Generation of Union Closed Sets and Moore families

TL;DR

This paper presents an algorithm for enumerating non-isomorphic Union closed Sets and Moore families, confirming known counts for small sets and providing new counts for larger sets, highlighting computational challenges for larger sizes.

Contribution

The authors develop a constructive enumeration algorithm for Union closed Sets and Moore families, confirming existing counts and extending to n=7 elements, revealing growth challenges.

Findings

Confirmed counts for n<=6 elements

Provided counts for n=7 elements

Highlighting computational complexity for n>=8

Abstract

In this article we will describe an algorithm to constructively enumerate non-isomorphic Union closed Sets and Moore sets. We confirm the number of isomorphism classes of Union closed Sets and Moore sets on n<=6 elements presented by other authors and give the number of isomorphism classes of Union closed Sets and Moore sets on 7 elements. Due to the enormous growth of the number of isomorphism classes it seems unlikely that constructive enumeration for 8 or more elements will be possible in the foreseeable future.

Tables

Table 1. Table 1: The number of Union Closed Sets and labeled Union Closed Sets.

$n$	Union Closed Set	labeled Union Closed Set
1	1	1
2	3	4
3	14	45
4	165	2.271
5	14.480	1.373.701
6	108.281.182	75.965.474.236
7	2.796.163.091.470.050	14.087.647.703.920.103.947

Table 2. Table 2: The number of Moore families and labeled Moore families.

$n$	Moore families	labeled Moore families
1	2	2
2	5	7
3	19	61
4	184	2480
5	14.664	1.385.552
6	108.295.846	75.973.751.474
7	2.796.163.199.765.896	14.087.648.235.707.352.472

Table 3. Table 3: The number of sparse Union Closed Sets.

$n$	sparse Union Closed Set
1	0
2	0
3	0
4	2
5	27
6	3.133
7	5.777.931

Full text

Introduction

All sets in this article are finite. A Union Closed Set is a set ${\cal U}$ of sets with the property that for all $A,B\in{\cal U}$ we have that $A\cup B\in{\cal U}$. We call $\Omega_{{\cal U}}=\bigcup_{A\in{\cal U}}A$ the universe of ${\cal U}$. Two Union Closed Sets with universe $\Omega_{{\cal U}}$, resp. $\Omega_{{\cal U}^{\prime}}$ are defined to be isomorphic if there is a bijective mapping $\Omega_{{\cal U}}\to\Omega_{{\cal U}^{\prime}}$ inducing a bijection between the Union Closed Sets.

As we are only interested in isomorphism classes, we may assume $\Omega_{U}=\Omega_{n}=\{1,\dots,n\}$ for some $n$. While the whole universe $\Omega_{U}$ is by definition an element of a Union Closed Set ${\cal U}$, this is not the case for the empty set. As the empty set has no impact on the structure of a Union Closed Set, one often either requires the empty set to be an element of each Union Closed Set or forbids it to be an element. We choose for the first convention, so our Union Closed Sets contain $\Omega_{n}$ as well as the empty set. We denote a set containing one representative of each isomorphism class of Union Closed Sets with universe $\Omega_{n}$ as ${\cal R}_{n}$.

Although the famous Union Closed Sets conjecture (or Frankl’s conjecture) is exactly about the structures we generate here, our approach is not suitable for testing this conjecture. A lot is known about the structure of possible counterexamples to the Union Closed Sets conjecture (see [Bruhn and Schaudt(2015)] for a survey), so any approach to extend the knowledge on the smallest size of a possible counterexample by constructive enumeration must focus on the subset of Union Closed Sets with those additional structural properties (e.g. with small average size of the sets, without some subconfigurations like singletons, etc.).

Union Closed Sets are closely related to Moore Families. A Moore family for a universe $\Omega_{n}$ is a set of subsets of $\Omega_{n}$ that is closed under intersection and contains $\Omega_{n}$. It is easy to see that for a Union Closed Set ${\cal U}$ the set ${\cal U}^{c}=\{\Omega_{n}\setminus A|A\in{\cal U}\}$ is a Moore family. For a Moore family ${\cal M}$ the set ${\cal M}^{c}=\{\Omega_{n}\setminus A|A\in{\cal M}\}$ is closed under union, but as the empty set is not necessarily contained in ${\cal M}$, it is a Union Closed Set for $\Omega_{n}\setminus\bigcap_{A\in\cal M}A$, which is isomorphic to a Union Closed Set for some $\Omega_{n^{\prime}}$ with $n^{\prime}\leq n$.

A set ${\cal M}_{n}$ of representatives of Moore families (with the canonical definition of isomorphism) for the universe $\Omega_{n}$ can be obtained from sets ${\cal R}_{0},\dots,{\cal R}_{n}$ of representatives of Union Closed Sets containing the empty set as

${\cal M}_{n}=\bigcup_{i=0}^{n}\{{\cal U}^{c}|{\cal U}\in{\cal R}_{i}\}$

if the complement is in each case taken in the universe $\Omega_{n}$.

In [Higuchi(1998)], [Habib and Nourine(2005)] and [Colomb et al.(2010)Colomb, Irlande, and Raynaud] Moore families are enumerated. In the most advanced of these articles – [Colomb et al.(2010)Colomb, Irlande, and Raynaud] – all Moore families for $n\leq 6$ were counted and representatives of isomorphism classes were generated. For $n=7$ the approach is not suitable for generating a set of representatives and only the number of labeled Moore families – that is without considering isomorphisms – was determined by clever use of the structure of representatives of Moore families for $n=6$. In our algorithm we determine the number of labeled Union Closed Sets resp. labeled Moore families for $n=7$ from representatives and their automorphism groups of the Union Closed Sets for $n=7$, resp. $n\leq 7$. This gives a very good independent test for the implemetation in [Colomb et al.(2010)Colomb, Irlande, and Raynaud] as well as for our implementation. When computing the number of labeled Moore families for $\Omega_{7}$ from the number of labeled Union Closed Sets with $n\leq 7$, for those Union Closed Sets with $n<7$, the possible ways to choose $n$ vertices from $\{1,\dots,7\}$ are also taken into account.

The algorithm

A subset $A\subseteq\Omega_{n}$ is represented as a number $b(A)$ given as the binary number $b_{n-1}\dots b_{0}$ – possibly with leading zeros – with $b_{i}=1$ if $(i+1)\in A$ and $b_{i}=0$ otherwise.

We use an ordering of the subsets of $\Omega_{n}$. For $A,B\subseteq\Omega_{n}$ we define $A>B$ if $|A|<|B|$ (so sets with more elements are considered smaller in this order) or $|A|=|B|$ and $b(A)>b(B)$. Whenever we refer to larger or smaller sets, we refer to this ordering.

The construction algorithm generates Union Closed Sets recursively based on the following easy lemma:

Lemma 1**.**

If ${\cal U}\not=\{\Omega_{n}\}$ is a Union Closed Set and $A$ is the largest non-empty element in ${\cal U}$, then ${\cal U}\setminus\{A\}$ is also a Union Closed Set.

This implies that Union Closed Sets for universe $\Omega_{n}$ can be recursively constructed from the Union Closed Set $\{\Omega_{n},\emptyset\}$ of smallest size by successively adding subsets of $\Omega_{n}$ that are larger than the largest non-empty set already present. Of course it is not assured that adding a smaller set to a Union Closed Set does not violate the condition that the set must be closed under unions.

In order to turn this into an efficient algorithm, two tests that are (in principle) applied to each structure generated must be very fast:

(i)

The test whether the set that has been constructed by adding a new set is closed under union.

(ii)

The test for isomorphisms.

We will first discuss (i). A straightforward way to test (i) for a Union Closed Set ${\cal U}$ to which a new set $A$ is added would be to form all unions $A\cup B$ with $B\in{\cal U}$ and test whether they are in ${\cal U}\cup\{A\}$. Although all these steps can be implemented as efficient bit-operations, their number would slow down the program. We define:

Definition 1**.**

For a Union Closed Set ${\cal U}$ we define the reduced set ${\mbox{red}}({\cal U})$ as

${\mbox{red}}({\cal U})=\{A\in{\cal U}|A\not=\emptyset\mbox{ and there is no }A_{1}\not=\emptyset\mbox{ in }{\cal U},A_{1}\subsetneq A\}$**

Lemma 2**.**

Let ${\cal U}$ be a Union Closed Set for a universe $\Omega_{n}$ and let $A\subset\Omega_{n}$, that is larger than any non-empty set in ${\cal U}$. Then ${\cal U}\cup\{A\}$ is closed under union if and only if $A\cup B\in{\cal U}$ for all $B\in{\mbox{red}}({\cal U})$.

Proof.

Assume first that ${\cal U}\cup\{A\}$ is closed under union and let $B\in{\mbox{red}}({\cal U})$. Then $A\cup B\in({\cal U}\cup\{A\})$ and as $A$ is larger than $B$, we have $A\cup B\not=A$, so $A\cup B\in{\cal U}$.

For the other direction assume that $A\cup B\in{\cal U}$ for all $B\in{\mbox{red}}({\cal U})$ and let $D\in{\cal U}$.

Choose any $D^{\prime}\subset D$ so that $D^{\prime}\in{\mbox{red}}({\cal U})$. Then $A^{\prime}=A\cup D^{\prime}\in{\cal U}$ and therefore also $A^{\prime}\cup D\in{\cal U}$ as ${\cal U}$ is closed under union, but $A^{\prime}\cup D=A\cup D$ – so $A\cup D\in{\cal U}\cup\{A\}$ and ${\cal U}\cup\{A\}$ is closed under union.

∎

It would be inefficient to compute ${\mbox{red}}({\cal U})$ each time a new Union Closed Set is constructed, but as a new Union Closed Set ${\cal U}^{\prime}$ is constructed by adding a new smallest element $A$ to ${\cal U}$, the set ${\mbox{red}}({\cal U}^{\prime})$ can easily be constructed from ${\mbox{red}}({\cal U})$ by adding $A$ and removing elements that contain $A$. Nevertheless the few lines of code testing whether the potential Union Closed Set is closed under union take more than $50\%$ of the total running time when computing Union Closed Sets for $\Omega_{6}$, which is the largest case that can be profiled.

In order to solve problem (ii) efficiently – that is avoid the generation of isomorphic copies, we use a combination of Read/Faradžev type orderly generation [Faradžev(1976)][Read(1978)] and the homomorphism principle (see e.g. [Brinkmann(2000)]).

Our first aim is to define a unique representative for each isomorphism class – called the canonical representative – and then only construct Union Closed Sets that are the the canonical representatives of their class. We represent a Union Closed Set ${\cal U}$ with $k+1$ elements $A_{1}<A_{2}<\dots<A_{k}<\emptyset$ as the string $s({\cal U})=b(A_{1}),\dots,b(A_{k})$ of numbers. For a given isomorphism class of Union Closed Sets for a universe $\Omega_{n}$ we choose the Union Closed Set with the lexicographically smallest string as the representative.

It is in principle easy to test whether a given Union Closed Set ${\cal U}$ is the representative of its class by applying all $n!$ possible permutations to ${\cal U}$ and comparing the strings. As $n\leq 7$ this would not be extremely expensive, but due to the large number of times that this test has to be applied, still too expensive to construct the Union Closed Sets for $\Omega_{7}$. In the sequel we will describe a way how this can be optimized.

In order to increase the efficiency by making it an orderly algorithm of Read/Faradžev type, we will use the canonicity test not only for structures we output, but also during the construction: non-canonical structures are neither output nor used in the construction. This will only lead to a correct algorithm if we can prove that canonical representatives are constructed from canonical representatives:

Theorem 1**.**

Let ${\cal U}\not=\{\Omega_{n},\emptyset\}$ be a Union Closed Set for the universe $\Omega_{n}$ that is the canonical representative for its isomorphism class. If ${\cal U}=\{A_{1},A_{2},\dots,A_{k},\emptyset\}$ with $A_{1}<A_{2}<\dots<A_{k}$ and $1\leq m\leq k$, then $\{A_{1},A_{2},\dots,A_{m},\emptyset\}$ is also the canonical representative of its class.

Proof.

We prove the result for $m=k-1$. For $k=m$ it is the assumption and for $m<k-1$ it then follows by induction.

Let $s({\cal U})=(s_{1},\dots,s_{k})$. For a permutation $\Pi$ of $\Omega_{n}$ and a Union Closed Set ${\cal U}$ we write $\Pi({\cal U})$ for the Union Closed Set obtained by replacing each element of a set in ${\cal U}$ by its image under $\Pi$. Assume that ${\cal U}^{\prime}=\{A_{1},A_{2},\dots,A_{k-1},\emptyset\}$ is not the canonical representative of its class. So there is a permutation $\Pi$ of $\Omega_{n}$ with $s(\Pi({\cal U}^{\prime}))<s({\cal U}^{\prime})$. Let $s(\Pi({\cal U}^{\prime}))=(p_{1},\dots,p_{k-1})$ and we have $s({\cal U}^{\prime})=(s_{1},\dots,s_{k-1})$. Let $j$ be the first position so that $p_{j}<s_{j}$. Let us now look at $s(\Pi({\cal U}))=(p^{\prime}_{1},\dots,p^{\prime}_{k})$ and let $r$ be the position of $\Pi(A_{k})$ in this string. If $r>j$ then $p^{\prime}_{i}=p_{i}=s_{i}$ for $1\leq i<j$ and $p^{\prime}_{j}=p_{j}<s_{j}$ – so there is a smaller representative for the isomorphism class of ${\cal U}$. If $r\leq j$ then $p^{\prime}_{i}=p_{i}=s_{i}$ for $1\leq i<r$ and $p^{\prime}_{r}<p_{r}\leq s_{r}$ and again we have found a smaller representative contradicting the minimality of $s({\cal U})$.

∎

This theorem proves that the algorithm can reject non-canonical Union Closed Sets and is a correct orderly algorithm, but the cost of the canonicity test would make it impossible to determine the number of Union Closed Sets for $\Omega_{7}$.

For a given Union Closed Set ${\cal U}$ with universe $\Omega_{n}$ and $1\leq m\leq n$ we write ${\cal U}_{m}$ for the subset containing all sets of size $k\geq m$ and $\Pi_{m}({\cal U})$ for all permutations $\Pi$ of $\Omega_{n}$ with the property that $\Pi({\cal U}_{m})={\cal U}_{m}$.

Lemma 3**.**

Let ${\cal U}\not=\{\Omega_{n},\emptyset\}$ be a non-canonical Union Closed Set for the universe $\Omega_{n}$ with sets $A_{1}<A_{2}<\dots<A_{k}<\emptyset$, so that $\{A_{1},A_{2},\dots,A_{k-1},\emptyset\}$ is canonical and $|A_{k}|=m$. Then all permutations $\Pi$ with $s(\Pi({\cal U}))<s({\cal U})$ are in $\Pi_{m+1}({\cal U})$.

Proof.

Any permutation $\Pi$ not in $\Pi_{m+1}({\cal U})$ would by definition give an isomorphic but different Union Closed Set $\Pi({\cal U}_{m+1})$. As due to Theorem 1 $s({\cal U}_{m+1})$ is minimal, $s(\Pi({\cal U}_{m+1}))$ would be larger and therefore also the part of the string of $s(\Pi({\cal U}))$ describing sets of size at least $m+1$ would imply $s(\Pi({\cal U}))>s({\cal U})$.

∎

Lemma 3 speeds up the canonicity test dramatically: We start with a list of all $n!$ permutations as $\Pi_{n}({\cal U})$. When testing canonicity of a Union Closed Set with the smallest set of size $k<n$, we only apply permutations from $\Pi_{k+1}({\cal U})$. During this application, we can already compute $\Pi_{k}({\cal U})$ by simply adding exactly those permutations to the initially empty set $\Pi_{k}({\cal U})$ that fix ${\cal U}$. As we work with small sets, it is no problem to store and use the set of all group elements instead of just a set of generators.

The impact is immediately clear: the number of permutations that has to be computed is much smaller and as soon as some $\Pi_{k}({\cal U})$ contains only the identity – which happens very fast – no canonicity tests have to be performed, so that the total time for isomorphism checking is only about $7\%$ of the total running time when computing Union Closed Sets for $\Omega_{6}$.

The implementation

The algorithm was implemented in $C$. Next to an efficient algorithm, of course also implementation details are of crucial importance to be able to compute the results for $\Omega_{7}$. We precomputed the action of all permutations on all sets, so that they could be applied very fast and used data structures that allow to check whether a set is contained in a Union Closed Set in constant time. Special functions were written that add sets with only one element. As no sets of smaller size will be added, no updates of the automorphism groups are necessary and it turned out that at this stage it is also not efficient any more to remove sets from ${\mbox{red}}(\cal U)$ that are a superset of another element. Details on the implementation can best be seen in the code which can be obtained from the authors.

Results

Tables 1 and 2 give the numbers of isomorphism classes of Union Closed Sets and Moore families as well as the numbers of labeled structures. Up to $5$ elements the running times are less than $0.01$ seconds. For $n=6$ it is $8.2$ seconds on a Xeon(R) CPU E5-2690 0 running with $2.90$ GHz and a high load (which can make a difference for these processors). For $n=7$ the jobs were run in parallel on different machines and some parts had to be divided further in order to finish, so it is hard to give precise times. Estimating the total running time from those parts that were run on the same machine used for $n=6$, the total time on this type of machine should be around $10$ to $12$ CPU-years.

A Union Closed Set on $n$ elements is called sparse if the average number of elements in a set – not counting the empty set – is at most $\frac{n}{2}$. For Union Closed Sets that are not sparse, the Union Closed Sets conjecture is trivially true. The following table gives the number of sparse Union Closed Sets. These numbers were computed once by filtering all Union Closed Sets and once by an independent implementation using special bounding criteria described in [Deklerck(2016)].

Testing

In [Deklerck(2016)] an independent implementation of the algorithm together with special bounding criteria for sparse Union Closed Sets was developed. The implementation was used to generate all isomorphism classes of Union Closed Sets for $\Omega_{1},\dots,\Omega_{6}$, and – using special bounding criteria – to confirm the numbers of isomorphism classes of sparse Union Closed Sets for $\Omega_{7}$.

A further and independent confirmation of the numbers for $\Omega_{1},\dots,\Omega_{6}$ and also an independent confirmation for $\Omega_{7}$ was obtained by computing the number of labeled structures corresponding to each representative from the size of the automorphism group. Note that as the size of the automorphism group is known in the algorithm anyway, the additional costs for this test can be neglected. From this we computed the number of (labeled) Moore families and got complete agreement with [Colomb et al.(2010)Colomb, Irlande, and Raynaud] for $\Omega_{1},\dots,\Omega_{7}$. Due to the completely different approaches this makes implementation errors leading to the same incorrect results in both cases extremely unlikely.

Acknowledgements

We want to thank Craig Larson for introducing us to these interesting structures and for interesting discussions. Furthermore we want to thank Anastasiia Zharkova and Mikhail Abrosimov. The first version of this algorithm was intended as a hands-on example to illustrate isomorphism rejection techniques in a course they attended at Ghent university. Without their visit, this algorithm might not have been developed.