1 Introduction

The publication of the theoretical convergence between Typical Testors and Minimal Transversal concepts [²] opened new possibilities for the study, development, and application within the hypergraph and testor theories. Particularly, algorithms for computing the set of all typical testors, as well as those for computing minimal transversals, can now be applied interchangeably in any of these áreas.

Graph theory is one of the most relevant fields of discrete mathematics because of its ability to model a wide range of problems.

The concept of minimal transversal, also known as minimal hitting sets [⁵], has been applied in relevant áreas such as artificial intelligence, reliability theory, database theory, integer programming, and learning theory [⁸, ¹²].

Whereas in Pattern Recognition, testor theory is a useful tool for feature selection and evaluation solving various practical problems like medical diagnosis [¹⁷, ²³, ²⁴, ¹⁰], text categorization [⁶], document summarization [¹³], document clustering [¹⁴], stellar structure [¹⁷], dimensión reduction in image databases [¹⁹], reduction of neural network models [²⁵], number recognition [¹⁸], etc.

However, the high time complexity that minimal transversal finding algorithms have [¹³, ⁹] limits the possibility of applying these concepts in situations that require handling large amounts of data, cases that would paradoxically be the most interesting and useful ones. The above limitation has encouraged the search for alternatives that allow an increase in the performance of these algorithms, so that the lowest possible computational cost can be reached.

The LEX algorithm [²²] is a particular case of these algorithms: it imposes a lexicographic order over the power set of all vertices/features and uses that order for generating combinations of columns from its input Boolean matrix, which represents an incidence matrix of a hypergraph.

In order to deal with its exponential complexity and facilitate LEX application in a wider range of problems, improving its efficiency becomes a matter of major significance.

In that same context, the recent initiative of embedding learning strategies into minimal transversal computation algorithms [¹¹] also becomes relevant.

This technique can be applied to a wide range of algorithms for computing typical testors and minimal transversals.

The symbolic model learned, as the host algorithm progresses, guides the search process for these algorithms by adding rules that allow a faster traversal of the search space, thus offering the possibility of a more efficient exploitation of the background knowledge, as well as of the newly acquired knowledge.

Since its publication, this technique has been tested only on two algorithms, BR [¹⁵] and YYC [⁴], in the same article where the technique was originally proposed.

However, in that article there is no description of the methodology for using it with other algorithms, nor are the modifications to the original algorithms made explicit.

Consequently, a more detailed explanation of the integration between a learning strategy and a host algorithm can help pave the way for using this technique on other algorithms.

On the other hand, since the BR and YYC algorithms use a very different search strategy, it is completely unknown whether a learning strategy can benefit the LEX algorithm in the same way.

Thus, in this article we incorporate a learning strategy into the LEX algorithm and test the result to assess its performance improvement over the original version of the same algorithm.

First, we determine the stages of the original algorithm in which the learning strategy can be used, then we proceed to integrate it and we show the code of the new algorithm, which we have called LEX*.

For testing purposes, we designed a set of benchmark matrices [¹] varying the number of rows and columns. Among the test matrices there are some with high density of minimal transversals, that is a high number of minimal transversals with respect to the cardinality of the power set of features.

Also there are others with large dimensions but few minimal transversals. Also, the comparison criteria for assessing the performance of LEX* is clearly defined and uniformly used in all experiments. Finally, results from the experiments are reported and conclusions are drawn.

2 Conceptual and Theoretical Background

In graph theory, an hypergraph is a generalization of the graph concept, i.e. an ordered pair ℋ=(V,ℰ) where V={v1,…,vn} is a finite set of objects and ℰ={ℰ1,…,ℰn} a covering of V(ℰi≠0,i=1,…,m and ∪i=1mℰi=V), where each ℰi is called an hyperedge. ℋ is simple if for every pair (ℰi,ℰj), ℰj⊆ℰi⇒j=i.

A transversal or hitting set of a simple hypergraph ℋ is a subset of vertex that intersects all edges, i.e. a subset τ⊆V such that ∀ℰi∈ℰ, τ∩ℰi≠0. τ is minimal if no proper subset of τ is a transversal of ℋ.

The incidence matrix A=[aij]m×n of ℋ is a matrix whose rows and columns correspond to the vertices and hyperedges of ℋ respectively, in such a way that aij=1 if vi∈ℰj and aij=0 .

2.2 Definitions and Properties of the LEX Algorithm

The strategy of LEX is to construct vertex lists that are minimal, i.e. lists where none if its subsets intersects the same amount of edges, and check whether this set is a transversal (originally presented as properties of typicality and testor respectively). LEX imposes a lexicographic order over the power set of all vertices, which is the reason of its name.

The order is denoted with the symbol ≪. For example, if we consider the set of vertices V={v1,v2,v3} and the lists [v1,v3], [v2,v3], [v1,v2,v3] then [v1,v2]≪[v1,v2,v3]≪[v2,v3].

Let L be a vertex list and vt a vertex of an incidence matrix A, we use the following notation:

• – F(vt,L): the set of hyperedges that contain vt but do not contain any other vertex of L.

• – r(i,L): the total of vertices v in L such that v belongs to ℰi.

• – Gap of L: the maximum of the indexes in L such that the index of the next element is not consecutive.

For example, consider the following matrix:

v1v2v3v4v5ℰ1ℰ2ℰ3 [0 0 1  0 10 1 0 1 110 1  0 0]. (1)

If L=[v1,v3] then F(v3,L)={ℰ1}, r(1,L)=1, r(3,L)=2 and L has gap 1. In [²²], it is proven that the following statements are true:

Let be L and vt a vertex not belonging to L.

• (1) Let U be the set of hyperedges that contain vt. If there is vk∈L such that F(L,vk)⊆U, then vt will not be part of any minimal transversal together with all the elements of L.

• (2) If F(vt,L) is empty then vt will not be part of any minimal transversal together with all the elements of L.

If either of the above cases is met, vt is said to be exclusive with respect to L. Let L be a vertex list that contains the vertex of the last column in the incidence matrix. In all of the following the symbol ¡¡ is used with the semantics of not including ends.

• – If L is a minimal transversal and has a gap p. Let L′ be a vertex list obtained removing from L all the vertices from vp up to the end and adding the vertex vp+1. A list λ such that L≪λ≪L′ is a subsets of L and thus, λ is not a minimal transversal.

• – If L is a minimal transversal that has no gaps. A list λ such that L≪λ is a subset of L and is not a minimal transversal.

• – If L is not a transversal and p is a gap of L. Let L′ be a vertex list obtained removing from L all the vertices from vp up to the end and adding the vertex vp+1, then a list l such that L≪λ≪L′ is a subset of L+[v] and it is not a transversal.

The original LEX algorithm is outlined in Algorithm 1.

Algorithm 1 LEX (original) 

3 Experimental Assesment

3.2 Partition of the Search Space in LEX

In LEX, the candidate evaluation (step 3 in algorithm 1.) is the key to the classification of vertex sets in an incidence matrix through proposition (1) and (2). The current list and vertex are saved as indicated in rule 4, in a table of accumulated knowledge named K, for later use of the algorithm.

The combination between the background knowledge and the learned knowledge makes possible a smarter selection for the next vertex to evaluate.

When an exclusive vertex exists in LEX, it means the combination evaluated is a transversal not minimal, or an incompatible set. To fill K, the second case is of major interest.

• – When a vertex set meets (1), it means the current candidate removes the minimal property from the analyzed vertex list. In other words, let i and L be the vertex and the list being reviewed in LEX respectively, where L+i meet (1), then there is a subset S⊆L that dominates i (definition 1).

• – Another incompatible pair can be found by applying a similar reasoning to case (2). When the candidate to evaluate fits in this case means it has 1’s where one or more vertex of L already have unit values. Therefore, there is a subset S⊆L such that S+i dominates some vertex j in L, with j not in S.

Briefly, the candidate evaluation process for any vertex i and list L is explained in figure 1.

Fig. 1 Evaluation candidate i in LEX using the table of accumulated knowledge K 

3.3 The New Algorithm

Finally, we show the LEX algorithm with the learning strategy using the established partition of its search space explained above. The pseudo code is presented in algorithm 2.. We call the new algorithm LEX*.

Algorithm 2 LEX* 

The host algorithm selects a vertex using its predefined lexicographic search order. With the acquired knowledge stored in table K, the tested subset is classified. If it includes a pair of exclusive vertex and sublist ((1) and (2)) the pair is classified as incompatible and K is updated.

Since the new steps reduce the number of elements tested by the algorithm, its performance is enhanced.

4 Discussion

In order to properly assess the experimental results, we present tables containing the number of rows and columns, the number of minimal transversals, the number of registered incompatibilities (knowledge), and the proportion of omitted sets for each operator and each matrix.

In addition, two graphs are included, the first one corresponds to the run time of the LEX algorithm versus the time of LEX*, the other graph is the total number of tests in LEX and LEX*.

In both graphics, the dotted line with circular marker corresponds to LEX, and the dashed line with cross marker corresponds to LEX*.

As it was expected, the gain resulting from using the learning strategy increases when the dimensions of the input matrix increase. Here is a brief description of all experimental results tables, starting with matrix A.

In table 2, we observe that for operator θ from N=2 between 20% and 30% of tests were reduced. For operator γ applied to matrix A there is up to 37% of omitted sets.

We see that there is more accumulated knowledge than in operator θ. The results for operator φ unveil that the number of sets in accumulated knowledge increases.

Also, up to N=10 there is more than 40% of omitted tests. If we compare with previous operators up to N=5 we see that more tests are dismissed than in θ but less than γ.

The results for matrix B are presented in table 3. Notably, it can be seen that, thanks to the learning strategy, the percentage of tests omitted exceeds 80%.

Although the same amount of knowledge has been accumulated in θ and γ matrices, the proportion of reduced test in γ is slightly higher.

Again, for φ there are more sets in accumulated knowledge. Fewer combinations are excluded than in the previous matrices generated from B, but we still get a good percentage of dismissed sets.

The difference between the performance of both algorithms for φ, is appreciated in Figures 2b and 3b. In table 4 is also shown that for the matrices obtained from basic matrix C the learning strategy reduce the number of reviewed candidates to minimal transversals in any of the three operators.

Fig. 2 Run time comparison between LEX and LEX* 

Fig. 3 Tests comparison between LEX and LEX* 

As in A and B scenarios, the highest percentage of reduced tests is obtained for the γ operator. Lastly, Figures 2c and 3c evidence the benefit of the strategy in matrices constructed from C for operator γ.

Finally, LEX* results for the real-world datasets matrices are in table 5. As in the synthetic matrices, there is a percentage of reduced tests for all matrices, which is higher for the matrices with more transversals.

5 Conclusions and Future Research

Results showed in the above section suggest how the supplementary knowledge about the problem search space supplied by the learning strategy enhances LEX’s general efficiency.

The latter is measured through the number of tests performed and the run-time achieved using a set of designed benchmark matrices. However, we have not limited ourselves to presenting evidence of it, but also to studying how the strategy acts within LEX through these results. In all experiments the learning strategy identified incompatibilities which make possible that the host algorithm reduce further unnecessary set testing.

Although there is no procedure that allows to establish which algorithm is better overall, we can consider certain parameters that lead us to determine certain interesting behaviors.

In our experiments, we have focused on matrices with a large quantity of minimal transversals and matrices with large dimensions and a small number of minimal transversals.

Additionally, we have varied the number of rows and columns of the base matrices to obtain incidence matrices with diverse structures.

The knowledge table is consulted by the host algorithm in accordance with its search method, in the case of the LEX algorithm, the lexicographic order enables the number of learned sets to be small compared to the number of avoided tests.

An interesting observation is that all matrices to which the γ operator is applied report a higher percentage of dismissed sets.

It would not be appropriate to conclude that matrices from a specific operation are handled more efficiently by LEX*, as in practice not all matrices come from any of the matrix operations used in the experiments.

Nevertheless, the γ operator produces matrices with considerable amount of testors. Thus, according to the results obtained in this paper, the incorporated learning strategy in LEX reduces in greater proportion the number of tests within matrices with large numbers of minimal transversals in comparison with those with large dimensions and few minimal transversals which were obtained by using the operator θ. We also notice a similar behavior in the real-world data matrices from table 5.

Conversely, in θN matrices the number of incompatible sets in the knowledge table is less than or equal to the number of sets registered in the remaining operators. This may be due to their large size, or more specifically to the large number of rows that characterize these matrices. In most cases of φN matrices, whose distinctiveness is a higher number of columns than rows, LEX* has learned more than in θN and γN matrices.

In addition, matrices coming from B, which contains a small number of rows and several columns, show to be the ones where the strategy is the most advantageous.

The number of revisions has decreased more respect to the matrices coming from other base matrices. Matrices generated from A, whose structure is similar to a square matrix have a smaller percentage of reduced tests in contrast with matrices coming from B. Lastly, those created from matrix C, which contain more rows than columns, are the ones with the lowest percentage.

Probably LEX* can reduce its search space in matrices which number of columns is higher than its number of rows, but also saves more incompatibilities in the knowledge table.

The results reached in these experiments can serve as a guide for future research work that contribute to the theoretical development in the testors field and, due to their theoretical convergence between concepts, also in the field of hypergraphs. Some of the following options may be considered:

Finally, besides providing a more detailed description of the functioning of the learning strategy, and a demonstration of its impact, this work motivated to contribute to the theoretical development of the fields involved, makes available the LEX* algorithm for applications in various problems capable of being modeled through minimal transversals or typical testors when needed.