Teaching dimension, VC dimension and critical sets for Latin squares
Teaching dimension, VC dimension and critical sets for Latin squares
A critical set in an Latin square is a minimal set of entries that uniquely identifies it among all Latin squares of the same size. It is conjectured by Nelder in 1979, and later independently by Mahmoodian, and Bate and van Rees that the size of the smallest critical set is . We prove a lower-bound of for sufficiently large , and thus confirm the quadratic order predicted by the conjecture. This improves a recent lower-bound of due to Cavenagh and Ramadurai.
From the point of view of computational learning theory, the size of the smallest critical set corresponds to the minimum teaching dimension of the set of Latin squares. We study two related notions of dimension from learning theory. We prove a lower-bound of for both of the VC-dimension and the recursive teaching dimension.
Latin square, critical set, VC-dimension, teaching dimension, recursive teaching dimension, defining set, forcing set.
Latin squares and critical sets:
Recall that a Latin square of order is an array filled with elements from the set such that every element occurs exactly once in each row and each column. Note that a Latin square can also be represented as the set of the triples
Following the notation of computational learning theory, we call a set of entries in a Latin square that uniquely identifies it among all Latin squares of order , a teaching set for . The minimal teaching sets in Latin squares were introduced and studied under the name critical set by statistician John Nelder [Nel77], and they have been studied extensively since then. We refer the reader to the two surveys [Kee96] and [Cav08] for more on this topic.
Note that a partially filled Latin square can determine the values of certain empty entries in a straightforward manner: if all the values already appear in the -th row and -th column, then the -th entry is determined to be . One can start from a partially filled Latin square and iteratively set the values of the entries that are determined in this manner. If this finally leads to a full Latin square , then is called a strong teaching set for . Obviously every strong teaching set is also a teaching set. Bate and van Rees [BvR99] showed that every strong teaching set is of size at least . Figure 1 illustrates an example of a strong teaching set of size for a Latin square.
Moreover, Bate and van Rees [BvR99] conjectured that this bound holds for every teaching set. This was also independently conjectured earlier by Nelder111John Nelder: Private communication to Jennifer Seberry (1979)., and Mahmoodian [Mah95].
Every critical set for a Latin square of order is of size at least .
The existence of critical sets of size was shown by Curran and van Rees [CVR79] and Cooper, Donovan and Seberry [CDS91]. However, despite several efforts, there has been little progress towards resolving this conjecture. Fu, Fu, and Rodger [FFR97] showed a lower-bound of for . This bound was improved by Horak, Aldred, and Fleischner [FFR97] to for . Cavenagh [Cav07] gave the first superlinear lower-bound of in 2007. Recently, Cavenagh and Ramadurai [CR16] improved this bound to .
In Theorem 1.2 below, we use recent results of Barber, Kühn, Lo, Osthus and Taylor [BKL16] and Dukes [Duk15] about edge-decomposition of graphs into triangles to show that for sufficiently large , every critical set in a Latin square is of size at least , thus establishing that as it was predicted in Conjecture 1.1, the size of the smallest critical set is of quadratic order.
For sufficiently large , every critical set for a Latin square of order is of size at least .
VC, teaching, and recursive teaching dimensions:
The notion of teaching set for Latin squares, as defined above, is quite natural, and can be easily defined for other combinatorial objects. Indeed similar notions have been defined and studied independently under various names in different contexts. For example, the term defining set is used for block designs and graph colorings, and the term forcing set, coined by Harary [Har93], is used for other concepts such as perfect matchings, dominating sets, and geodetics (see the survey [DMRS03]).
The general concept of identifying an object by a small set of its attributes arises naturally in the area of computational learning theory. Consider a finite set , and let denote the power set of . In computational learning theory, a subset is refered to as a concept class, and the elements are called concepts. A set is called a teaching set for a concept if uniquely identifies among all other concepts. In other words, for every concept . The notion of a teaching set was independently introduced by Goldman and Kearns [GK95], Shinohara and Miyano [SM91] and Anthony et al. [ABCST92]. It has also been studied under the names witness set by Kushilevitz et al. in [KLRS96], discriminant in [Nat91], and specifying set in [ABCST92].
Recall from (1.0.1) that every Latin square of order can be represented as a subset of . Hence the set of all Latin squares of order can be considered as a concept class. It is worth noting that our definition of a teaching set for a Latin square coincides with its definition when is considered as a concept class.
The concept of a teaching set naturally gives rise to various notions of dimension associated to concept classes. Let denote the smallest size of a teaching set for a concept . The teaching dimension and the minimum teaching dimension of a concept class are respectively defined as and . It turns out that for some purposes, due to its local nature, the minimum teaching dimension do not capture the characteristics of teaching and learning, and thus the related notion of recursive teaching dimension is often considered:
Note that for every concept class .
Finally let us recall one of the most celebrated notions of dimension associated to a concept class, i.e. its VC dimension (for Vapnik-Chervonenkis dimension). A subset is said to be shattered by if for every there exists a concept with . The size of the largest set shattered by is called the VC-dimension of . Recently in [CCT16], using a surprisingly short argument, Chen, Cheng and Tang showed that , where .
VC, teaching, and recursive teaching dimensions for Latin Squares:
Our main result, Theorem 1.2, says that for sufficiently large . Ghandehari, Hatami and Mahmoodian [GHM05] showed that every Latin square contains a critical set of size at most , and moreover there are Latin squares with no critical sets of size smaller than . In other words, for sufficiently large , we have
On the other hand, does not seem to correspond to any of the previously studied parameters related to critical sets. In Theorem 1.4 below, we show that one can adopt the argument of [GHM05] to obtain a stronger result that . Surprisingly, a similar argument combined with a lemma of Pajor (Lemma 2.6) implies the same bound for the VC-dimension.
The VC-dimension of the class of Latin squares of order is at least .
The recursive teaching dimension of the class of Latin squares of order is at least .
2 Proof of Main Theorems
2.1 The size of the smallest critical set, Theorem 1.2
We give some remarks before proceeding to the proof of Theorem 1.2. A graph has a -decomposition if its edge set can be partitioned into (edge-disjoint) copies of . We call a -partite graph balanced if each part has the same number of vertices, and we call it locally balanced if every vertex of has the same number of neighbours in each of the other two parts (however, these numbers might be different for different vertices). The following theorem is immediate from results of Barber, Kühn, Lo, Osthus and Taylor [BKL16] and Dukes [Duk15].
Let and . Every balanced and locally balanced -partite graph on vertices with minimum degree at least , admits a -decomposition.
Corollary 2.2 ([Bkl16]).
Let be a partial Latin square of order such that every row, column, and symbol is used at most times. Then can be completed to a Latin square.
We will take a similar approach to prove Theorem 1.2.
Proof of Theorem 1.2.
Set . A partial Latin square of order is a partially filled array with elements chosen from such that each element occurs at most once in every column and at most once in every row. In other words, some of the cells of the array are empty and the filled entries agree with the Latin property. The size of , denoted by , is the number of filled entries. We need to show that providing is sufficiently large, if a partial Latin square of size at most can be completed to a Latin square , then can also be completed to a different Latin square .
For such a , let be respectively the set of all rows, columns and symbols in that have at least filled entries, where . We extend to a larger partial Latin square by completing all those rows, columns and symbols by filling the empty cells with the entries of . Let , and note . We obtain by filling additional rows, additional columns, and additional symbols with entries of . Since , exactly rows, columns, and symbols are all fully filled in .
Let . Such an element exists because . Let be any symbol such that for all . Such a exists because the number of symbols in the -th row and the number of symbols in the -th column of are each at most , and thus there are in total at most symbols appearing in the -th row and the -th column.
Let and we claim that can be completed to a Latin square. Note that still has exactly completed rows, columns and symbols as filling in cannot create another complete row, column or symbol. Start from the complete -partite graph where the vertices of each part are labeled with , and for every entry remove the three edges of the triangle from the graph. Let be the resulting graph. Note that has vertices of degree corresponding to the completed rows, columns and symbols in . Ignoring the -degree vertices, is balanced and locally balanced, and it is of minimum degree at least . Hence by Theorem 2.1, it admits a -decomposition, which in turn corresponds to a completion to a Latin square . Note that as the two Latin squares disagree on the -th entry. ∎
A conjecture of Daykin and Häggkvist [DH84] (see [BKL16, Conjecture 1.3]) suggests that Theorem 2.1 holds under the weaker condition that the minimum degree of is at least . If this is true, the proof of Theorem 1.2 provides a better lower-bound of on the size of the smallest critical set. However, this is still far from the conjectured bound of .
Lemma 2.4 ([vLW92, Theorem 17.2]).
Let be the set of all Latin squares of order . Then
Lemma 2.5 ([Ghm05, Theorem 3]).
Let be the set of all partial Latin squares of order and of size . Then
The VC-dimension of Latin squares
The most basic result concerning VC-dimension is the Sauer-Shelah lemma. This lemma that has been independently proved several times (e.g. in [Sau72]), provides an upper-bound on the size of a concept class in terms of and . Formally it says where . Note that for the set of Latin squares , we have . Then it is not difficult to see that the Sauer-Shelah lemma together with Lemma 2.4 implies . The factor in this bound is due to the cubic size of in terms of . To obtain the bound of Theorem 1.3, we will use the following strengthening of the Sauer-Shelah lemma due to Pajor [Paj85].
Lemma 2.6 ( [Paj85]).
Every finite set family shatters at least sets.
Proof of Theorem 1.3.
We will prove that for sufficiently large . Note that if a set is shattered by , then in particular for some , and thus . Hence every shattered set corresponds to a partial Latin square. By Lemma 2.6, the set of all Latin squares of order shatters at least sets. It follows that for , we have
Hence to prove , it suffices to show that for every , we have , or equivalently implies .
Setting , and using , we obtain
Using , we obtain
Fix a sufficiently large . If , then , and moreover is an increasing function of in . So inequality (2.2.3) implies , which in turn shows as desired.
The recursive teaching dimension of Latin squares
Proof of Theorem 1.4.
Recall that denotes the set of all Latin squares of order , and denotes the set of all partial Latin squares of order and of size . Set , and while there are partial Latin squares that have unique extensions to full Latin squares , remove such ’s from . Repeat this process with the updated until no such partial Latin square can be found. Denote by the set of all Latin squares that are removed from the initial , and note that . Note further that if is not empty, then its minimum teaching dimension is at least . We know from the proof of Theorem 1.3 that if , and thus as desired. ∎
3 Concluding Remarks
In Theorems 1.3 and 1.4 we established a lower-bound of for both VC-dimension and the recursive teaching dimension of the set of Latin squares of order . One can easily obtain an upper-bound of the form for the VC-dimension, but obtaining a stronger upper-bound, and more ambitiously, determining the exact asymptotics of the VC-dimension seems highly nontrivial. For the teaching dimension and consequently recursive teaching dimension, a stronger upper-bound of follows from the results of [GHM05]. Hence for sufficiently large ,
It would be interesting to improve either of the constants and appearing in the power of in the above bounds.
We wish to thank Yaqiao Li for bringing our attention to the notions of teaching dimension and recursive teaching dimension. We would also like to thank John Bate, John Van Rees and Nicholas Cavenagh for their valuable comments and suggestions.
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