Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [discrete-optimization]

For questions about discrete optimization, which is a branch of optimization with discrete variables, opposed to continuous optimization in applied mathematics and computer science.

Discrete optimization is a branch of optimization in applied mathematics and computer science. As opposed to continuous optimization, some or all of the variables used in a discrete mathematical program are restricted to be discrete variables — that is, to assume only a discrete set of values, such as integers.

Two notable branches of discrete optimization are: combinatorial optimization, which refers to problems on graphs, matroids and other discrete structures, and integer programming.

These branches are closely intertwined however since many combinatorial optimization problems can be modeled as integer programs (e.g. shortest path) and conversely, integer programs can often be given a combinatorial interpretation.

933 questions
63
votes
11 answers

7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.

$7$ fishermen caught exactly $100$ fish and no two had caught the same number of fish. Prove that there are three fishermen who have captured together at least $50$ fish. Try: Suppose $k$th fisher caught $r_k$ fishes and that we…
nonuser
  • 86,352
  • 18
  • 98
  • 188
40
votes
4 answers

Belt Balancer problem (Factorio)

So this question is inspired by the following thread: https://forums.factorio.com/viewtopic.php?f=5&t=25008 In it, the poster is examining an $8$-belt balancer (more on that to come) which he shows fails to satisfy a desirable property, which he…
Justin Benfield
  • 3,872
  • 2
  • 11
  • 26
25
votes
4 answers

square cake with raisins

Alice bakes a square cake, with $n$ raisins (= points). Bob cuts $p$ square pieces. They are axis-aligned, interior-disjoint, and each piece must contain at least $2$ raisins. Note that a single raisin can be shared by two pieces (if it is on their…
Erel Segal-Halevi
  • 9,751
  • 3
  • 35
  • 88
19
votes
1 answer

Choose signs such that $\pm\sqrt{1}\pm\sqrt{2}\pm\dots\pm\sqrt{2022}$ is as close as possible to $0$.

Choose signs such that $\pm\sqrt{1}\pm\sqrt{2}\pm\dots\pm\sqrt{2022}$ is as close as possible to $0$. I tried looking at examples for small $n$ (up to $8$) for inspiration: $$\begin{align} &1: -\sqrt{1} = -1 &(0, 0) \\ &2: +\sqrt{1}-\sqrt{2} =…
Darius
  • 1,071
  • 3
  • 14
18
votes
0 answers

Maximizing $\sum_{i,j=1}^{n}|\operatorname{deg}\ x_{i}-\operatorname{deg}\ x_{j}|^{3}$ over all simple graphs with $n$ vertices

For a simple graph $G$ on $n$ vertices, let us define $$\mathcal{I}_{n}(G)=\sum_{i,j=1}^{n}|\operatorname{deg}\ x_{i}-\operatorname{deg}\ x_{j}|^{3}$$ I am highly interested in finding $\sup \mathcal{I}_{n}$ over all graphs with $n$ vertices (or at…
John Tuwim
  • 1,187
  • 2
  • 8
  • 23
18
votes
2 answers

Largest rectangle not touching any rock in a square field

You want to build a rectangular house with a maximal area. You are offered a square field of area 1, on which you plan to build the house. The problem is, there are $n$ rocks scattered in unknown locations throughout the field. The rocks are…
Erel Segal-Halevi
  • 9,751
  • 3
  • 35
  • 88
16
votes
1 answer

Domination problem with sets

Let $M$ be a non-empty and finite set, $S_1,...,S_k$ subsets of $M$, satisfying: (1) $|S_i|\leq 3,i=1,2,...,k$ (2) Any element of $M$ is an element of at least $4$ sets among $S_1,....,S_k$. Show that one can select $[\frac{3k}{7}] $ sets from…
nonuser
  • 86,352
  • 18
  • 98
  • 188
15
votes
3 answers

Maximizing the value of a determinant

Given the entries of a matrix how can we optimize its determinant? So, if the entries of a $n\times n$ matrix belong to the set $\{a_1,a_2,\ldots ,a_p\}$, how to arrange them to maximize or minimize the determinant? I have seen a result concerning …
Soumyadip Sarkar
  • 597
  • 3
  • 17
15
votes
4 answers

We have $n$ charged and $n$ uncharged batteries and a radio which needs two charged batteries to work.

We have $n$ charged and $n$ uncharged batteries and a radio which needs two charged batteries to work. Suppose we don't know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the…
nonuser
  • 86,352
  • 18
  • 98
  • 188
15
votes
3 answers

Greatest number of parts in which n planes can divide the space

Find the greatest number of parts including unbounded in which n planes can divide the space. I am trying like this, since it is very hard to visualize( or draw in paper). Equation of plane in 3 space could be ax + by + cz +d = 0. If I could get…
Amrita
  • 840
  • 1
  • 6
  • 19
13
votes
2 answers

Find the minimum number of edges in a graph with $3n+1$ vertices if ...

Let $G$ be a simple graph with $3n+1$ vertices. For any vertex $v$, there exists $n$ disjoint $K_3$ (i.e. triangle) such that none of them contains $v$. Find minimum number of edges of graph $G$. If we take $n$ triangles $v_iu_iw_i$, where $1\leq…
nonuser
  • 86,352
  • 18
  • 98
  • 188
13
votes
0 answers

Maximum number of permutations not repeating smaller permutations

There are $n$ soldiers, lining up every morning for their military service. The commander demands that the morning lineup of these soldiers be arranged differently for every next day according to the following rule: "Any $m$ soldiers cannot be…
Vepir
  • 11,937
  • 4
  • 27
  • 78
13
votes
0 answers

$200n$ diagonals are drawn in a convex $n$-gon. Prove that one of them intersects at least $10000$ others.

$200n$ diagonals are drawn in a convex $n$-gon. Prove that one of them intersects at least $10000$ others. There was no information about $n$ in a original problem. Attempt: Choose at random and uniform a diagonal with a probability $p={1\over…
nonuser
  • 86,352
  • 18
  • 98
  • 188
13
votes
1 answer

What is the name of this class of (combinatorial?) problems?

Judging from the number of similar questions, I've found myself in a rather common situation: I've come up with a problem, encountered a dead end and am now searching for the name of the problem in order to learn more about it. An example of the…
ZyTelevan
  • 746
  • 6
  • 19
12
votes
1 answer

Maximize the trace of a matrix by permuting its rows

I have been struggling with a combinatorial problem that eventually translates to the following: Given an $n \times n$ nonnegative matrix, find a permutation of the rows that maximizes the trace. I could compute all $n!$ permutations and find the…
Keivan
  • 916
  • 1
  • 7
  • 20
1
2 3
…
62 63