Questions tagged [heuristics]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2 votes
1 answer
113 views

Constructing optimal Hamilton cycles from optimal Hamilton paths

Question: can the shortest Hamilton cycle in a complete symmetric graph with weighted edges be constructed from the shortest Hamilton path in the same graph by connecting its ends and then exchanging ...
Manfred Weis's user avatar
  • 12.6k
0 votes
0 answers
21 views

Calculation of cardinality constrained minimum weight matchings

Given a complete weighted graph $G(V,E),\ |V|=2n$, calculating a minimum weight matching with $n-k$ edges can be reduced to calculating a perfect matching in $H(V+U,E+F),\ |U|=2k,\ F=(u\in U,v\in V),\ ...
Manfred Weis's user avatar
  • 12.6k
0 votes
0 answers
13 views

Identifying optimization-potential in weighted Hamilton cycles

From the (possibly incorrect) assumption that inoptimality of weighted Hamilton cycles, that resemble a heuristic solution to a TSP problem, can be identified by calculating the MST (minimum spanning ...
Manfred Weis's user avatar
  • 12.6k
0 votes
1 answer
70 views

Optimality of a "shopping" heuristic

Suppose the following situation: we have to buy $n$ goods $g_1,\,\dots,\,g_n$ starting at day 1 and we can't buy more than one good per day. On day $d$ the prices are $p_1^d,\,\dots,\,p_n^d;\quad p_i^...
Manfred Weis's user avatar
  • 12.6k
9 votes
0 answers
178 views

Cyclic numbers of the form $2^n + 1$

A cyclic number (or cyclic order) is a number $m$ such that the only group of order $m$ is the cyclic group $\mathbb{Z}/m\mathbb{Z}$. The set of cyclic numbers admits a couple of cute number-theoretic ...
Z. A. K.'s user avatar
  • 596
0 votes
0 answers
20 views

LP formulation of $k$-opt moves

Question: what is known about formulating $k$-opt moves that strive for improving the length of Hamilton cycles by means of exchanging $k$ of the tour edges with $k$ non-tour edges? Specifically: are ...
Manfred Weis's user avatar
  • 12.6k
-2 votes
2 answers
144 views

Greedy euclidean tour expansion - a case of unexpected hanging?

In the euclidean plane an common heuristic for the TSP is to start with the convex hull of the point set and then successively integrate as the next point and insertion position the combination that ...
Manfred Weis's user avatar
  • 12.6k
0 votes
1 answer
34 views

Edge-length constraints from greedy matching

The subject of this question are perfect matchings of a complete undirected graph $G(V,E), n:=\mathrm{card}(V)=2k$, without self-loops or parallel edges and $n=2k$ vertices. The objective is to ...
Manfred Weis's user avatar
  • 12.6k
2 votes
2 answers
335 views

Heuristic model for Lehmer pairs?

Rodgers and Tao proved that the De Bruijn–Newman constant $\Lambda$ is non-negative. The study of $\Lambda$ goes back at least to Lehmer's paper, On the roots of the Riemann zeta-function, whose ...
Timothy Chow's user avatar
  • 78.1k
0 votes
0 answers
26 views

Are there any examples of "autonomous" TSP heuristics

By "autonomous" TSP heuristic I mean algorithms whose reported edge-set for a short Hamilton cycle is invariant under the addition of vertex weights; the terminology is borrowed from ...
Manfred Weis's user avatar
  • 12.6k
1 vote
0 answers
254 views

Heuristics for minimum path cover of undirected graph

Suppose you would like to find a set of paths on an undirected connected graph that ensures every vertex is visited exactly once while minimising the number of paths used. In this case, a "path&...
skytect's user avatar
  • 139
1 vote
1 answer
513 views

Factorial primes: expected finite or infinite?

A factorial prime is of the form $n! \pm 1$. The first $14$ factorial primes are listed in the Online Integer Sequences (OEIS) A088054: $$ 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, ...
Joseph O'Rourke's user avatar
0 votes
1 answer
129 views

Heuristics for lightweighted "cubic" spanning trees

I have the problem of calculating a good approximation of the minimimum-weight spanning tree with vertex-degrees in $\lbrace 1,3\rbrace$ of a complete symmetric graph, without parallel edges or self-...
Manfred Weis's user avatar
  • 12.6k
1 vote
0 answers
41 views

Vertex cover via maximally unbalanced spanning trees

The vertex cover problem asks for a smallest subset $U\subseteq V$ that is adjacent to all edges of a symmetric graph $G(V,E)$. Inspired by the observation that led to this question Perfectly balanced ...
Manfred Weis's user avatar
  • 12.6k
2 votes
1 answer
136 views

Generating short Hamilton cycles from complete graphs

Let $G(V,E)$ be a complete symmetric graph without self-loops or parallel edges; depending on the context the edges may however be interpreted as a pair of antiparallel arcs of equal weight. A vertex ...
Manfred Weis's user avatar
  • 12.6k
5 votes
0 answers
184 views

Heuristics for the very little torsion in the cohomology of Shimura variety

Consider the following statement which is a part of Conjecture 1.3 in the paper titled "The asymptotic growth of torsion homology for arithmetic groups" authored by N. Bergeron and A. ...
random123's user avatar
  • 411
0 votes
0 answers
49 views

Does LKH perform best with $\mathrm{1\unicode{x2013}trees}$

The LKH heuristic essentially generates sequence connected graphs with $n$ edges by means calculating minimum-weight spanning trees of $n-1$ of the vertices and connects the unspanned vertex to the ...
Manfred Weis's user avatar
  • 12.6k
0 votes
1 answer
123 views

Nicely motivated papers or book chapters on the formula for the sum of the $k$-th powers of the first natural numbers [closed]

Do you know of a text where I can find a nicely motivated proof of the formula for $1^{k}+2^{k}+\cdots+n^{k}$? At the very beginning of page 68 of Professor H. S. Wilf's generatingfunctionology, one ...
Jamai-Con's user avatar
-1 votes
1 answer
239 views

Helsgaun's $k$-Opt moves

In his 2009 paper General k-opt submoves for the Lin–Kernighan TSP heuristic, Helsgaun defines the local tour improvements on which the LKH heuristics are based as: with a cycle defined here: which ...
Manfred Weis's user avatar
  • 12.6k
3 votes
0 answers
161 views

Are class numbers of number fields with prime degree often $1$?

I have taken a look at the class number statistics of the L-functions and Modular Forms Database: https://www.lmfdb.org/NumberField/stats, table "Distribution by class number". It appears ...
wandersam's user avatar
  • 125
1 vote
1 answer
103 views

$\mathrm{LP}$ formulation for $\mathrm{k}$-$\operatorname{opt}$ moves

$\mathrm{k}$-$\operatorname{opt}$ moves are an idea to improve non-optimal Hamilton cycles in weighted symmetric graphs by exchanging $\mathrm{k}$ tour-edges with $\mathrm{k}$ edges that do not belong ...
Manfred Weis's user avatar
  • 12.6k
0 votes
0 answers
23 views

Complexity of heaviest 2-optimal vertex-disjoint cycle covers

Calculating lightest vertex-disjoint cycle covers of finite complete symmetric graphs with weighted edges can be done efficiently and also renders the edge set of the calculated cycles free of pairs ...
Manfred Weis's user avatar
  • 12.6k
1 vote
1 answer
533 views

Calculating vertex weights

Vertex weights are a metaphor for a constant value $\pi_i$ that is added to the weight of every edge $e_{ij}$ that is adjacent to vertex $v_i$ in a symmetric graph $G(V,E)$ with weighted edges. The ...
Manfred Weis's user avatar
  • 12.6k
59 votes
9 answers
5k views

Examples of back of envelope calculations leading to good intuition?

Some time ago, I read about an "approximate approach" to the Stirling's formula in M.Sanjoy's Street Fighting Mathematics. In summary, the book used a integral estimation heuristic from ...
1 vote
0 answers
33 views

Calculating vertex weights via mutually tangent circles of triangles

given a metric graph with positive edge weights $\left|e_{ij}\right|$ a standard task, especially in the context of the Traveling Salesman Problem, is to calculate $\max\sum\limits_{i=1}^n\omega_i:\ \...
Manfred Weis's user avatar
  • 12.6k
9 votes
3 answers
1k views

Books on the relationship between the Socratic method and mathematics?

Apart from books on heuristics by George Polya. When trying to engage with and understand mathematical concepts and when applying abstract mathematical concepts to model "continuum" or real ...
James Fife's user avatar
3 votes
0 answers
80 views

I have a question on the definition of 'good' primes in the paper of Cohen and Martinet

I'm reading the paper of Cohen and Martinet 'Etude heuristique des groups de classes'. In the section 6, for an central idempotent $e$ of $\mathbb{Q}[\Gamma]$ and a prime $p$, the 'goodness' of $p$ is ...
gualterio's user avatar
  • 1,043
2 votes
0 answers
95 views

Is there any good reference on the Bayesian view that can be helpful for reading papers on the number theory using heuristic arguments?

Nowadays there are many papers on the number theory using heuristics. I have read some of them. But I have no clear understanding of the Bayesian Probability(subjective probability). The concept of ...
gualterio's user avatar
  • 1,043
5 votes
0 answers
313 views

Heuristic for a density conjecture related to the Collatz $(3x+1)$-problem

First, some notation. Define $T(n)$ over $n\in \mathbb{N}$ as: $$ T(n) = \left\{ \begin{array}{} 3n+1, & \text{if $n$ is odd}\ \\ n/2, & \text{if $n$ is even} \end{array} \right. $$ ...
mhum's user avatar
  • 1,625
4 votes
1 answer
260 views

Idea behind Carleson's theorem modern proof "intitial reductions"

I'm having troubles to understand the philosophy behind the modern proof of Carleson's theorem. For convenience, let me state precisely what I am asking for. For any $f \in L^2(\mathbb{R})$, let $\...
J.Mayol's user avatar
  • 489
-2 votes
1 answer
165 views

What is known about iterated matching as a TSP heuristic

A fairly wellknown heuristic for TSP that is based on matching is described in the 2003 paper Match twice and stitch: a new TSP tour construction heuristic by Andrew B. Kahng and Sherief Reda. Its ...
Manfred Weis's user avatar
  • 12.6k
0 votes
0 answers
33 views

Reasons for inapplicability of complete induction to tour expansion

It is known that tour expansion is a rather poor heuristic for generating short Hamilton cycles even in the planar Euclidean case. That comes as a surprise when learning of that for the first time. ...
Manfred Weis's user avatar
  • 12.6k
1 vote
0 answers
26 views

Complexity of tour-expansion heuristic for the planar Euclidean TSP

This is a followup question to this one: Computational Geometric Aspects of Greedy Tour Expansion. Assume that the candidate point, whose insertion into current incurs the least tour-length increase, ...
Manfred Weis's user avatar
  • 12.6k
3 votes
1 answer
335 views

Is there a clear criterion or rule about when one can use the heuristic given by Cramér random model for prime numbers?

I don't know if I have misunderstood the information from the section about the heuristic related to Cramér's conjecture and the known facts about Maier's theorem, from the Wikipedia Cramér's ...
user142929's user avatar
3 votes
1 answer
218 views

How can we justify the use of Example 5,4 (of Cohen, Lenstra) assuming their heuristics

In these days, I'm studying Cohen-Lenstra heuristics to understand the paper of Rene Schoof "Class Numbers of Real Cyclotomic Fields of Prime Conductor". On page 932 of Schoof's paper, there is a ...
gualterio's user avatar
  • 1,043
3 votes
0 answers
95 views

Matching of two weighted graphs allowing one-to-many mapping

I am looking for a heuristic for a graph matching problem as follows. Given two graphs: $A$ (consisting of nodes $a_i$) and $B$ (consisting of nodes $b_i$). Typically the size of $B$ is larger than ...
Trung's user avatar
  • 31
4 votes
1 answer
674 views

Does multiplication increase entropy?

Does multiplication increase entropy? The Shannon entropy of a number $k$ in binary digits is defined as $$ H = -\log(\frac{a}{l})\cdot\frac{a}{l} - \log(1-\frac{a}{l})\cdot (1-\frac{a}{l})$$ where $...
user avatar
2 votes
0 answers
222 views

Heuristic for lower bounding the time for integer factorization?

I am posting this question here in hope that someone finds this heuristic useful, and maybe someone with more experience will make use of this: As @GerryMyerson suggested here is a statement of what ...
user avatar
0 votes
1 answer
90 views

Can time-scale calculus be used to derive a counterpart theorem of discrete-time dynamic systems directly from continuos-time dynamics systems?

From what I read of time-scale calculus literature, most results of continuous-time and discrete-time systems can be generalized to arbitrary time-scales by considering the generalized derivative ...
Tadashi's user avatar
  • 1,580
0 votes
0 answers
32 views

Generating Biconnected Graphs from Spanning Trees

Background of my question is an idea for generating an initial subtour for general symmetric TSPs: Add to a MST a set of edges with minimal weight sum, that renders the resulting graph free of ...
Manfred Weis's user avatar
  • 12.6k
4 votes
2 answers
848 views

Heuristics behind the Circle problem?

Is there a heuristic argument behind the exponent in the circle problem? The problem that I am referring to is the following: Consider a circle of radius $R$ centered at the origin in the plane and ...
Mustafa Said's user avatar
  • 3,679
57 votes
16 answers
8k views

What are examples of books which teach the practice of mathematics?

One may classify the types of mathematics books written for students into two groups: books which merely teach mathematics (i.e., they present theorems and proofs, ready-made, as it were) and those ...
11 votes
1 answer
1k views

Why do we mainly integrate with respect to martingales?

Although my resarch focuses on PDEs (optimal transport, these days), I am currently trying to learn stochastic calculus and integration. I am just beginning in this topics, but I was wondering: why do ...
leo monsaingeon's user avatar
1 vote
1 answer
127 views

Computational Geometric Aspects of Greedy Tour Expansion

Has the following problem already been investigated from the Computational Geometry point of view and what are the results regarding worst case complexity? Given a finite set $\mathcal{P}...
Manfred Weis's user avatar
  • 12.6k
1 vote
0 answers
118 views

Heuristics for this "subset" traveling salesman problem

Are there any known heuristics for the following variation of the traveling salesman problem: given $n$ sets of points $S_1,\dots,S_n$, and $n$ integers $k_i$ such that $k_i \leq |S_i|$, find the ...
Tom Solberg's user avatar
  • 3,929
3 votes
2 answers
206 views

(How) do Better TSP Heuristics help in Answering the $NP=P$ Question?

This question is motivated by my impression, that finding better heuristics for the TSP problem (or any other $NP$-complete problem) is "only" of practical interest, but doesn't provide any progress ...
Manfred Weis's user avatar
  • 12.6k
5 votes
0 answers
237 views

The set of numbers $a+b$ such that $ma^2+nb^2$ is prime

Conjecture: If $m,n$ are coprime it exist a minimal natural number $N_{mn}$ such that: $\{a+b>N_{mn}\mid a,b\in\mathbb N^+\wedge ma^2+nb^2\in\mathbb P_{>2}\} = \{ k > N_{mn} \mid \...
Lehs's user avatar
  • 852
2 votes
1 answer
258 views

Worst Case Region for a Convex Hull Heuristic

I am currently implementing a heuristic algorithm for planar convex hulls hand would like to know, for which kind of strictly convex region it exhibits worst performance. I know that there are many ...
Manfred Weis's user avatar
  • 12.6k
2 votes
1 answer
179 views

Heuristics for counting degrees of freedom

I have recently learned about the representation theorem for isotropic, linear operators, which says the following: Defintion: Let $M_n$ be the vector space of $n \times n$ real matrices. We say a ...
Asaf Shachar's user avatar
  • 6,611
2 votes
0 answers
46 views

Performance guarantee of RLF [closed]

I cannot manage to find the performance guarantee of the Recursive Largest First (RLF) algorithm for approximating the chromatic number of a graph. I know DSATUR has a $\mathcal{O}(n)$ guarantee, ...
Kuifje's user avatar
  • 225