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Bramble for Submodular Partition Function

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04 June 2023

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06 June 2023

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Abstract
In this compact and focused paper, we revisit the interplay between Bramble and Filter through the lens of a submodular partition function.
Keywords: 
Subject: Computer Science and Mathematics  -   Mathematics

1. Short Introduction

In this short paper, we revisit and contemplate the interplay between Bramble and Filter through the lens of a submodular partition function.
The submodular partition function, Bramble, and graph width parameter are well-known parameters that have received significant attention due to their importance, leading to numerous research studies (e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]). While the level of novelty may be limited, our objective is to make a modest contribution to the future research on graph width parameters.

2. Preliminaries

In this section, we present the essential definitions required for this paper. Throughout the paper, we utilize a finite set (referred to as the underlying set) X, a set of partitions P, and natural numbers i, k, and p.
Moreover, in this paper, we utilize the symbol α to denote a collection of subsets. It is worth noting that this notation is adopted from reference [22]. For instance, α represents a collection A1, ..., Ak of subsets of a finite set X. The collection α is considered a partition if the sets Ai are mutually disjoint and their union forms the set X. We introduce the following notation: if α represents the collection A1, ..., Ak, and A is another subset, then α ∩ A denotes the collection A1 ∩ A, ..., Ak ∩ A. Similarly, we use α \ A as a related notation. Lastly, [B1, ..., Bp, α] represents the collection obtained by inserting sets B1, ..., Bp into the collection α.

2.1. Submodular Partition Functions and Brambles

We will explain about submodular partition functions. The definition of a partition function and a submodular partition function of separations is provided below:
Definition 1 [21,22]. A partition function is a function that maps the set of all partitions to non-negative integers, satisfying the condition ψ([∅, α]) = ψ(α) for every partition α. In other words, inserting an empty set into a collection does not alter the value of the partition function. A partition function ψ is submodular if the following holds for every two partitions [A, α] and [B, β]:
ψ([A, α]) + ψ([B, β]) ≥ ψ([A ∪ (X\B), α ∩ B]) + ψ([B ∪ (X\A), β ∩ A]) .
We will further assume that ψ([X]) = 0 since shifting all values of a submodular partition function by a constant does not break the property. Pk[ψ] denote the set of partitions α of X such that ψ(α) ≤ k.
The submodular partition function possesses the following properties.
Lemma 2 [22]. Let ψ be a submodular partition function on X and [A, α] a partition. Then ψ([A, α]) ≥ ψ([A, X\A]) .
Lemma 3 [25]. Let ψ be a submodular partition function on X. Then ψ([A, X\A]) = ψ([X\A, A]).
Lemma 4 [25]. Let ψ be a submodular partition function on X. Then ψ(∅) = 0.
Inspired by reference [22], we define the concept of Bramble, which serves as a fundamental dual concept to width parameters such as Tree-width and branch-width, and tree-cut width [3,4,5,7,29,30,31,32,33,34,35].
Definition 5 [21]: Let ψ be a submodular partition function on a finite set X. A (non-principal) k-bramble, denoted as L, is a nonempty family of subsets of X satisfying the following conditions:
(B1) For any A and B belonging to L, their intersection A ∩ B is not empty.
(B2) For every [A1, . . . , An] ∈ Pk[ψ], there exists Ai in L.
(B3) For all e ∈ X, if the partition [{e}, X\{e}] belongs to Pk[ψ], then {e}∉ L,
In the case of a non-principal k-bramble, the following holds true.
Lemma 6: Let X be a finite set. A (non-principal) k-bramble satisfies following conditions:
(B4) If A1 ∈ F, A1 ⊆ A2, [A2, X\(A2)] ∈ Pk[ψ], then A2 ∈ L,
(B5) ∅ ∉ L.
Proof of Lemma 6: We show that axiom (B4) holds. Suppose A1 is a subset in L and A1 ⊆ A2. Additionally, we are given that [A2, X\A2] is in Pk[ψ]. According to the definition of a (non-principal) k-bramble (Definition 5), for every [A1, ..., An] in Pk[ψ], there is some Ai in L (by Condition (B2)). Since [A2, X\A2] is in Pk[ψ], and A1 ⊆ A2, we can conclude that A2 must be in L.
We show that axiom (B5) holds. The condition (B1) of Definition 5 states that the intersection of any two subsets in L is not empty. If we assume that the empty set is in L, then the intersection of any subset in L with the empty set would also be empty, contradicting condition (B1). Thus, we conclude that the empty set is not in L.
This concludes the proof of Lemma 6.
Furthermore, it is known that the following holds true in the context of Bramble.
Lemma 7 [21]: Let L be a k-bramble corresponding to the partition function. For every A, B, C in L, the intersection A ∩ B ∩ C is non-empty.

2.2. UltraFilter of partitions

The definition of Filter for submodular partition functions is below.
Definition 8 [25]: Let ψ be a submodular partition function on a finite set X. A Pk[ψ]-(non-principal) filter of partitions is a family F satisfying the following four axiom:
(F1) For all e ∈ X, if the partition [{e}, X\{e}] belongs to Pk[ψ], then {e}∉ F,
(F2) If A1 ∈ F, A1 ⊆ A2, [A2, X\(A2)] ∈ Pk[ψ], then A2 ∈ F,
(F3) If A1, A2, . . .,Ai ∈ F for i = 1, . . ., p, [X\A1, . . .,X\ Ap, X\(X\A1 ∪ . . . ∪ X\Ap)] ∈ Pk[ψ], then A1 ∩ ... ∩ Ap ∈ F,
(F4) ∅ ∉ F.
In this paper, we introduce an additional axiom (F5) for the Pk[ψ]-(non-principal) filter of partitions. We refer to this as a Pk[ψ]-(non-principal) ultrafilter.
(F5) If [A1, X\A1] ∈ Pk[ψ], either A1 ∈ F or X \ A1 ∈ F
Lemma 9: Let L be a Pk[ψ]-(non-principal) ultrafilter corresponding to the partition function. For every A, B, C in L, the intersection A ∩ B ∩ C is non-empty.
Proof of Lemma 9: Proof of Lemma 9 can be established similarly to Lemma 7.

3. Result: Filter of partitions and Bramble of partitions

The main result of this paper is presented below.
Theorem 10. Let ψ be a submodular partition function on a finite set X. T is a k-Bramble iff T is a Pk[ψ]-(non-principal) filter.
Proof of Theorem 10:
The proof of Theorem 10 proceeds in two parts.
Step 1: A k-Bramble is a Pk[ψ]-(non-principal) filter.
Let L be a k-Bramble. We need to show that L satisfies the conditions (F1) to (F5) of a Pk[ψ]- (non-principal) ultrafilter (Definition 8).
We show that axiom (F1) holds. This condition is precisely the same as condition (B3) of Definition 5.
We show that axiom (F2) holds. By Lemma 6, condition (B4) implies condition (F2).
We show that axiom (F3) holds. Let's suppose, for contradiction, that the opposite is true: there exist A1, A2, . . .,Ap in L such that A1 ∩ ... ∩ Ap is not in L.
Define a partition β = [X\A1, . . .,X\ Ap, X\(X\A1 ∪ . . . ∪ X\Ap)]. Given that all Ai are in L, by our supposition, β ∈ Pk[ψ].
Since L is a k-bramble, there must exist some B in L such that B is a subset of a part of β, i.e., B ⊆ X\Ai for some i ∈ {1, ..., p}, or B ⊆ X\(X\A1 ∪ . . . ∪ X\Ap).
Consider two cases:
Case 1: B ⊆ X\Ai for some i: In this case, we also have Ai ⊆ X\B due to the properties of set subtraction. By the definition of a k-bramble (Condition (B4)), if A1 ⊆ A2 and [A2, X\(A2)] ∈ Pk[ψ], then A2 ∈ L. Here, Ai ∩ ... ∩ Ap (which is a subset of Ai and thus a subset of X\B) must be in L. This contradicts our original supposition that A1 ∩ ... ∩ Ap is not in L.
Case 2: B ⊆ X\(X\A1 ∪ . . . ∪ X\Ap): By definition, B is disjoint from each Ai. Hence, the intersection A1 ∩ ... ∩ Ap is empty, which contradicts the property of a bramble, that for any A and B in L, the intersection A ∩ B is not empty (Condition (B1) from the properties of a bramble).
We show that axiom (F4) holds. Condition (B5) of Lemma 6 corresponds directly to condition (F4) in the definition of Pk[ψ]-(non-principal) ultrafilter.
We show that axiom (F5) holds. If [A1, X\A1] ∈ Pk[ψ], then we know by the properties of a k-bramble that either A1 ∈ L or X\A1 ∈ L (Condition (B2)).
Hence, all the conditions for L to be a Pk[ψ]-(non-principal) ultrafilter are satisfied.
Step 2: A Pk[ψ]-(non-principal) filter is a k-Bramble.
Now, suppose F is a Pk[ψ]-(non-principal) ultrafilter. We will show that F satisfies the properties of a k-Bramble.
We show that axiom (B1) holds. Condition (F3) of Definition 8 and Lemma 9 ensure the non-emptiness of the intersection of any subsets in F, hence satisfying Condition (B1) of Definition 5.
We show that axiom (B2) holds. If [A1, . . . , An] ∈ Pk[ψ], we know from condition (F5) that there must exist some Ai in F, satisfying condition (B2).
We show that axiom (B3) holds. Condition (F1) of Definition 8 is precisely condition (B3) of Definition 5.
Therefore, all conditions for F to be a k-bramble are satisfied.
Hence, we conclude that a family of subsets of X is a k-Bramble if and only if it is a Pk[ψ]-(non-principal) ultrafilter. This completes the proof of Theorem 10.

4. Future tasks: Single Filter and Weak Filter

We will define a single filter as defined below and investigate its connection to graph width parameters as needed.
Definition 11: Let ψ be a submodular partition function on a finite set X. A Pk[ψ]-(non-principal) single filter of partitions is a family F satisfying the following four axiom:
(F1) For all e ∈ X, if the partition [{e}, X\{e}] belongs to Pk[ψ], then {e}∉ F,
(F2) If A1 ∈ F, A1 ⊆ A2, [A2, X\(A2)] ∈ Pk[ψ], then A2 ∈ F,
(F3) If X\{e1}, X\{e2}, . . ., X\{ei} ∈ F for i = 1, . . ., p, [{e1}, . . .,{ep}, X\({e1} ∪ . . . ∪ {ep})] ∈ Pk[ψ], then X\{e1} ∩ ... ∩ X\{ep} ∈ F,
(F4) ∅ ∉ F.
And we will consider about Weak filter of submodular partition function. Weak filter is a concept used in the world of logic [36,37,38]. Definition of Weak filter of submodular partition function is below.
Definition 12: Let ψ be a submodular partition function on a finite set X. A Pk[ψ]-(non-principal) weak filter of partitions is a family F satisfying the following four axiom:
(F1) For all e ∈ X, if the partition [{e}, X\{e}] belongs to Pk[ψ], then {e}∉ F,
(F2) If A1 ∈ F, A1 ⊆ A2, [A2, X\(A2)] ∈ Pk[ψ], then A2 ∈ F,
(F3) If A1, A2, . . .,Ai ∈ F for i = 1, . . ., p, [X\A1, . . .,X\ Ap, X\(X\A1 ∪ . . . ∪ X\Ap)] ∈ Pk[ψ], then A1 ∩ ... ∩ Ap ≠ ∅,
(F4) ∅ ∉ F.

Acknowledgements

I humbly express my sincere gratitude to all those who have extended their invaluable support, enabling me to successfully accomplish this paper.

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