Questions related to NP-hardness and NP-completeness.

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3
votes
1answer
136 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
vote
0answers
69 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 ...
1
vote
0answers
47 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 ...
0
votes
0answers
44 views

Capacitated min-k-cut problem [on hold]

In the capacitated min-$k$-cut problem we are given a graph (hypergraph) with non-negative edge (hyperedge) weights. The task is then to find a partitioning of the graph's vertices into $k$ sets of ...
1
vote
1answer
153 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
108 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
202 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
106 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
107 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
130 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
101 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
votes
0answers
495 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
vote
1answer
151 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
188 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
vote
0answers
113 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
126 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
78 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 ...
0
votes
0answers
59 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
votes
0answers
117 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., ...
7
votes
1answer
165 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 ...
10
votes
1answer
220 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
votes
1answer
247 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
177 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
votes
4answers
408 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
138 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
200 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
votes
0answers
96 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
144 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 ...
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. $P(w_e=i)=p(1-p)^{i-1}, i\geq 1$. $s,t$ ...
1
vote
0answers
43 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
255 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 ...
1
vote
0answers
96 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
220 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
votes
0answers
458 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
votes
0answers
30 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
votes
1answer
230 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). ...
2
votes
1answer
170 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 ...
0
votes
0answers
36 views

Complexity of fixed fractional relaxations

I recently stumbled upon a curious relaxation of a combinatorial problem that was restricted to a fixed subset of fractional solutions. I am now wondering if the complexity of such constrained ...
2
votes
1answer
112 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 ...
21
votes
1answer
408 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 ...
3
votes
0answers
43 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 ...
4
votes
0answers
230 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 ...
8
votes
1answer
95 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 ...
3
votes
2answers
279 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, ...
1
vote
0answers
187 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}\;\; ...
-1
votes
1answer
183 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
136 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 $D'$ ...
0
votes
0answers
144 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
0answers
82 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 ...
3
votes
1answer
399 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 ...