All Questions
Tagged with co.combinatorics computational-complexity
216 questions
9
votes
0
answers
2k
views
Exactly Counting the Number of Lattice Points in an $n$-Dimensional Sphere
Let $S_n(R)$ denote the number of lattice points in an $n$-dimensional "sphere" with radius $R$. For clarification, I am interested in lattice points found both strictly inside the sphere, and on its ...
- 311
-4
votes
1
answer
220
views
What is the computationally simplest way to universally index the set of simple graphs?
If given a simple, integer-labeled, but not necessarily connected, graph $G := (V,E)$ consisting of at least one vertex, i.e. $\lvert \rvert V \lvert \rvert \geq 1$, then is there a function to ...
- 387
1
vote
1
answer
82
views
The complexity on calculation of the Graev metric on the free Boolean group of a metric space
For a set $X$ by $B(X)$ we denote the family of all finite subsets of $X$ endowed with the operation $\oplus$ of symmetric difference. This operation turns $B(X)$ into a Boolean group, which can be ...
- 41.8k
2
votes
0
answers
77
views
Confirming existence in polynomial time while solution finding is NP-complete
Assume P≠NP.
Say there's an NP-complete decision problem:
Does $P$ have a $Q$ ?
And we have a proposition $F$ computable in polynomial time, where $F(P)$ implies the existence of a solution in ...
- 9,501
1
vote
0
answers
29
views
Existence of Costas array with specified displacment vectors?
Costas array is a set of $n$ points lying on the square of a $n×n$ checkerboard, such that each row or column contains only one point, and that all of the $n(n − 1)/2$ displacement vectors between ...
- 2,993
6
votes
1
answer
173
views
What is the minimum worst-case length of an element removal game?
A game is played as follows. There is a set $X = \{1, \ldots, n\}$. Player 1 is trying to find a "locally minimal subset" $M \subseteq X$ - that is, player 2 has said that $M$ is good, and also that ...
- 1,331
44
votes
4
answers
2k
views
A curious process with positive integers
Let $k > 1$ be an integer, and $A$ be a multiset initially containing all positive integers. We perform the following operation repeatedly: extract the $k$ smallest elements of $A$ and add their ...
- 5,590
7
votes
0
answers
93
views
Combinatorial region-halfplane incidence structures
I've seen a bunch of similar MO questions, yet hopefully this is not a complete duplicate.
Consider $n$ halfplanes in $\mathbb{R}^2$ with their borders in general position, that is, no point of $\...
- 5,590
1
vote
0
answers
78
views
Bipartite clustering is NP-hard?
Let $G = (A\cup B, E)$ be a bipartite graph with edge weights $w: E\to \mathbb{R}$. Find a partition $B_1, B_2$ of $B$ and a nonempty disjoint subsets $A_1, A_2$ of $A$ such that $w(A_1,B_1) + w(A_2, ...
- 221
21
votes
0
answers
441
views
Straight-line drawing of regular polyhedra
Find the minimum number of straight lines needed to cover a crossing-free straight-line drawing of the icosahedron $(13\dots 15)$ and of the dodecahedron $(9\dots 10)$ (in the plane).
For example, ...
- 4,535
29
votes
2
answers
1k
views
Determining if a rational function has a subtraction-free expression
This question was first asked by Mehtaab Sawhney in Alex Postnikov's combinatorics class.
Given a rational function $F=P(x_1,...,x_n)/Q(x_1,...,x_n)$ with (say) integer coefficients, it is often of ...
- 2,753
0
votes
1
answer
140
views
Maximum partition of bipartite graph
Let $G = (U, V, E)$ be a bipartite graph. Let $w: E \to \mathbb{R}$ be a weight function on the edge set $E$. Given subsets $U_1,\ldots, U_k \subset U, U_i\cap U_j = \emptyset$ and a partition $V_1,\...
- 221
0
votes
0
answers
149
views
Is 3-coloring bounded degree graphs subexponential: $O(\exp{(\sqrt{n}\log^2{n})})$?
We got an argument that 3-coloring bounded degree graphs is subexponential
with complexity $O(\exp{(\sqrt{n}\log^2{n})})$.
The treewidth of a planar graphs on $n$ vertices is $O(\sqrt{n})$
and 3-...
- 25.4k
2
votes
1
answer
339
views
Bit complexity of Barvinok's algorithm
I have seen many references which state Barvinok's algorithm has polynomial time complexity for counting integer points of polytopes in fixed dimension.
What exactly is this arithmetic complexity?
...
- 13.9k
0
votes
1
answer
245
views
Modification of the subset sum problem - "perfect coverage" of the set with good solutions
I have a problem.
We have a set.
$S = (a_1, a_2 ... a_k)$ and an integer $x$.
We know that there is a sum of elements to $x$ in it.
We also know that:
if there is only one sum to $x$ then it must ...
- 133
2
votes
0
answers
113
views
How many bits/questions does it take to identify the most frequent number in an array?
Note that "most frequent" here means "any of the most frequent, don't care which".
Example $n=3$. Consider the Bell partitions
aaa
aab
aba
baa
abc
which subsume all possibilities of a 3 element array ...
- 4,829
2
votes
0
answers
254
views
maximum independent set in d-regular graphs
Does anyone know whether the maximum independent set problem is NP-hard in triangle free d-regular graphs and if it's NP-hard for all d larger than some threshold t? Can anyone provide a reference ...
- 213
7
votes
0
answers
342
views
Multidimensional hook length formula
A well-known hook length formula states that the number of ways to arrange the elements of $[n]$ in a Young tableau with $n$ cells so that all columns and rows are increasing is $\frac{n!}{\prod_c h(c)...
- 5,590
7
votes
2
answers
324
views
Graph isomorphism problem for minimally strongly connected digraphs
A minimally strongly connected digraph (MSC) is strongly connected (SC), while removal of any arc destroys this. That is, between any two vertices a, b there exists a directed path from a to b, while ...
19
votes
1
answer
616
views
How hard is it to tell when a finite set tiles the integers?
Given a nonempty set $B$ of integers between 1 and $n$, we wish to determine whether or not $\mathbb{Z}$ can be tiled with translates of $B$ (that is, covered by disjoint translates of $B$). I know an ...
- 19.7k
8
votes
0
answers
88
views
Is recognizing if a Latin square is isotopic to its transpose more efficient than computing its symmetry group?
Ihrig and Ihrig (2007) described a mathematical method for determining if a Latin square is isotopic to its transpose (where isotopic Latin squares vary by permuting the rows, columns and symbols). ...
- 1,327
4
votes
0
answers
84
views
Complexity of counting colorings of co-bipartite graphs?
A graph is co-bipartite if it is the complement of bipartite graph.
What is the complexity of counting colorings of co-bipartite graphs?
Unlike split graphs, the chromatic polynomial isn't of ...
- 25.4k
2
votes
1
answer
240
views
valuation from $\mathbb{Z}^n$ into $\mathbb{Z}$. Easy way?
Let $\mathcal{F}:=\{f_a\}_{a\in\mathbb{Z}}$ be a set of symbols indexed by the integers and satisfying the rules:
$$f_a=f_{-a} \qquad \text{and} \qquad f_af_b=f_{a+b}+f_{a-b}.$$
Define a linear ...
- 43.2k
1
vote
1
answer
206
views
Show $0-1$ Knapsack is polynomially reducible to this problem
I have already posted this question here but have not received an answer so I am cross-posting with hope to reach a larger amount of mathematicians:
Let $T=\{1,\cdots,n\}$ and consider the ...
- 225
19
votes
2
answers
1k
views
Explicit invariant of tensors nonvanishing on the diagonal
The group $SL_n \times SL_n \times SL_n$ acts naturally on the vector space $\mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$ and has a rather large ring of polynomial invariants. The element $$\...
- 148k
5
votes
2
answers
413
views
An interesting variant on the maximum independent set problem.
Suppose i have a graph $G=(V,E)$ with $|V|=n$. Furthermore suppose i give you a maximum independent set $\mathcal{I}$ in $G$. Now suppose i obtain a new graph $G'$ from $G$ by removing a single vertex ...
- 213
3
votes
1
answer
158
views
Matroids of hypercubes
Let $M_k$ be the (oriented) matroid of the $2^k$ points $B_k = \{-1, 1\}^k$ in $\mathbb R^k$. In other words, the (oriented) circuits of $M_k$ are the minimal (signed) linear dependences among $B_k$.
...
- 459
3
votes
0
answers
98
views
Is the Graph Isomorphism problem in βP class?
βP is the limited nondeterminism NP, cf. https://complexityzoo.uwaterloo.ca/Complexity_Zoo:B#betap
Last year Laslo Babai proved that the GI problem can be solved in (deterministic) time
$\exp(\log^c(...
- 193
3
votes
2
answers
362
views
Sparse graphs that are hard to color
I am interested in knowing if there are any types of graphs that are very sparse, perhaps consisting of just connections between paths and cycles, and for which $k$-coloring is $\mathsf{NP}$-hard for ...
- 213
11
votes
3
answers
3k
views
Do you know a faster algorithm to color planar graphs?
while studying the four color theorem, I implemented an algorithm (in Python and Sage) that can color planar graphs much faster than the implementations I found around on internet.
The program can be ...
- 351
1
vote
0
answers
1k
views
Number of different combinations in a 0-1 knapsack problem with integer weights [closed]
My question is actually very similar to this other one: Given a vector of positive integers, count the number of combinations which have a sum that produces a different value. But, since this previous ...
- 153
3
votes
1
answer
497
views
Which is the most time efficient algorithm for having a Tait Coloring (edge-3-coloring) of planar cubic graphs?
Crossposted from: https://math.stackexchange.com/questions/1964486/which-is-the-most-time-efficient-algorithm-for-having-a-tait-coloring-edge
I wasn't able to find an efficient algorithm nor an ...
- 351
2
votes
0
answers
146
views
Is pos(n) an algorithmic counter?
Let $\ \mathbf N = \{1\ 2\ \ldots\}\ $ be the set of natural numbers. Let $\ f : \mathbf N\rightarrow\mathbf N\ $ be an arbitrary function, and $\ \forall_{n\in\mathbf N}\, F(n)\ :=\ \max_{k = 1\ldots ...
- 7,500
1
vote
1
answer
357
views
Analysis of a partition algorithm
EDIT:
I realized that if $J$ is not a solution so is $J^c$. I updated the algorithm because of this.
Given some positive integers $x_1,\cdots, x_n$.
The following algorithm is for solving the ...
user6671
3
votes
0
answers
230
views
On weight enumerators of codes
Are there $[n,k]_q$ constant rate $\frac kn$ and constant alphabet linear code families with automorphism group of size $\Omega((n-n^\beta)!)$ that have minimum distance $d=O(n^\alpha)$ and number of ...
- 13.9k
11
votes
1
answer
950
views
Computational complexity of computing simplicial homology
Is there any literature regarding the fastest known algorithm to compute the homology groups of a simplicial complex (on n vertices)? What about computing the fundamental group? The context is to tell ...
- 123
4
votes
0
answers
155
views
Effective "almost enumeration" of monotone boolean functions
Denote by $\mathcal{M}(n)$ the set of all monotone functions $\{0,1\}^n \to \{0,1\}$.
Can $\mathcal{M}(n)$ be represented as $\mathcal{M}(n) = \{ f(t) | t\in \{0,1\}^k \}$ such that:
1) $k = \log |\...
- 2,576
1
vote
0
answers
139
views
bounded degree graph colouring.
I was wondering if anyone could provide references on the following:
Is determining the chromatic number of a bounded degree graph APX-complete?
2.I've seen the result that states it is NP-hard to ...
- 903
3
votes
0
answers
181
views
Hypergraph edge colouring
I'm interested in knowing if finding the edge-chromatic number of a $k$-uniform $k$-partite hypergraph is NP-hard for $k\geq 3$ Could anyone provide a reference for the result? By edge-chromatic ...
- 903
12
votes
2
answers
292
views
Permutation search problems with no known $o(n!)$ algorithms
I am looking for problems for whose solution no known subfactorial algorithms are known. I am particularly interested in questions of isomorphism; that is, is there a permutation that converts one ...
- 333
5
votes
0
answers
145
views
Complexity of $\mathbb{Z}^n$ tilings
Let $\mathcal{T} \subset \mathbb{Z}^n$ be a finite set. Let $\Lambda \subset \mathbb{Z}^n$ be a full rank lattice.
We say that $\mathcal{T}$ is a $\Lambda$-tile for $\mathbb{Z}^n$ if the following ...
- 800
5
votes
0
answers
222
views
Littlewood-Richardson rule for the complete flag variety: GapP complete?
The cohomology ring of a complete flag variety $X$ has a basis of Schubert classes $S_u$ for permutations $u$. Define the Littlewood-Richardson coefficient $c_{uv}^w$ for permutations $u,v,w$ to be ...
- 2,168
1
vote
1
answer
332
views
Graph colouring for bounded degree graphs
I'm fairly new to colourings on bounded degree graphs i'm interested in the following questions,
For planar graphs with bounded degree $4$ is finding the colouring number $NP$-hard? So is ...
- 903
2
votes
1
answer
93
views
Directed edge-colouring
I'm interested to know whether the following problem is NP-complete or if there is an algorithm to solve it.
Suppose we are given a directed graph $G=(V,E^{\rightarrow})$ and we want to colour the ...
- 903
9
votes
1
answer
326
views
Factoring a multiset into a product of two multisets
Given a multiset $S$ of $mn$ numbers, how hard is to find multisets $A$ and $B$, of $m$ and $n$ numbers respectively, such that
$$ S = \{ xy \mid x\in A, y\in B \}~~~~\text{(multiset sense)}, $$
or ...
- 37.7k
1
vote
0
answers
28
views
The complexity of Max-K interval selection
I came up with the following problem, but do not know how to analyze it.
Let $S$ be an ordered set of integers with size $n$ (i.e., $S=\{1,2,...,n\}$). An interval $INV(a,b)$ covers the elements in $...
- 11
1
vote
0
answers
34
views
Some confusion regarding the definition of NPO reduction
I've seen the following definition in a paper on approximation preserving reductions.
Definition:Let $\pi_{1}$ and $\pi_{2}$ be two NPO maximization problems. Then we say that $\pi_{1} \leq_{R} \pi_{...
- 213
4
votes
1
answer
140
views
Complexity of this minimization
For integer $N$ consider the mapping $$f : (0,1)^N \to \mathbb{R}, \quad x \mapsto \min_{b \in \{0,1\}^N} \left\{ x^b + x^{1-b} \right\},$$
where $x^b = x_1^{b_1} \cdots x_N^{b_N}$ and $1-b = (1-b_1, \...
- 171
6
votes
0
answers
208
views
Is there a decomposition strengthening of the Sauer-Shelah Lemma?
Let $S \subset \{-1,1\}^n$. For a subset $A \subset [n]$ let $P_A$ denote the coordinate projection operator on S; in other words let $P_A(S)$ be the coordinate projection of $S$ onto the coordinates ...
- 13k
6
votes
2
answers
2k
views
How can I prove that these two graph coloring problems are polynomial time equivalent?
Given a graph $G(V,E)$. The standard $k$-coloring problem consists in finding a feasible coloring (no two adjacent nodes share the same color) of the nodes with $k$ colors. Let this problem be $P_1$.
...
- 225