All Questions
Tagged with co.combinatorics computational-complexity
216 questions
2
votes
0
answers
90
views
Computing basis of a lower set given basis of complementary upper set
In a poset $P$, $U\subseteq P$ is an upper set when for all $x\in U$, we have $y\ge x$ implies $y\in U$. Any subset of $P$ generates an upper set, and the basis of an upper set $U$ is the smallest ...
- 455
12
votes
2
answers
589
views
Faster multiplication with a restricted set of multiplicands?
Let $A$ be a set of $k>1$ distinct elements from a semigroup. We wish to compute the product
$$ p=b_1 b_2 \cdots b_n$$
where each $b_i\in A$.
Clearly $n-1$ multiplications suffice to compute $p$; ...
- 3,028
2
votes
1
answer
115
views
enumeration of connected blocks in finite size square
Given a square of size n by m, how many ways could we choose sites, such that all the sites are connected?
By "connected" we mean "connected" by adjacent sites. We will illustrate by example, say, we ...
- 8,086
5
votes
1
answer
213
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Aperiodic set of corner Wang Tile [closed]
There is quite some reference on aperiodicity of the edge-type of Wang Tile. But I could not yet find aperiodic corner type of Wang Tiles... Could someone provide me some instances (better with ...
CommunityBot
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1
vote
0
answers
68
views
Polynomial problems in graph classes defined by forbidden induced cyclic subgraphs
Let $C$ be a graph class defined by a finite
number of forbidden induced subgraphs, all
of which are cyclic (contain at least one cycle).
Are there graph problems that can be solved in
polynomial ...
- 25.4k
0
votes
0
answers
165
views
Resources about integral maximization problem
I am looking at the following problem. Given an interval I, and a function f over that interval, find sub-intervals for which:
The sum of the length of the sub-intervals is < k;
The sub-intervals ...
- 6,210
2
votes
3
answers
987
views
An established proof in Wang Tile which I doubt
When I was reading the paper:
Wang, Hao. "Notes on a class of tiling problems." Fundamenta Mathematicae 82.4 (1975): 295-305.
from http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82119.pdf
I could not ...
- 13.4k
3
votes
2
answers
297
views
Conjecture of a subset of Wang tile which might be decidable
From the two papers proving the undecidability of Wang tile in 1966 by Berger and in 1971 by RM Robinson, the tiles used in proving undecidability has a general common feature:
The left color and ...
CommunityBot
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2
votes
0
answers
131
views
characterization of all periodic tiling of a simple set of Wang Tile
Consider a set of Wang Tile such that all the edges are either 1 or 0.... there are 16 elements in such a set.
Now, I wish to characterize all the periodic tilings of this set (better if they are ...
- 867
1
vote
1
answer
631
views
relationship between corner tile and edge tile of wang tile
It is clear that any corner type of Wang Tile could be converted to edge type of Wang Tile by defining the edge color according to the corner color.
However, could we convert edge type of Wang Tile ...
- 236k
1
vote
0
answers
74
views
TSP: Approximation Ratio of the Double Tree Heuristic after Diagonals have been Removed
In their article "Double-Tree Approximations for the Metric TSP: Is the Best One Good Enough?", Vladimir Deineko and Alexander Tiskin derive a lower bound for the approximation ratio of the double-...
CommunityBot
- 1
7
votes
1
answer
288
views
What is the Essential Reason that allows a PTAS for the EUCLIDEAN TSP?
Questions:
Is there some understanding of the reason, why the euclidean TSP allows a PTAS, whereas the metric TSP in general does not and, is the PTAS stable under sufficiently small perturbation of ...
- 82.7k
30
votes
1
answer
3k
views
An edge partitioning problem on cubic graphs
Hello everyone,
I already asked this question on the TCS Stack Exchange, but it has not been resolved yet. Maybe readers of this forum will have other ideas or information, although I suspect that ...
CommunityBot
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3
votes
1
answer
509
views
Application of Combinatorics, Logic and computability theory in physical science: Tiling of Wang Tile with proportionality
The original problem of Domino Tiling and Wang Tile has great theoretical interest on computability theory... However, the great emerging problem on application of Wang Tile in material science and ...
CommunityBot
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4
votes
3
answers
405
views
powers in strings
I have a feeling that the following question might have been studied: Suppose I have a finite alphabet $A,$ with $|A| = n,$ and a string $S$ of length $N.$ A string can be said to contain a $k$-th ...
- 24.8k
2
votes
1
answer
183
views
Is the domination number NP for non-bipartite graphs?
Calculating the domination number is an NP-Hard problem. Does it remain NP-Hard if we restrict it to non-bipartite graphs?
- 44.4k
7
votes
1
answer
484
views
Seeming contradiction about P vs NP between graphclasses.org and at least two papers about clique in even-hole-free graphs
I believe correctness about clique in even-hole-free graphs
of graphclasses.org
and the paper Vertex elimination orderings for hereditary graph classes, Pierre Aboulker, Pierre Charbit, Nicolas ...
- 24.8k
8
votes
1
answer
418
views
A combinatorial problem concerned with logic circuits
Consider a logic circuit with two-bit gates only. The length of each gate is the number of bit lines that the gate crosses. How hard is to compute the maximum length for a given circuit? Notice that ...
CommunityBot
- 1
3
votes
2
answers
420
views
Making a graph claw-free by adding as few edges as possible
Independent set is polynomial in claw-free graphs,
so I am wondering if this can approximate independent set.
By adding enough edges to $G$ and gets claw-free $G'$.
IS in $G'$ is IS in $G$, so this ...
- 18.6k
6
votes
0
answers
154
views
Complexity of finding three perfect matchings with no edge in common in a bridgeless cubic graph
According to a conjecture:
Conjecture (Fan & Raspaud, 1994) Every bridgeless cubic graph contains three perfect matchings with no edge in common.
Equivalent statement here
Main question:
...
- 25.4k
1
vote
1
answer
603
views
The smallest altitude amongst the triangles formed by points in the unit circle
Let $S$ be a finite set of points inside the unit circle. Consider all possible triangles formed by three distinct points in $S$, and among all such triangles find the smallest altitude. Denote this ...
CommunityBot
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5
votes
2
answers
450
views
A simple language and systematic computations
The following somewhat popular simple computer language was enjoyed on sci.math, sci.math.research, pl.sci.matematyka, and perhaps before and after at several places (I wish I knew it's exact history)....
- 851
0
votes
0
answers
87
views
maximum weight k-edge problem
Given positive integer $k$ and an undirected graph $(V,E)$, with nonnegative (non-uniform) weights on the nodes. Find $k$ edges whose spanning nodes have the maximum weight.
Is this in P or NP? I ...
- 101
5
votes
1
answer
301
views
NP-hardness of sparsest cut
Consider bipartitioning the vertices of a graph $(V,E)$ into $V = P \cup Q$ to minimize $$\frac{|E(P,Q)|}{|P| |Q|},$$ where $E(P,Q)$ denotes the set of edges in the cut. The usual citation for NP-...
- 11
6
votes
0
answers
346
views
Enumerating (generalized) de Bruijn tori
Given a cyclic word $w$ of length $N$ over a $q$-ary alphabet and $k \in \mathbb{Z}_+$, consider the directed multigraph $G_k(w) = (V,E)$ with $V \subset$ {$1,\dots,q$}$^k$ given by the $k$-lets (i.e.,...
- 13.9k
1
vote
0
answers
157
views
Indecomposability of image transformations (pure algebra). Open questions
W-transformations -- definitions
We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of ...
- 7,500
4
votes
0
answers
73
views
Is the $d$-dimensional Arrangement of Trees still $NP$-hard?
The $d$-dimensional Arrangement Problem for general graphs is known to be $NP$-hard since the special case $d=1$ (OLA) already is (Garey et al, [1976]). For Trees however, the one dimensional case can ...
- 41
17
votes
0
answers
449
views
Splay trees and Thompson's group $F$
( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but:
Question: Is there a reformulation of the Dynamic ...
- 18.7k
5
votes
1
answer
475
views
Maximal chain of 1s in binary strings
Let $S$ be the set of $2^n$ binary $n$-bit strings. For every $x\in S$, let $f(x)$ is the maximal chain of bits 1 in $x$. So Can we find a good upper bound of $$F(n)=\frac{\sum_{x\in S}f(x)}{2^n}$$
Of ...
- 28k
17
votes
1
answer
960
views
Polynomial-time algorithm to compare numbers in Conway chained arrow notation
I am looking for a polynomial-time algorithm which, given a character string containing two numbers in Conway's chained arrow notation for large numbers, indicates whether the first number is less ...
CommunityBot
- 1
1
vote
1
answer
107
views
Provided a list of sets, $L$, computing an array where each entry $q_i \in Q$ is the family of sets in $L$ that have intersection $k$ with $l_i \in L$
I have a set of $(l_1, ..., l_N) \in L$ smaller sets, each with $(r_1, ..., r_M) \in R$ integer elements. I would like create an ordered array of $(q_1, ..., q_N) \in Q$ sets s.t.:
(1) Each $q_i \in ...
CommunityBot
- 1
1
vote
0
answers
112
views
Did anybody come across a computational problem which is related to the notion of cartesian product and is at least NP-hard?
Did anybody come across a computational problem which is related to the notion of cartesian product and is at least NP-hard?
Equally interesting would be to learn about such problems with a non-...
- 41
19
votes
1
answer
3k
views
Möbius Randomness of the Rudin-Shapiro Sequence
The Rudin-Shapiro sequence (also known as the Golay-Rudin-Shapiro sequence) is defined as follows.
Let $a_n = \sum \epsilon_i\epsilon_{i+1}$ where $\epsilon_1,\epsilon_2,\dots$ are the digits in the ...
CommunityBot
- 1
0
votes
1
answer
377
views
Algorithm for vector space
I have $n$ vectors $e_1 \in (\mathbb Z/2 \mathbb Z)^m,\dots,e_n \in (\mathbb Z/2 \mathbb Z)^m $
and a vector $ v \in (\mathbb Z/2 \mathbb Z)^m $
I need to find the better algorithm which answers ...
- 1,683
4
votes
1
answer
1k
views
Complexity of convex polytope volume calculation ? (Volume of Voronoi cell) (Error probability)
Assume I have polytope in R^k given by N (k<< N) linear inequalities (A_i x < b_i).
I guess complexity of its volume calculate is higher than linear in "N", am I right ?
(Is the complexity ...
CommunityBot
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1
vote
3
answers
5k
views
How to get the largest subset of a set of sets of intervals with no overlapping intervals
Given: Set of Set of Intervals. Example {{(1,2), (3,4)}, {(1, 3)}, {(13,14)}}
Wanted Result: Largest Subset, in which no Interval overlaps with another from every Set pairwise.
Example:
Input {{(1,...
CommunityBot
- 1
1
vote
1
answer
1k
views
k-uniform k-partite hypergraph matching in polynomial time
I have what seems like an elementary question, but google didn't throw up any answers for it. I would appreciate any pointers that MO users may provide.
It is well known that for $k\geq 3$ finding ...
- 1,288
7
votes
1
answer
535
views
Vertex transitive graphs
Does having vertex transitivity make the problem of calculating independence and chromatic numbers easier?
- 12.1k
1
vote
0
answers
266
views
Multiobjective semidefinite programming
Let $C$ be size $n \times n^{2}$.
Let $B$ be size $2^{g(n)} \times n^{2}$ where $g(n) > n$.
There is only one $\mathcal{1}$ per row of $C$ and remaining entries of $C$ are $\mathcal{0}$.
$B$ is ...
- 800
2
votes
2
answers
837
views
Enumerating all Hamiltonian Cycles in a Bipartite Vertex Transitive Graph
Hi everyone!
This is my first post, apologies if I made any mistakes anywhere.
Here goes the question:
Consider all length 7 binary sequences.
Let $X$ be the set of sequences with hamming weight 3 ...
- 360
5
votes
1
answer
780
views
Algorithmic war
No, not the war on drugs, but the game of War considered in
Does War have infinite expected length?
As noted in that discussion, the game of war can go on forever, but my question is: can it be ...
CommunityBot
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3
votes
3
answers
2k
views
Problem regarding subsets that sum to 0
Let $X=\{x_1,...,x_n\}$ be a multiset of $n$ real numbers, and let $x_1+\dots+x_n = 0$. Is there a way to find the maximum number of unique subsets any $X$ can have given $n$, such that each subset ...
- 9,114
5
votes
1
answer
269
views
Simplices in convex polytopes
This question is a direct generalization of:
Counting the (additive) decompositions of a quadratic, symmetric, empty-diagonal and constant-line matrix into permutation matrices
Given a convex ...
CommunityBot
- 1
5
votes
2
answers
612
views
A Boolean function that is not constant on affine subspaces of large enough dimension
I'm interested in an explicit Boolean function $f \colon \{0,1\}^n \rightarrow \{0,1\}$ with the following property: if $f$ is constant on some affine subspace of $\{0,1\}^n$, then the dimension of ...
CommunityBot
- 1
2
votes
0
answers
642
views
Hamiltonian paths in subgraphs of rectangular lattice graphs
Is following decision problem NP-hard / NP-complete:
Having vertex-induced subgraph of rectangular lattice graph determine if any Hamiltonian path exists
Having vertex-induced subgraph of rectangular ...
- 47.3k
20
votes
4
answers
870
views
Enumeration and random selection
In Peter J. Cameron's book "Permutation Groups" I found the following quote
It is a slogan of modern enumeration theory that the ability to count a set is closely related to the ability to pick a ...
- 37.7k
4
votes
1
answer
365
views
Computing Simultaneous Hamming Neighborhood for a Set of Strings
Let $S = \lbrace s_1, s_2 \ldots s_n \rbrace$ be a set of strings each of length $k$ from an alphabet $\Sigma$, $h(s_i, s_j)$ denote the hamming distance between two strings. The simultaneous hamming ...
2
votes
2
answers
4k
views
Greedy approach to 0-1 Knapsack problem in specific instances
The 0-1 knapsack problem is known to be NP-complete, and the greedy approach by Dantzig (based on choosing on the basis of density or value/weight) can be shown to be suboptimal using counterexamples. ...
- 3,934
8
votes
3
answers
947
views
Boolean Cube of Primes
For a large enough $n$, and a parameter $ m $ I'm looking for a subset of the prime numbers in the range $[n,2n]$ with a unique structure. I am looking for a prime $p$ and a set of $m$ positive (not ...
- 1
3
votes
3
answers
852
views
A conjecture on a Subset Power Sum Problem motivated by Computer Science
Let $X=\{x_{1}, \cdots , x_{n}\}$ be a set of $n$ positive integers and integer $i \ge 1$. Let’s say that the set $X$ is $i$-sum-avoiding if for any nonnegative integers $c_{1}, \cdots, c_{n}$ such ...
- 1