edge in F. Suppose that G is a weighted undirected graph with positive ...
6
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2answers
155 views

Solution/Hardness of the following (integer) budgeted problem?

I have no idea how to solve the following INTEGER problem or prove its hardness. Thanks for any help/comment/open discussion! Assume there are $N$ startups. For each startup $i$, you can invest $x_i\...
7
votes
1answer
151 views

What is known about the hardness of the chromatic index for restricted graph classes?

There is a nice paper from 1991 that contains three diagrams about different graphclass-families showing what is known about the hardness of determining the chromatic index for them. Are there any ...
4
votes
1answer
129 views

Kth best problem that is NP-hard for K polynomial

A Kth best problem is, given some constraint, to find a solution that has the Kth best value compared to all solutions that meet the constraint. One way to write this as a decision problem is to ...
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0answers
20 views

Reference Request : Computing directed domination number of Oriented graphs

An oriented graph ($\overrightarrow{G}$) is a directed graph having no symmetric pair of directed edges (arcs). A directed dominating set in a directed graph $D$ is a subset $S$ of vertex set $V$ ...
2
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1answer
70 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 ...
7
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1answer
197 views

NP-hardness on Cayley graphs

What is known about complexity of NP-hard problems on Cayley graphs? Suppose that the graph is given explicitly as the multiplication table of the group and the list of generators. So the input ...
1
vote
1answer
48 views

Constant Width Max Sum Product Multi-objective Shortest path problem

This question is a follow-up on the question I asked three days ago here. For convenience I restate it here. I am given a graph. Each edge is labelled by a vector of numbers, called weights. They ...
5
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1answer
63 views

Max Sum Product Multi-objective Shortest path problem

Is anything known about the following problem: I am given a graph. Each edge is labelled by a vector of numbers, called weights. They are numbers between 0 and 1. A path is first assigned a vector, ...
1
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0answers
35 views

About increasing the objective values of certificates for Max-Clique SDP

Say you write a round-k Lasserre (or any other hierarchy!) SDP relaxation of the Max-Clique problem. Lets say one now finds (or knows how to sample with high probability) a graph $G_1$ of size $n$ and ...
2
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0answers
71 views

Max common sub forest on $k$ graphs

Not sure how to phrase this really, but here goes. Suppose you are given $k$ simple graphs, each having exactly $m$ edges. The edges in each graph are labeled from 1 to $m$. The problem is to find ...
9
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0answers
98 views

NP-hardness of approximation for unconstrained submodular maximization

The problem of unconstrained submodular maximization can be phrased as follows: Given a non-negative submodular function $f$ on a domain $D$ find a set $S \subseteq D$ maximizing $f(S)$. Here a ...
2
votes
0answers
136 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 ...
3
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1answer
177 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 $$\max_{S\subseteq \{1,\ldots,T\}} \sum_{i\in S} a_i$$ $$...
1
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0answers
73 views

About representability of optimization versions of NP-complete questions as polynomial optimization over the hypercube

Naively, in my very limited awareness, it feels that the Max-CUT is a very "special" NP-Hard problem because for a graph with edge-set $E$, it can be written as the question of trying to maximize a $n$...
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0answers
53 views

About complexity of recovering or learning Bayesian networks

Are there complexity theoretic results about recoverability or learnability of the marginals (of the source vertices) and the conditionals (along each of the edges) of a Bayesian network from having ...
1
vote
1answer
166 views

Packing sets to maximize overlap

For a set of sets $A$, let $\cup A := \cup_{S \in A} S$. Consider the following problem: Input: a list of $m$ weights $w = (w_1, \ldots, w_m)$, a list of $n$ distinct subsets $T = (S_1, \ldots, ...
4
votes
2answers
112 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 ...
8
votes
1answer
218 views

On reducing the hardness of CNF-SAT to k-Clique

CNF-SAT refers to the following problem: Given a boolean formula $\phi$ in conjunctive normal form, does there exist an assignment to the variables that satisfies $\phi$. There are several ...
6
votes
1answer
113 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
110 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
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0answers
135 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
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0answers
105 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 ...
17
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0answers
505 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 ...
1
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1answer
154 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 ...
10
votes
1answer
193 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 $...
1
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0answers
117 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?
1
vote
1answer
136 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 ...
-4
votes
1answer
95 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+...
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0answers
62 views

On NP-hardness of list decoding of RS codes

Given an $[n,k,n-k+1]_q$ Reed Solomon code we know that . Unique decoding upto half minimum distance can be done in polynomial time. . List decoding upto $n-\sqrt{nk}$ can be done in polynomial ...
3
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0answers
118 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
173 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 $(...
9
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1answer
229 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 ...
2
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1answer
251 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 ...
4
votes
1answer
179 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 ...
12
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4answers
430 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
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2answers
140 views

Popular average-case complexity assumptions

Except for planted clique and random 3SAT, what are the other commonly used average-case complexity assumptions?
4
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2answers
201 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?
2
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0answers
97 views

Empty sudoku and NP-completeness [closed]

My question is straightforward: Is an empty sudoku grid (not partially completed) still NP-complete?
4
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1answer
148 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-...
1
vote
1answer
148 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
44 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
256 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 ...
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0answers
100 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 $...
7
votes
0answers
250 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 ...
14
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0answers
463 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, ...
0
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0answers
31 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 ...
11
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1answer
234 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 ...