Two Combinatorial Problems involving Lottery Schemes: Algorithmic Determination of Solution Sets A.P. Burger Department of Mathematics and Statistics University of Victoria Victoria, BC, Canada
[email protected] W.R. Gr¨ undlingh & J.H. van Vuuren Department of Applied Mathematics University of Stellenbosch Matieland, South Africa
[email protected],
[email protected]
Abstract. Suppose a lottery scheme consists of randomly selecting a winning n–set from a universal m–set, while a player participates in the scheme by purchasing a playing set of any number of n–sets from the universal set prior to the winning draw, and is awarded a prize if k (or more) elements in the winning n–set match those of at least one of the player’s n–sets in his playing set (1 ≤ k ≤ n ≤ m). Such a prize is called a k–prize. The player may wish to construct a smallest playing set for which the probability of winning a k–prize is at least ψ (0 < ψ ≤ 1), no matter which winning n–set is chosen from the universal set. Alternatively, the player might only be able to purchase a playing set of cardinality `, in which case he may wish to construct his playing set so as to maximise his chances of winning a k–prize. In this paper these two combinatorial optimisation problems are considered. The aim of the paper is twofold: (i) to derive growth properties of and establish bounds on solutions to these problems analytically, and (ii) to develop a number of algorithmic approaches toward finding respectively upper and lower bounds on the solutions to these problems numerically.
Keywords: Lottery, Incomplete lottery problem, Resource utilisation. AMS Subject Classification: 05C69.
Utilitas Final Copy, 38 pages
File: finalLottery.tex
1
Introduction
Suppose a participant of a lottery scheme wishes to play in a manner so as to be at least 100ψ% sure of winning a (lesser) prize, 0 < ψ ≤ 1. Suppose the lottery scheme hm, n; ki consists of randomly selecting a winning n–set w from the universal set Um = {1, 2, . . . , m}, while a player participates in the scheme by purchasing a playing set L of any number of n–sets from Um prior to the winning draw, and is awarded a prize if at least k elements of w match those of at least one of the player’s n–sets in L. In such a case the player may wish to construct his playing set so that the probability of having at least one n–set in L overlapping in at least k elements with w is at least ψ, no matter which winning n–set w is chosen from Um . With this objective in mind, the player may well wonder what the cardinality of such a smallest feasible set L might be and how to go about constructing such a set of smallest cardinality. To make the above description more precise, let Φ(A, s) denote the set of all (unordered) s–sets from a set A, so that |Φ(A, s)| = |A| s . Also, let the neighbourhood of v, N [v] denote the set of all n–sets from Um having at least one k–subset in common with v ∈ Φ(Um , n), that is, N [v] = {φ ∈ Φ(Um , n) | Φ(φ, k) ∩ Φ(v, k) 6= ∅} . We consider the following problem. Definition 1 (Incomplete lottery problem) Define a 100(1 − ψ)%incomplete lottery set for hm, n; ki as a subset Lψ (Um , n; k) ⊆ Φ(Um , n) with the property that [ N [l] l∈Lψ (Um ,n;k)
m n
has cardinality at least dψ e. Then the incomplete lottery problem is: what is the smallest possible cardinality of a 100(1−ψ)%–incomplete lottery set Lψ (Um , n; k)? Denote the answer to this problem by the incomplete lottery number Lψ (m, n; k)1 . An incomplete lottery set Lψ (Um , n; k) of minimum cardinality Lψ (m, n; k) will be referred to as an Lψ (m, n; k)–set for hm, n; ki. 1 In the case where ψ = 1 the incomplete lottery number is referred to as the complete lottery number or simply the lottery number for hm, n; ki, and is usually denoted L(m, n, n; k) in the literature. The reason for including four parameters in the notation is that authors usually allow the governing body of the lottery to draw a t–set as winning set, while players of the lottery draw n–sets, with n 6= t in general, and denote the corresponding (complete) lottery number by L(m, n, t; k). However, in this paper we shall always take n = t, thereby considering only lottery numbers of the form L(m, n, n; k). Hence a four parameter notation is superfluous in our case.
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The incomplete lottery problem is certainly of interest from a combinatorial point of view, but perhaps the following problem, which is essentially the inverse of the incomplete lottery problem, is of more interest from a practical point of view. Suppose a participant of a lottery scheme wishes to play in a manner so as to maximise his chance of winning a k–prize in the lottery hm, n; ki, by constructing a playing set L of fixed cardinality (perhaps as dictated by resources available to the participant). Using the same notation as above, this alternative problem may be defined formally as follows. Definition 2 (Resource utilisation problem) The resource utilisation of a playing set L` = {v 1 , . . . , v` } ⊆ Φ(U m , n) in the lottery hm, n; ki is defined as the proportion ∪`i=1 N [vi ] / m n , and the resource utilisation problem is: given a fixed playing set cardinality 1 ≤ ` ≤ L1 (m, n; k), what is the maximum resource utilisation that may be achieved by some set L` ? Denote the answer to this problem by the resource utilisation number, S ` i=1 N [vi ] Ψ` (m, n; k) = max . m v1 ,...,v` ∈Φ(Um ,n)
n
A playing set of cardinality ` that realises this maximum resource utilisation of Ψ` (m, n; k) will be referred to as a Ψ` (m, n; k)–set for hm, n; ki. Two different, well–studied combinatorial problems, closely related to the complete lottery problem, are the packing and covering problems [2, 3, 5, 9, 11, 17, 18]. Although further investigations into the determination of packing and covering numbers will not be considered in this paper, these definitions are presented to clarify their distinction from the complete lottery problem (a special case of the problem in Definition 1), and to obtain upper bounds for incomplete lottery numbers, as will be demonstrated in the next section.
Definition 3 (Packing problem) Define a packing set for hm, n; ki as a subset P ⊆ Φ(Um , n) with the property that, for any elements p1 , p2 ∈ P, it holds that Φ(p1 , k) ∩ Φ(p2 , k) = ∅. Then the packing problem is: what is the largest possible cardinality of a packing set P for hm, n; ki? Denote the answer to this problem by the packing number P (m, n; k). Definition 4 (Covering problem) Define a covering set for hm, n; ki as a subset C ⊆ Φ(Um , n) with the property that, for any element φk ∈ Φ(Um , k), there exists an element c ∈ C such that {φk } ∩ Φ(c, k) 6= ∅. Then the covering problem is: what is the smallest possible cardinality of a covering set C for hm, n; ki? Denote the answer to this problem by the covering number C(m, n; k).
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We illustrate the two newly defined incomplete lottery and resource utilisation problems (and the subtle difference between the complete lottery problem and the covering problem) by means of a brief example. Example 1 Consider the small lottery scheme h7, 3; 2i. A player constructs a playing set by selecting 3–sets from U7 = {1, . . . , 7} and wins a prize if at least one 3–set exhibits a 2–subset match with the winning 3–set. In order to ensure that every possible pair from U7 is in the player’s playing set, the player might play the complete lottery set L1 = {{1, 2, 3}, {1, 4, 5}, {1, 6, 7}, {2, 4, 6}, {2, 5, 7}, {3, 4, 7}, {3, 5, 6}}. Since every pair from U 7 appears exactly once in L1 , the scheme h7, 3; 2i is called a Steiner system, in which case it is known that P (7, 3; 2) = C(7, 3; 2) = 7. However, the player may achieve the same goal by playing the much smaller complete lottery set L2 = {{1, 2, 3}, {1, 5, 7}, {2, 5, 7}, {3, 4, 6}}, from which it follows that L1 (7, 3; 2) ≤ 4. By exhaustively verifying that no complete lottery set of cardinality 3 exists for h7, 3; 2i, one may deduce that indeed L1 (7, 3; 2) = 4. L2 is therefore an L1 (7, 3; 2)–set for h7, 3; 2i. Now suppose the player is not interested in a guarantee (100% assurance) of winning a 2–prize, but will instead settle for a 90% assurance of winning a prize. One incomplete lottery set that would achieve this goal, is given by L3 = {{1, 4, 5}, {2, 4, 5},{3, 6, 7}}. This is true, because |N [{1, 4, 5}] ∪ N [{2, 4, 5}] ∪ N [{3, 6, 7}]|/ 73 = 32 35 ≥ 0.9. Although other incomplete lottery sets of cardinality 3 yield a similar result, no such sets of cardinality 2 exist, from which we conclude that L0.9 (7, 3; 2) = 3. Therefore L3 is an example of an L0.9 (7, 3; 2)–set for h7, 3; 2i. In fact, L3 is also an 32 (7, 3; 2)–set for h7, 3; 2i. L 35 Finally, consider the playing set L4 = {{1, 5, 7}, {3, 4, 6}} of cardinality ` = 2 < L1(7, 3; 2) = 4. Since |N [{1, 5, 7}] ∪ N [{3, 4, 6}]|/ 73 = 26 35 , it follows 26 . It may be verified exhaustively, by considering all 35 that Ψ2 (7, 3; 2) ≥ 35 2 possible 3–set combinations of cardinality 2 from U7 , that Ψ2 (7, 3; 2) ≤ 26 35 . Hence Ψ2 (7, 3; 2) = 26 and thus L is an example of a Ψ (7, 3; 2)–set for 4 2 35 h7, 3; 2i. The incomplete lottery and resource utilisation problems are inverses of each other in the sense of the following proposition, which is easy to prove. Proposition 1 Let 0 < ψ ≤ 1 be a real number and let m, n, k, ` be natural numbers such that 1 ≤ k ≤ n ≤ m. Then Lψ (m, n; k) ≤ ` if and only if Ψ` (m, n; k) ≥ ψ. Note that if Ψ` (m, n; k) = ψ in the above proposition, then it also holds that Lψ (m, n; k) = `, but that the converse of this special case is not necessarily true.
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To the best knowledge of the authors only the complete lottery problem (i.e., the case where ψ = 1) appears in the combinatorial literature (as early as 1964). The reader is referred to [4] for a brief survey of work on the complete lottery problem. However, the incomplete lottery problem and the resource utilisation problem seem to be novel contributions to the body of literature on lottery schemes.
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Properties of Parameters Lψ (m, n; k) and Ψ` (m, n; k)
The following preliminary result may be derived via a simple counting argument (see, for example, [4] and [10]). Proposition 2 The number of n–sets from Um that have at least one k– subset in common with some n–set v ∈ Φ(Um , n) (excluding v itself ) is given by n−1 X nm − n r= (1) i n−i i=k
for all 1 ≤ k ≤ n ≤ m.
Using the above result it is possible to derive bounds for the parameters Lψ (m, n; k) and Ψ` (m, n; k) from the known existence (and hence boundedness) of the complete lottery number L1 (m, n; k). Theorem 1 (Boundedness of Lψ (m, n; k) and Ψ` (m, n; k)) Both the incomplete lottery number Lψ (m, n; k) and the resource utilisation number Ψ` (m, n; k) exist, and, in fact, & ' ψ m m n ≤ Lψ (m, n; k) ≤ P (m, n; k) ≤ C(m, n; k) ≤ (2) r+1 k and
) ( r+` (r + 1)` ≤ Ψ` (m, n; k) ≤ min 1, m m n
(3)
n
for all 1 ≤ k ≤ n ≤ m, all 0 < ψ ≤ 1 and all 0 < ` ≤ L1 (m, n; k), where r is given in (1). Proof: Suppose Lψ = {l1 , l2 , . . . , lLψ (m,n;k) } is an Lψ (m, n; k)–set for the lottery hm, n; ki and that the set Vi contains all elements φn ∈ Φ(Um , n) such that Φ(φn , k) ∩ Φ(li , k) 6= ∅, excluding li itself, for all i = 1, . . . , Lψ (m, n; k). Then |Vi | = r according to Proposition 2. At best,
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the sets {Vi ∪ li } and {Vj ∪ lj } are disjoint for all i 6= j, in which case (r + 1)Lψ (m, n; k) ≥ ψ m n , from which the first inequality in (2) follows. For the second inequality in (2) it suffices to show that every maximal packing set is also an incomplete lottery set. Suppose P is a packing set of cardinality P (m, n; k) for hm, n; ki. Then, for every p ∈ Φ(Um , n)\P, it holds that Φ(p, k) ∩ Φ(p0 , k) 6= ∅ for some p0 ∈ P, implying that P is indeed also a (possibly non–minimal) incomplete lottery set, for any 0 < ψ ≤ 1. Let C be a covering set of cardinality C(m, n; k) for hm, n; ki, but suppose that P (m, n; k) > C(m, n; k). Then there exists a φ∗n ∈ Φ(Um , n)\C such that Φ(φ∗n , k) ∩ Φ(c, k) = ∅ for all c ∈ C, contradicting the fact that C is a covering set for hm, n; ki and therefore yielding the third inequality in (2). The final upper bound in the inequality chain (2) is achieved by the existence of a covering set C ∗ of cardinality m k for hm, n; ki, where the m ∗ first k entries of elements of C consist of all k elements of Φ(Um , k), and the remaining n − k entries consist of randomly chosen elements from the remaining m − k elements from Um in each case. Then C∗ is a covering set for hm, n; ki by construction, and hence C(m, n; k) ≤ m k .
The lower bound on Ψ` (m, n; k) in (3) may be obtained by constructing a playing set L of cardinality 0 < ` ≤ L1 (m, n; k) for hm, n; ki as follows: Select any n–set l1 ∈ Φ(Um , n) and suppose l1 shares k–subsets with the subset V1 of elements from Φ(Um , n), excluding l1 itself. Now L0 = Φ(Um , n)\V1 is clearly a complete lottery set for hm, n; ki. This is true, because if the winning n–set, w, is an element of L0 , then certainly there exists an element l 0 ∈ L0 such that Φ(l0 , k) ∩ Φ(w, k) 6= ∅, namely the element l0 = w. Otherwise, if w ∈ V1 , then Φ(l1 , k) ∩ Φ(w, k) 6= ∅. Therefore ` ≤ L1 (m, n; k) ≤ |L0 | = m n − r. Hence there exist at least ` − 1 elements in the set Φ(Um , n)\{V1 ∪ {l1 }}. Choose any ` − 1 elements l2 , l3 , . . . , l` from this last set to complete the construction of L. Then the resource utilised by L is at least (r + `)/ m n .
The trivial upper bound Ψ` (m, n; k) ≤ 1 in (3) is obtained by any L1 (m, n; k)–set for hm, n; ki (i.e., by letting ` = L1 (m, n; k)). The non– trivial upper bound Ψ` (m, n; k) ≤ (r + 1)`/ m n in (3) is attained when it is possible to construct a playing set L = {l1 , l2 , . . . , l` } for hm, n; ki with the property that {Vi ∪ {li }} ∩ {Vj ∪ {lj }} = ∅ for all i 6= j = 1, . . . , `, where Vi denotes the set of all n–sets from Um that share a k–subset with li (excluding li itself), for all i = 1, . . . , `. In this case the resource utilised by L is ` ` X (r + 1)` |Vi ∪ {li }| X r + 1 = = m m m i=1
n
i=1
6
n
n
by Proposition 2. In cases where it is not possible to construct a playing set L in a manner that all sets Vi are pairwise disjoint, it follows by the inclusion–exclusion principle that the resource utilised by any playing set of cardinality ` for hm, n; ki is strictly less than (r + 1)`/ m n .
The bounds on Lψ (m, n; k) and Ψ` (m, n; k) in Theorem 1 serve the purpose of establishing the existence of solutions to the incomplete lottery and resource utilisation problems, but are typically very weak (wide apart) for large values of m, prompting an investigation into algorithmic methods for improving these bounds. This will be the topic of the next section. However, we conclude this section with a number of growth properties of the parameters Lψ (m, n; k) and Ψ` (m, n; k) with respect to variations of their arguments, as well as with an isomorphism result showing that certain different lottery schemes are in fact equivalent. Theorem 2 (Growth properties of Lψ (m, n; k) and Ψ` (m, n; k)) (a) Lψ (m0 , n; k) ≤ Lψ (m, n; k) for all 1 ≤ k ≤ n ≤ m0 < m and 0 < ψ ≤ 1. (b) Lψ (m, n0 ; k) ≥ Lψ (m, n; k) for all 1 ≤ k ≤ n0 < n ≤ m and 0 < ψ ≤ 1. (c) Lψ (m, n; k 0 ) ≤ Lψ (m, n; k) for all 1 ≤ k 0 < k ≤ n ≤ m and 0 < ψ ≤ 1. (d) Lψ0 (m, n; k) ≤ Lψ (m, n; k) for all 1 ≤ k ≤ n ≤ m and 0 < ψ 0 < ψ ≤ 1. (e) Ψ` (m0 , n; k) ≥ Ψ` (m, n; k) for all 1 ≤ ` ≤ L1 (m0 , n; k) and 1 ≤ k ≤ n ≤ m0 < m. (f ) Ψ` (m, n0 ; k) ≤ Ψ` (m, n; k) for all 1 ≤ ` ≤ L1 (m, n; k) and 1 ≤ k ≤ n0 < n ≤ m. (g) Ψ` (m, n; k 0 ) ≥ Ψ` (m, n; k) for all 1 ≤ ` ≤ L1 (m, n; k 0 ) and 1 ≤ k 0 < k ≤ n ≤ m. (h) Ψ`0 (m, n; k) ≥ Ψ` (m, n; k) for all 1 ≤ `0 < ` ≤ L1 (m, n; k) and 1 ≤ k ≤ n ≤ m. Proof: (a) Suppose Lψ is an Lψ (m, n; k)–set for hm, n; ki, and that m0 < m, for some 0 < ψ ≤ 1. Consider the following reduction method to obtain a (possibly non–minimal) 100(1 − ψ)%–incomplete lottery set for hm − 1, n; ki from Lψ . Reduction method: Consider a two–dimensional tabular representation of Lψ , consisting of Lψ (m, n; k) rows denoting the n–sets in Lψ and m columns denoting the elements of Um , in which the (i, j)–th cell contains a cross if j ∈ Um is an element of the i–th n–set of Lψ , and is empty otherwise. The remaining part of Lψ , obtained by temporarily disregarding some j–th column in the above tabular representation (some of the resulting rows will represent n–subsets of Um \{j} and
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some might represent (n−1)–subsets of Um \{j}) will collectively share k–subsets of Um \{j} with a proportion, ψj , of elements from Φ(Um \{j}, n). Note that an n–set that has a k–match with Lψ , will have a k–match with exactly m − n of the m possible reductions. Thus counting the total number P of n–sets with a m k–match over all possible reductions, we have j=1 m−1 ψj = n m n ψ × (m − n). The average of the proportions ψ1 , . . . , ψm (i.e., when removing columns 1, . . . , m) is given by Pm m−1 Pm m ψj j=1 j=1 ψj n n ψ(m − n) = = ψ. = m−1 m−1 m m m n n Thus there exists a j ∗ ∈ Um such that ψj ∗ ≥ ψ.
Now permanently remove column j ∗ from the tabular representation of Lψ , and place a cross in any empty cell of each row that now contains only n − 1 crosses as a result of the permanent column deletion. The result is a tabular representation of a 100(1 − ψj ∗ )%–incomplete lottery set for hm − 1, n; ki, which is also a 100(1 − ψ)%–incomplete lottery set for hm − 1, n; ki. By (possibly repeated) application of the above reduction method to Lψ , a 100(1 − ψ)%–incomplete lottery set for hm0 , n; ki may be obtained, implying that Lψ (m0 , n; k) ≤ Lψ (m, n; k) = |Lψ |. (b) Suppose L0ψ = {T10 , . . . , TL0 ψ (m,n0 ;k) } is an Lψ (m, n0 ; k)–set for hm, n0 ; ki, and that n0 < n, for some 0 < ψ ≤ 1. We show, by considering the construction method outlined below, that a (possibly non–minimal) 100(1 − ψ)%–incomplete lottery set for hm, n0 + 1; ki may be constructed from L0ψ . Construction method: Append, to each n0 –set, Ti0 , in L0ψ an arbitrary element of Um \Ti0 to form an (n0 + 1)–set Ti , for each i = 1, . . . , Lψ (m, n0 ; k). Define the set Lψ (m,n0 ;k)
Lψ =
[
Ti .
i=1
It is easy to see that if any w ∈ Φ(Um , n0 ) has a k–match with L0ψ , then w also has a k–match with Lψ . By the definition of L0ψ it follows that at least 100ψ% of the elements of Φ(Um , n0 ) have a k–match with L0ψ . It remains to be shown that at least 100ψ% of the elements of Φ(Um , n0 + 1) also have a k–match with L0ψ and hence with Lψ . Append to each φn0 ∈ Φ(Um , n0 ) all possible single elements from Um \φn0 to form m − n0 new (n0 + 1)–sets.
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Now, if a particular n0 –set, φn0 , has a k–match with L0ψ , then each of its associated (n0 + 1)–sets (m − n0 of them) will also have a k–match with L0ψ . Therefore at least 100ψ% of all the appended (n0 + 1)–sets have a k–match with L0ψ . Note that every element of Φ(Um , n0 +1) occurs exactly n0 +1 times amongst the set of appended (n0 + 1)–sets, because each of the elements of Φ(Um , n0 + 1) is constructed from a different n0 –set φn0 ∈ Φ(Um , n0 ). Therefore, if we remove all duplicates amongst the appended (n0 + 1)– sets, the proportion of (n0 + 1)–sets that have a k–match with L0ψ remains unchanged. Hence at least 100ψ% of the elements of Φ(Um , n0 + 1) have a k–match with Lψ . Therefore Lψ is a 100(1 − ψ)%–incomplete lottery set for hm, n0 + 1; ki. By (possibly repeated) application of the above construction method to the set L0ψ , a 100(1 − ψ)%–incomplete lottery set for hm, n; ki may be obtained, hence Lψ (m, n; k) ≤ Lψ (m, n0 ; k). (c) Suppose Lψ is an Lψ (m, n; k)–set for hm, n; ki, and that k 0 < k, for some 0 < ψ ≤ 1. Therefore there exists a subset Vψ ⊆ Φ(Um , n) of e with the property that, for any φn ∈ Vψ , there cardinality at least dψ m n exists an l ∈ Lψ such that Φ(φn , k) ∩ Φ(l, k) 6= ∅. But for any φk ∈ Φ(φn , k) ∩ Φ(l, k) it holds that Φ(φk , k 0 ) ∩ Φ(l, k 0 ) 6= ∅ (or equivalently that Φ(φn , k 0 ) ∩ Φ(l, k 0 ) 6= ∅), implying that Lψ is also a (possibly non–minimal) 100(1 − ψ)%–incomplete lottery set for hm, n; k 0 i. Hence Lψ (m, n; k 0 ) ≤ Lψ (m, n; k). (d) Suppose Lψ is an Lψ (m, n; k)–set for hm, n; ki, where 1 ≤ k ≤ n ≤ m, and that 0 < ψ 0 < ψ ≤ 1. Therefore there exists a subset Vψ ⊆ Φ(Um , n) of cardinality at least dψ m n e with the property that, for any φn ∈ Vψ , there exists an l ∈ Lψ such that Φ(φn , k) ∩ Φ(l, k) 6= ∅. This condition also holds 0 for any φ0n ∈ Vψ0 ⊆ Vψ (where Vψ has a cardinality of at least dψ 0 m n e), implying that Lψ is a (possibly non–minimal) 100(1 − ψ 0 )%–incomplete lottery set for hm, n; ki. Hence Lψ0 (m, n; k) ≤ Lψ (m, n; k). (e) Suppose, to the contrary, that Ψ` (m0 , n; k) < Ψ` (m, n; k) for some m < m. Then it follows, by Proposition 1, that LΨ` (m,n;k) (m0 , n; k) > `. However, due to the (trivial) inequality Ψ` (m, n; k) ≤ Ψ` (m, n; k), it follows (again by Proposition 1) that LΨ` (m,n;k) (m, n; k) ≤ `. Therefore LΨ` (m,n;k) (m, n; k) ≤ ` < LΨ` (m,n;k) (m0 , n; k), which contradicts the result of Theorem 2(a). This implies that Ψ` (m0 , n; k) ≥ Ψ` (m, n; k). 0
(f) Suppose, to the contrary, that Ψ` (m, n0 ; k) > Ψ` (m, n; k) for some n0 < n. Then it follows, by Proposition 1, that LΨ` (m,n0 ;k) (m, n; k) > `. However, due to the (trivial) inequality Ψ` (m, n0 ; k) ≤ Ψ` (m, n0 ; k), it follows (again by Proposition 1) that LΨ` (m,n0 ;k) (m, n0 ; k) ≤ `. Therefore LΨ` (m,n0 ;k) (m, n0 ; k) ≤ ` < LΨ` (m,n0 ;k) (m, n; k), which contradicts the result of Theorem 2(b). This implies that Ψ` (m, n0 ; k) ≤ Ψ` (m, n; k).
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(g) Consider a Ψ` (m, n; k)–set L for hm, n; ki, and suppose that ` ≤ L1 (m, n; k 0 ) for some k 0 < k. Then it follows, from (1), that r is a decreasing function of k, implying that every l ∈ L shares common k 0 –subsets with more elements of Φ(Um , n) than it shares k–subsets with elements of Φ(Um , n). Consequently Ψ` (m, n; k 0 ) ≥ Ψ` (m, n; k). (h) Let L0 be a Ψ`0 (m, n; k)–set for hm, n; ki and suppose that 0 < `0 < ` ≤ L1 (m, n; k). By adding any ` − `0 n–sets l 6∈ ∪l∈L0 N [l] to L0 , a new playing set L (of cardinality `) is constructed with at least one more element in ∪l∈L N [l] than in ∪l0 ∈L0 N [l0 ]. Hence Ψ`0 (m, n; k) ≤ Ψ` (m, n; k). The following result enables us to focus our attention on only half of the parameter values m, n and k; the other half being accounted for by the proposition, the proof of which may be found in [10]. Although the result for the case of complete lottery numbers is well-known, the result may be generalised as below, because the proof is independent of the value of ψ; it only involves the overlapping n–set structure of Φ(Um , n), essentially showing that hm, n; ki ' hm, m − n, m + k − 2ni. Proposition 3 (Isomorphism results) Suppose 0 < ψ ≤ 1 is a real number and that ` is a natural number satisfying ` ≤ L1 (m, n; k). Then Lψ (m, n; k) = Lψ (m, m − n; m + k − 2n) Ψ` (m, n; k) = Ψ` (m, m − n; m + k − 2n) for all 1 ≤ k ≤ n ≤ m satisfying 2n < m + k.
Due to the divergence of the upper and lower bounds in Theorem 1 for realistic values of the lottery parameters hm, n; ki, the remainder of this paper is devoted to various algorithmic approaches towards improving these analytic bounds numerically.
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Algorithmic Bounds on Lψ (m, n; k) and Ψ` (m, n; k)
In this section we consider a number of algorithmic approaches towards establishing upper and lower bounds for respectively incomplete lottery and resource utilisation numbers. A total of six algorithms were implemented, and this section has been organised into subsections accordingly, presenting the algorithms in order of increasing level of performance. The lottery h20, 4; 3i, for which the complete lottery number is known to satisfy 111 ≤ L1 (20, 4; 3) ≤ 148 (see [11]), was chosen as example to demonstrate the working and efficiency of the algorithms throughout this section.
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The main approaches behind the different algorithms presented in this section may be classified as follows: • Repetitive generation of elements in a playing set L(i) of cardinality ` performed independently from preceding playing sets L(i−1) (Classical random algorithm in §3.1 and Distributed random algorithm in §3.2). • Iterative construction of a playing set L(i) of cardinality i from a previous playing set L(i−1) of cardinality i − 1 (Minimal overlapping algorithm in §3.3 and Neighbourhood removal algorithm in §3.4). • Successive modification of a playing set L of cardinality ` (Tabu search algorithm in §3.5 and Intelligent genetic algorithm in §3.6). The numerical work presented in this section was in certain instances computationally rather taxing, and hence all implemented algorithms were run on a Linux–based MOSIX cluster of workstations. MOSIX is a set of enhancements of the Linux kernel for supporting cluster computing. The core of MOSIX consists of adaptive (on–line) resource sharing (load–balancing, memory ushering and I/O optimisation) algorithms and a pre–emptive process migration mechanism that allows a cluster of workstations and servers (nodes) to work cooperatively as if part of a single system. The cluster may include a large number (up to 65 535) of nodes, which can be any combination of workstations, servers or single–processors of any speed. MOSIX is implemented within the Linux kernel and it maintains compatibility with standard Linux, hence there is no need to modify any user–level files, programs or binaries. When activated, normal Linux processes may be allocated and migrate automatically and transparently to other nodes within the cluster in order to achieve a better resource usage of the cluster. As the demands for resources change across the cluster, processes may migrate again, as many times as necessary, to continue optimising the overall resource usage. More specifically, the MOSIX cluster used for the numerical work in this and the following sections consisted of more than 50 CPUs that included Celeron 900 MHz and Intel Celeron 1 GHz processors (each with 256 MB memory) as well as a single Intel Pentium IV 1.5 GHz master node processor (with 512 MB memory), while the collection of stand–alone computers consisted of a Pentium III 800 MHz dual processor and a Pentium III 600 MHz dual processor (each with 512 MB memory), a Celeron 700 MHz (with 256 MB memory) and 59 Pentium 133 MHz processors (each with 64 MB memory). In order to derive bounds on the (worst case) complexity measure of the algorithms in this section, an upper bound on the complexity of determining the resource utilisation Ψ` (m, n; k) of a given playing set of cardinality ` is required. To determine whether two n–sets share a common k–subset, n comparisons are necessary. In a possible worst case scenario, it might
11
be necessary to examine all ` elements of a playing set in search of a k– subset intersection. We therefore deduce that the number of comparisons performed in order to determine the resource utilisation Ψ` (m, n; k) of a playing set of cardinality ` in a worst case scenario is given by m τΨ` (m,n;k) = O n ` . (4) n
3.1
Classical random algorithm
The classical random algorithm consists of repeatedly generating a random playing set of (user) pre–specified cardinality `. In the event that a cardinality ` playing set yields a resource utilisation of at least ψ (some user pre–specified value), an upper bound on Lψ (m, n; k) is obtained (the upper bound being Lψ (m, n; k) ≤ `), otherwise a lower bound on Ψ` (m, n; k) is established. Algorithm 1 (Classical random algorithm) INPUT: The lottery parameters hm, n; ki, a playing set cardinality `, a desired resource utilisation ψ, the number of iterations allowed Iterations. OUTPUT: Lψ (m, n; k) ≤ ` or Ψ` (m, n; k) ≥ ruBest. (1) index ← 1, ruBest ← 0. (2) Generate random playing set L of cardinality `. (3) Determine the resource utilisation ru of L. (4) If (ru ≥ ψ) then output Lψ (m, n; k) ≤ `. STOP. (5) If (ru > ruBest) then ruBest ← ru. (6) index ← index + 1. (7) If (index < Interations) then goto (2), otherwise output Ψ` (m, n; k) ≥ ruBest. STOP.
The implementation of Algorithm 1 is intuitive and elementary. With the reasonable assumption that the generation of a random playing set of cardinality ` has a worst case complexity of O(`), the worst case complexity of Algorithm 1 is O(` τΨ` (m,n;k) Iterations), with τΨ` (m,n;k) given in (4). Example 2 Consider the lottery h20, 4; 3i. Algorithm 1 was initialised with Iterations = 1 000 and desired resource utilisation of ψ = 0.9, generating playing sets of cardinality ` = 100. The best resource utilisation was achieved at iteration 173, yielding the lower bound Ψ100 (20, 4; 3) ≥ 34 752 845 ≈ 77.4247%. The graph in Figure 3.1 presents the (percentage) resource utilisation of the random playing sets (of cardinality 100) generated by Algorithm 1 at every iteration. No convergence is observed, as expected.
12
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Figure 3.1: Lower bound on the resource utilisation for h20, 4; 3i using a hundred 4–sets from U20 at every iteration, obtained by Algorithm 1. The upper, middle and lower solid (—) lines respectively represent the maximum, mean and minimum resource utilisation. The main advantage of this implementation is the fast unconditional playing set generation, although from Figure 3.1 the independence of successive iterations is evident, yielding no visible improvement trend during application of the algorithm. The algorithm is not expected to yield very good results in general, but it does yield a bench mark against which the performance of other algorithms may be tested.
3.2
Distributed random algorithm
As a possible improvement to the classical random algorithm in the previous section, the generation of playing sets (Step (2) in Algorithm 1) may be constrained. Let fL (i) denote the frequency of occurrence of the element i ∈ Um in a playing set L. Then the constraint to mgenerate m a playing set L −1, such that fL (i) may only take the value m n n or n +1 (depending on the divisibility properties of m by n) may be added to Algorithm 1, thus forcing the elements of Um to be utilised approximately equally in L. This constraint on the generation of n–sets seems intuitively justifiable, due to the fact that “spreading out” n–sets (and hence distributing the elements of Um across the n–sets of L) in a playing set is expected to yield a greater resource utilisation. Algorithm 2 (Distributed random algorithm) INPUT: The lottery parameters hm, n; ki, a playing set cardinality `, a
13
desired resource utilisation ψ, the number of iterations allowed Iterations. OUTPUT: Lψ (m, n; k) ≤ ` or Ψ` (m, n; k) ≥ ruBest. (1) index ← 1, ruBest ← 0. (2) Generate random m m playing set L of cardinality ` such that fL (i) ∈ m − 1, , + 1 for all 1 ≤ i ≤ m. n n n
(3) Determine the resource utilisation ru of L.
(4) If (ru ≥ ψ) then output Lψ (m, n; k) ≤ `. STOP. (5) If (ru > ruBest) then ruBest ← ru. (6) index ← index + 1. (7) If (index < Interations) then goto (2), otherwise output Ψ` (m, n; k) ≥ ruBest. STOP.
The distributed random algorithm performs marginally better than the classical random procedure presented in Algorithm 1 in the sense that it yields a higher mean resource utilisation at a fractional increase in worst order complexity measure. Assuming the choice of an element for inclusion in any n–set has worst case complexity O(m) (i.e., examining all elements 1 ≤ i ≤ m in search of a minimum fL (i)), the worst case complexity measure of Algorithm 2 is O(m ` τΨ` (m,n;k) Iterations), with τΨ` (m,n;k) given in (4). Example 3 (cont. of Example 2) Reconsider the lottery h20, 4; 3i (in Example 2). Algorithm 2 was initialised with Iterations = 1 000, a desired resource utilisation of ψ = 0.9, and generating playing sets of cardinality ` = 100. The best resource utilisation was achieved at iteration 715, yield805 ≈ 78.5346% in comparison to ing the lower bound Ψ100 (20, 4; 3) ≥ 43 845 the bound Ψ100 (20, 4; 3) ≥ 77.4247% found by Algorithm 1. The graph in Figure 3.2 represents the (percentage) resource utilisation of the conditionally generated random playing sets (of cardinality 100) at every iteration of Algorithm 2. Again no convergence is observed, as expected. Although conditions are imposed on the generation of playing sets, there is still no improvement trend in the resource utilised by successively generated playing sets during the algorithm implementation.
3.3
Minimal overlapping algorithm
It is intuitively expected that the more elements of Um shared by any two elements of Φ(Um , n), the less resource is utilised. It is therefore reasonable
14
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Figure 3.2: Lower bound on the resource utilisation for h20, 4; 3i using a hundred 4–sets from U20 at every iteration, obtained by Algorithm 2. The upper, middle and lower solid (—) lines respectively represent the maximum, mean and minimum resource utilisation. to assume that, in order to increase the resource utilised by a given playing set L, the elements of L should be chosen so that the cardinality of intersection between any two n–sets in L is minimised. This is the principle behind the following heuristic algorithm. Algorithm 3 (Minimal overlapping algorithm) INPUT: The lottery parameters hm, n; ki, a playing set cardinality `, a desired resource utilisation ψ. OUTPUT: Lψ (m, n; k) ≤ ` or Ψ` (m, n; k) ≥ ruBest. (1) index ← 0, ruBest ← 0, L(0) ← {}, ovrlps ← 0. (2) vindex ← {}, ovrlps← min1≤j 9: ;
(a) Crossover procedure
M!NOP%Q R ? ?@ABACDA ?BAEAF D A ?*EAGAFDD SUTWV XY Z[2[7\] ^_` bdcRNeR Jf Q HIJ KL ? ?@ABACDA ? @ABAED A ?*EAGAFDD SUTWV XY Z[2[7\a ^_ ` (b) Mutation procedure
Figure 3.7: Illustration of the (a) crossover procedure performed on two parent chromosomes yielding two (fitter) children (Child 1 [2] is obtained from Parent 1 [2] by the exchange of gene 2 [3] from Parent 2 [1] with gene 2 [1] from Parent 1 [2]) and (b) mutation procedure on a single chromosome yielding a (weaker) child (the Child is obtained from the Parent by replacing gene 2 with the random gene {1, 2, 4}) for h7, 3; 2i. a random proportion of genes in a random proportion of solution candidates (using a uniform distribution). These proportions are given respectively by the parameters gMutate and cMutate in Algorithm 6. Figure 3.7(a) [(b)] presents an illustrative example of the crossover [mutation] operation performed on chromosomes for h7, 3; 2i. Algorithm 6 (Intelligent genetic algorithm) INPUT: The lottery parameters hm, n; ki, a playing set cardinality `, a desired resource utilisation ψ, a candidate population cardinality |M|, the percentage of chromosomes to mutate per generation cMutate, the percentage of genes to mutate per chromosome gMutate, the number of generations Generations. OUTPUT: Lψ (m, n; k) ≤ ` or Ψ` (m, n; k) ≥ ruBest. to replace the corresponding parent in the population. A more faithful approach would have been to select one of the `2 offspring randomly, but this approach was found to exhibit far inferior convergence of mean fitness of the chromosome population over the generations.
28
(0)
(1) index ← 1, ruBest ← 0, Lj ← an `–set consisting of random n–sets in Φ(Um , n) for all j = 1, . . . , |M| (using uniform distribution). (index) (2) Update RU Lj for all j = 1, . . . , |M|. (index) > ruBest then (3) If RU Lj (index) index∗ ← argmaxj∈{1,...,|M|} RU Lj and ∗ (index ) . ruBest ← RU Lj (4) If (ruBest ≥ ψ) then output Lψ (m, n; k) ≤ `. STOP.
(5) Perform transition procedures: Crossover & Mutation (according to the parameters gMutate and cMutate). (6) index ← index + 1. (7) If (index ≤ Generations) then goto (2), otherwise output Ψ` (m, n; k) ≥ ruBest. STOP.
This process is iterated for a (user) pre–specified number of generations or until a certain fitness tolerance is obtained (for this specific implementation, the tolerance may be given by some minimally acceptable average population fitness or if one candidate represents a Lψ (m, n; k)–set for hm, n; ki). In Algorithm 6, chromosome i of a population M and its resource utilisation is represented by L(i) and RU(L(i) ), respectively. In the following example, the impact of individual parameter variations are considered. This sheds light on the effectiveness of genetic algorithms in the context of maximising resource utilisation or finding 100(1 − ψ)%– incomplete lottery sets. Example 7 (cont. of Examples 2–6) Reconsider the lottery h20, 4; 3i of Examples 2–6. Three independent parameter variations were considered (for every case Generations = 200): 1: Algorithm 6 was initialised with a random population consisting of (0) |M| = 20 candidate solutions, each representing a playing set Lj (j = 1, . . . , 20) of ` = 100 genes (4–sets). In every generation, cMutate = 5% of the population candidates were subjected to mutation. The parameter gMutate was initialised to take the values (i) 1%, (ii) 2%, (iii) 5%, (iv) 10% and (v) 25% respectively. The maximum population fitness (and hence the best resource utilisation) per generation is presented in Figure 3.8(a). The best resource utilisation was obtained with a gene mutation percentage of gMutate = 2%, yielding 250 the lower bound Ψ100 (20, 4; 3) ≥ 44 845 ≈ 87.7193%.
29
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Figure 3.8: Influence on the resource utilisation Ψ100 (20, 4; 3) due to a change in the genetic algorithm parameters: (a) gene mutation parameter gMutate, (b) candidate mutation parameter cMutate, (c) population parameter |M| and (d) utilising the best genetic parameters (gMutate = 2%, cMutate = 10% and |M| = 20) for the lottery h20, 4; 3i.
30
2: The parameter cMutate was initialised to take the values (i) 5%, (ii) 10%, (iii) 25% and (iv) 50% respectively. Algorithm 6 was further randomly initialised with a population consisting of |M| = 20 (0) candidate solutions, each describing a playing set Lj (j = 1, . . . , 20) of ` = 100 genes (4–sets). Every candidate subjected to mutation, had gMutate = 5% of its respective genes mutated. The maximum population fitness (and hence the best resource utilisation) per generation is presented in Figure 3.8(b). The best resource utilisation was obtained with a candidate mutation percentage of cMutate = 10%, yielding the 261 lower bound Ψ100 (20, 4; 3) ≥ 44 845 ≈ 87.9463%. 3: The number of candidate solutions |M| in a population was varied to take the values (i) 6, (ii) 10 and (iii) 20. A single chromosome 1 (cMutate = |M| ) of the |M| candidates in the population, was subjected to a mutation of gMutate = 5% genes. The maximum population fitness (and hence the best resource utilisation) per generation is presented in Figure 3.8(c). The best resource utilisation was obtained with a candidate population of |M| = 20, yielding the lower bound 260 ≈ 87.9257%. Ψ100 (20, 4; 3) ≥ 44 845 Using these results, the following additional case was considered. 4: Algorithm 6 was initialised with gMutate = 2%, cMutate = 10% in a population of |M| = 20 candidate solutions, thus utilising the best empirically obtained parameter values from Cases 1–3. Figure 3.8(d) represents the minimum, mean and maximum generation fitness (or equivalently the resource utilisation Ψ100 (20, 4; 3)) as a function of the population generation. The best resource utilisation was obtained 277 in generation 99, yielding the lower bound Ψ100 (20, 4; 3) ≥ 44 845 ≈ 88.2766%. Although the best lower bound on the resource utilisation Ψ100 (20, 4; 3) obtained by Algorithm 6 is weaker than that obtained by Algorithm 4, this is not the case in general, as will be demonstrated in §4. Consider the complexity of performing the crossover procedure (of Algorithm 6) between any two individual candidates of a population M. Every possible gene exchange is considered in the generation of child candidates, which may be performed in 2`2 |M| operations. For each of the possible future offspring, the resource utilisation is calculated in order to extract those children that are fittest, leading to a complexity of O Generations `2 |M|τΨ` (m,n;k) . Consider, on the other hand, the complexity of performing the mutation procedure on a generation of candidate solutions. A
31
random selection of gMutate×` genes in cMutate×|M| individuals are mutated (replaced by random n–sets). The choice of candidates and relevant genes may each be considered of constant complexity. This leads to the conclusion that the mutation procedure has a complexity of O(` gMutate × |M|cMutate) per generation. With both procedures being performed independently (and hence having complexity O(Generations (`2 |M|τΨ` (m,n;k) + ` gMutate|M|cMutate))), we deduce that the worst case complexity of Algorithm 6 is O Generations `2 |M|τΨ` (m,n;k) .
One serious disadvantage of the genetic algorithm implementation in determining lower bounds on the resource utilisation Ψ` (m, n; k), as opposed to the tabu search optimisation heuristic of Algorithm 5, is the crossover procedure making a considerable contribution to the computational complexity. Tabu search implementations for finding bounds on Lψ (m, n; k) and Ψ` (m, n; k) become less attractive as the cardinality of Φ(Um , n) increases. This limitation is even more acute in the case of the genetic algorithm.
4
Comparison of Algorithmic Performances
The relative performances of the six algorithms presented in §3 are compared in this section, so that a ranking of the algorithms may be achieved in terms of the quality of the bounds they produce (on average). This is done by applying the algorithms to a number of small lotteries in §4.1, after which we apply the algorithms, in §4.2, to arguably the most popular realistic lottery scheme in operation in the world.
4.1
Some small lotteries
Algorithms 1–6 were implemented with the intention of finding complete lottery sets for hm, 5; 2i, where 5 ≤ m ≤ 30, in order to provide the reader with an impression of the relative algorithmic performances. The smallest complete lottery set cardinalities determined after 1 000 iterations [generations] by the algorithms (hence upper bounds on L1 (m, 5; 2)) are summarised in Table 4.2. The table provides some evidence towards our claim that the six algorithms in §3 were presented in order of increasing level of average performance. Further evidence of this claim may be found in Table 4.3, where lower bounds are found on resource utilisation numbers Ψ` (9, 4; 3) = Ψ` (9, 5; 4) for all 2 ≤ ` ≤ 9 = L1 (9, 4; 3) = L1 (9, 5; 4) and on Ψ` (10, 4; 3) = Ψ` (10, 6; 5) for all 2 ≤ ` ≤ L1 (10, 4; 3) = L1 (10, 6; 5) ≤ 14, by virtue of Proposition 3.
32
(1) (2) (3) (4) (5) (6)
5 1 1 1 1 1 1
6 1 1 1 1 1 1
7 1 1 1 1 1 1
8 1 1 1 1 1 1
9 2 2 2 2 2 2
10 2 2 2 2 2 2
(1) (2) (3) (4) (5) (6)
18 6 6 4 4 4 4
19 8 8 4 4 4 4
20 10 8 4 4 4 4
21 13 11 9 9 6 6
22 13 13 9 9 6 6
23 16 16 12 11 8 9
Classical random (1) Minimal overlapping (3) Tabu search (5)
m 11 2 2 2 2 2 2 m 24 19 17 14 15 11 11
12 3 2 2 2 2 2
13 14 3 3 3 3 3 3 3 3 3 3 3 3
15 3 3 3 3 3 3
16 4 4 3 3 3 3
17 5 4 4 4 4 4
25 20 20 16 18 15 12
26 25 22 18 18 16 15
28 30 28 22 22 21 18
29 33 32 26 26 25 20
30 36 35 28 27 26 22
27 29 26 21 22 21 16
Distributed random (2) Neighbourhood removal (4) Intelligent genetic (6)
Table 4.2: A comparison between upper bounds on L1 (m, 5; 2), where 5 ≤ m ≤ 30, as determined by Algorithms 1–6. Bold faced entries represent exact values of L1 (m, 5; 2) [1].
33
`→ Algorithm Algorithm Algorithm Algorithm Algorithm Algorithm
1 2 3 4 5 6
2
3
4
5
6
7
8
9
42 126 42 126 42 126 42 126 42 126 42 126
63 126 63 126 48 126 63 126 63 126 63 126
80 126 80 126 65 126 80 126 80 126 80 126
93 126 93 126 82 126 93 126 93 126 93 126
100 126 101 126 90 126 103 126 104 126 104 126
109 126 106 126 100 126 110 126 115 126 115 126
114 126 112 126 108 126 116 126 121 126 121 126
119 126 117 126 112 126 120 126 125 126
1
(a) Lower bounds on Ψ` (9, 4; 3) = Ψ` (9, 5; 4) for all 2 ≤ ` ≤ 9 = L1 (9, 4; 3) = L1 (9, 5; 4)
`
2
3
4
5
6
7
8
9
10
11
12
13
14
1
50 210 50 210 50 210 50 210 50 210 50 210
75 210 75 210 75 210 75 210 75 210 75 210
100 210 100 210 96 210 96 210 100 210 100 210
117 210 121 210 117 210 117 210 125 210 125 210
131 210 134 210 127 210 138 210 139 210 139 210
145 210 148 210 141 210 151 210 156 210 156 210
156 210 161 210 155 210 164 210 170 210 170 210
169 210 169 210 167 210 175 210 180 210 181 210
178 210 174 210 177 210 185 210 191 210 193 210
184 210 182 210 185 210 193 210 198 210 199 210
189 210 188 210 193 210 198 210 201 210 204 210
191 210 193 210 198 210 202 210 206 210 208 210
200 210 201 210 201 210 204 210 208 210
2 3 4 5 6
1
(b) Lower bounds on Ψ` (10, 4; 3) = Ψ` (10, 6; 5) for all 2 ≤ ` ≤ L1 (10, 4; 3) = L1 (10, 6; 5) ≤ 14
Table 4.3: A comparison of the relative performances of Algorithms 1–6 in the determination of lower bounds on the resource utilisation numbers Ψ` (9, 4; 3), Ψ` (9, 5; 4), Ψ` (10, 4; 3) and Ψ` (10, 6; 5) for all feasible values of `.
34
4.2
The lottery h49, 6; 3i
The lottery scheme h49, 6; ki is perhaps the most popular one in operation across the world3 , where k ∈ {3, 4, 5, 6}. The set Φ(U49 , 6) has cardinality 13 983 816, and for this scheme r = 260 623 in (1). Due to the complexities of Algorithms 4–6, only Algorithms 1–3 could be applied to determine lower bounds on Ψ` (49, 6; 3). These lower bounds are presented in Figure 4.9, together with the best known bound of L1 (49, 6; 3) ≤ 163, available from Internet repository tables [11]. ;
