<p>This is not clear from your question, but I am going to assume that the groups are disjoint. I.e. $C = \bigcup_i C_i$ and $C_i \cap C_j = \emptyset$ for every $i \neq j$. Then the collection of sets $\mathcal{S} = \{S \subseteq C: |S \cap C_i| = k_i \ \ \forall i\}$ is the collection of bases of a <a href="https://en.wikipedia.org/wiki/Partition_matroid" rel="nofollow noreferrer">partition matroid</a>. In the context of combinatorial optimization then the constraint $S \in \mathcal{S}$ is often called a <em>partition constraint</em>. This is a special case of matroid constraints. There is, for example, an extensive body of work on maximizing a submodular function $f(S)$ subject to matroid constraints, i.e. subject to the set $S$ being a basis of a matroid.</p>

CC BY-SA 3.0
2017-10-25T20:47:22.860Z
Yes, I was thinking about disjoint groups.
polar_bear_cheese
CC BY-SA 3.0
2017-10-25T18:33:41.663Z
2017-10-25T22:27:32.340Z
2017-10-25T22:27:32.340Z
Sasho Nikolov
<p>To be clear that I understand the context, I'm going to first reproduce you situation worded differently.</p>

<blockquote>
  <p>Consider a sequence of sets, $C_i$, and a target function, $f$. Let $\cup C_i=C$. We wish to find a subset $S\subseteq C$ that maximizes $f(S)$ under the constraint that $\forall i, |S\cap C_i|&lt;k_i$ and $|S|&lt;k$.</p>
</blockquote>

<p>I would agree with the commenter that this should be referred to as a <strong>cardinality constraint or a collection of cardinality constraints</strong>. I might also call it a "structural constraint," but I would specifically avoid calling it a "group carnality constraint" as "group" already has a meaning. In general, I would advocate for avoiding referring to a collection as a "group," "set," or "class" unless your object actually satisfies the required axioms. "Collection" is a good, general word that doesn't have a mathematical meaning.</p>

<p>If you don't want to use the term "carnality constraint" you can always <strong>introduce a term to refer to this specific restriction in your context</strong>. This can be helpful because it allows you to highlight the context-sensitive meaning of the constraint in a way that a more general term doesn't. For example, I have a paper with a constraint of this form that I refer to as an "isolation condition" because, when $S$ satisfies it, $S$ is isolated from the structure of the rest of the graph in a relevant way. This gives me a simple and intuitive way to refer to the condition, while simultaneously telling the reader what is important about the condition every time it comes up.</p>

CC BY-SA 3.0
2017-10-26T01:21:26.793Z
I think it's fair to come up with a new terminology like you said. On the other hand, I want to make sure that I know the right or often used terminology for literature research as well.
polar_bear_cheese
CC BY-SA 3.0
2017-10-26T02:23:34.643Z
@HTV Indeed, that was meant as a side comment and not the focus of my answer.
Stella Biderman
CC BY-SA 3.0
2017-10-25T17:55:21.550Z
2017-10-26T02:23:18.037Z
2017-10-26T02:23:18.037Z
Stella Biderman
<p>If we want to pick a solution $S$ from a collection of items $C$ to maximize some function $f(S)$. </p>

<p>The constraint that we pick at most $k$ item, i.e., $|S| \leq k$, is called  the cardinality constraint.</p>

<p>If the items in $C$ belongs to different groups and we are allowed to picked at most $k_i$ items from group $i$, what is the right terminology for this constraint? I think some paper used to call it the group cardinality constraint but I'm not sure if it's still the popular terminology.</p>

CC BY-SA 3.0
2017-10-25T17:27:35.497Z
I'd say cardinality constraints are typically over a set of variables, e.g. Sudoku etc. So for me what you're describing is just a bunch of cardinality constraints.
Mikolas
CC BY-SA 3.0
2017-10-25T19:12:47.110Z
"If the items in C belongs to different groups" => C is a multiset. Perhaps you can go with "multisets with cardinality constraints" or "cardinality constraints on multisets".
Marzio De Biasi
<terminology>
Constraint terminology
polar_bear_cheese
# Constraint terminology

(Asked by: username_0 on 2017-10-25)

If we want to pick a solution $S$ from a collection of items $C$ to maximize some function $f(S)$.

The constraint that we pick at most $k$ item, i.e., $|S| \leq k$, is called the cardinality constraint.

If the items in $C$ belongs to different groups and we are allowed to picked at most $k\_i$ items from group $i$, what is the right terminology for this constraint? I think some paper used to call it the group cardinality constraint but I'm not sure if it's still the popular terminology.

**Tags:** terminology

> **Comments:**
> **username_1**: I'd say cardinality constraints are typically over a set of variables, e.g. Sudoku etc. So for me what you're describing is just a bunch of cardinality constraints.  
> **username_2**: "If the items in C belongs to different groups" => C is a multiset. Perhaps you can go with "multisets with cardinality constraints" or "cardinality constraints on multisets".  

---

## 2 answers

### Answer by username_3 on 2017-10-25. Score: 1 | Accepted answer

This is not clear from your question, but I am going to assume that the groups are disjoint. I.e. $C = \bigcup\_i C\_i$ and $C\_i \cap C\_j = \emptyset$ for every $i \neq j$. Then the collection of sets $\mathcal{S} = \{S \subseteq C: |S \cap C\_i| = k\_i \ \ \forall i\}$ is the collection of bases of a [partition matroid](https://en.wikipedia.org/wiki/Partition_matroid). In the context of combinatorial optimization then the constraint $S \in \mathcal{S}$ is often called a *partition constraint*. This is a special case of matroid constraints. There is, for example, an extensive body of work on maximizing a submodular function $f(S)$ subject to matroid constraints, i.e. subject to the set $S$ being a basis of a matroid.

> **Comments:**
> **username_0**: Yes, I was thinking about disjoint groups.  

---

### Answer by username_4 on 2017-10-25. Score: 1

To be clear that I understand the context, I'm going to first reproduce you situation worded differently.

> Consider a sequence of sets, $C\_i$, and a target function, $f$. Let $\cup C\_i=C$. We wish to find a subset $S\subseteq C$ that maximizes $f(S)$ under the constraint that $\forall i, |S\cap C\_i|

I would agree with the commenter that this should be referred to as a **cardinality constraint or a collection of cardinality constraints**. I might also call it a "structural constraint," but I would specifically avoid calling it a "group carnality constraint" as "group" already has a meaning. In general, I would advocate for avoiding referring to a collection as a "group," "set," or "class" unless your object actually satisfies the required axioms. "Collection" is a good, general word that doesn't have a mathematical meaning.

If you don't want to use the term "carnality constraint" you can always **introduce a term to refer to this specific restriction in your context**. This can be helpful because it allows you to highlight the context-sensitive meaning of the constraint in a way that a more general term doesn't. For example, I have a paper with a constraint of this form that I refer to as an "isolation condition" because, when $S$ satisfies it, $S$ is isolated from the structure of the rest of the graph in a relevant way. This gives me a simple and intuitive way to refer to the condition, while simultaneously telling the reader what is important about the condition every time it comes up.

> **Comments:**
> **username_0**: I think it's fair to come up with a new terminology like you said. On the other hand, I want to make sure that I know the right or often used terminology for literature research as well.  
> **username_4**: @HTV Indeed, that was meant as a side comment and not the focus of my answer.  

---

