- Email: [email protected]
The Football Pool Polytope
Electronic Notes in Discrete Mathematics 30 (2008) 75–80 www.elsevier.com/locate/endm
The Football Pool Polytope Javier L. Marenco 1 Computer Science Dept., FCEN, University of Buenos Aires, Argentina Sciences Institute, National University of General Sarmiento, Argentina
Pablo A. Rey 2 Departamento de Ingenier´ıa Industrial, FCFM, Universidad de Chile, Chile
Abstract The football pool problem asks for the minimun number of bets on the result on n football matches ensuring that some bet correctly predicts the outcome of at least n − 1 of them. This combinatorial problem has proven to be extremely difficult, and is open for n ≥ 6. Integer programming techniques have been applied to this problem in the past but, in order to tackle the open cases, a deep knowledge of the polytopes associated with the integer programs modeling this problem is required. In this work we address this issue, by defining and studying the football pool polytope in connection with a natural integer programming formulation of the football pool problem. We explore the basic properties of this polytope and present several classes of facet-inducing valid inequalities over natural combinatorial structures in the original problem. Keywords: football pool, polyhedral combinatorics
1
Partially supported by Ubacyt Grant X212, Foncyt Grant 11-09112 (Argentina), and CNPq under Prosul project Proc. 490333/2004-4 (Brazil). Email: [email protected] 2 Partially supported by Fondecyt Grant 1040727, Millennium Science Nucleus Complex Engineering Systems F-06-04 (Chile), and CNPq under Prosul project Proc. 490333/2004-4 (Brazil). Email: [email protected] 1571-0653/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2008.01.014
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J.L. Marenco, P.A. Rey / Electronic Notes in Discrete Mathematics 30 (2008) 75–80
Introduction
Let S = {0, 1, 2} be a set of symbols and, given a positive integer n, define An to be the set of all strings of length n over the alphabet S. The elements of An are called codewords. If a, b ∈ An , we say that a covers b if they differ in at most one symbol, i.e., if the Hamming distance between them is 0 or 1. The football pool problem asks for the minimum-cardinality subset C ⊂ An such that every codeword in An is covered by some codeword in C [1]. The football pool problem has been solved for n ≤ 5 [3], and for n = (3k − 1)/2 for any natural number k [4] (i.e., when there exists a ternary Hamming code of length n). The football poll problem for n ≥ 6 is an important open combinatorial problem [5]. Integer programming techniques have been applied to this problem in the past but, in order to tackle the open cases, a deep knowledge of the polytopes associated with the integer programs modeling this problem is required. In this work we address this issue, by defining and studying the football pool polytope in connection with a natural formulation of the football pool problem as an integer program. If the codeword length n is not explicitly needed, we write A instead of An . If a, b ∈ A, we denote by dist(a, b) the number of symbols in which a and b differ (i.e, the Hamming distance between a and b). For a ∈ A, we define the neighborhood of a to be N (a) = {x ∈ A : dist(a, x) = 1}. Moreover, if C ⊆ A, we define the neighborhood of C to be N (C) = {x ∈ A\C : dist(a, x) = 1 for some a ∈ C}. Finally, for a ∈ A we define the closed neighborhood of a to be N [a] = N (a) ∪ {a} and, similarly, for C ⊆ A we define N [C] = N (C) ∪ C.
2
The football pool polytope
If C ⊆ A is a set of codewords such that every codeword in A is covered by some codeword in C, we say that C is a feasible set. The incidence vector associated with a feasible set C is the vector xC ∈ {0, 1}|A| such that xC a = 1 if and only if a ∈ C, for every a ∈ A. Definition 2.1 [football pool polytope] We define the football pool polytope |A | Fppn ⊆ R n to be the convex hull of the incidence vectors associated to all the feasible sets of An . When the codeword length n is not explicitly needed, we write Fpp instead of Fppn . With this definition, we can state the football pool problem as min xa : x ∈ Fpp . a∈A
J.L. Marenco, P.A. Rey / Electronic Notes in Discrete Mathematics 30 (2008) 75–80
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The football poll polytope Fpp is a special set covering polytope, hence we can explore the basic properties and facets of Fpp by resorting to known results on set covering polytopes [2,6]. It is not difficult to verify that the polytope Fpp is full-dimensional [2]. For every a ∈ A, the binary bounds 0 ≤ xa and xa ≤ 1 are trivial valid inequalities of Fpp and, moreover, a straightforward argument shows that these inequalities define facets of Fpp. Definition 2.2 [point inequalities] If a ∈ A, we define (1) xz ≥ 1 z∈N [a]
to be the point inequality associated with the codeword a. Theorem 2.3 The point inequalities (1) are valid and facet-inducing for Fpp. Note that the point inequalities (1) together with the binary constraints xa ∈ {0, 1} for every a ∈ A define an integer programming formulation for Fpp, i.e., this polytope equals the convex hull of all points x ∈ {0, 1}|A| satisfying the point inequalities.
3
Box inequalities
We define a set B = {a, b, c, d} ⊆ A to be a box if dist(a, b) = 2, and c and d are the only two codewords with dist(a, c) = dist(c, b) = 1 and dist(a, d) = dist(d, b) = 1 (i.e., N (a) ∩ N (b) = {c, d}). For example, if a = 00000 and b = 11000, then c = 10000 and d = 01000. Note that c and d are the only two codewords which differ from a in only one symbol and differ from b in the other symbol separating a from b. Definition 3.1 [box inequalities] If B = {a, b, c, d} is a box, we define (2) xz + xz ≥ 2 z∈B
z∈N (B)
to be the box inequality associated with B. Theorem 3.2 The box inequalities (2) are valid and facet-inducing for Fpp. For x ∈ B, define NB (x) = N (x)\B and, for C ⊆ B, define NB (C) = ∪x∈C NB (x). Definition 3.3 [reinforced box inequalities] If B = {a, b, c, d} is a box, we define 2xd + (3) xz + xNB ({a,b,d}) ≥ 2 z∈B\{d}
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to be the reinforced box inequality associated with B and d. Theorem 3.4 The reinforced box inequalities (3) are valid and facet-inducing for Fpp. If πx ≤ π0 is a valid inequality for Fpp, we define its support to be supp(π) = {z ∈ A : πz = 0}, i.e., the set of codewords with nonzero coefficients in the inequality. Let B ⊆ A be a box, and let πx ≤ π0 be a valid inequality for Fpp. We say that πx ≤ π0 is contained in the box structure N [B] = B ∪ N (B) if supp(π) ⊆ N [B] and for every w ∈ B there exists αw ≥ 0 s.t. every z ∈ NB (B) has πz = max{αw : z ∈ N (w)}. Theorem 3.5 The only facet-inducing inequalities of Fpp contained in a box structure are the point inequalities (1), the box inequalities (2), and the reinforced box inequalities (3).
4
3D-Box inequalities
We define a 3D-Box to be a set T = B ∪ B , where B = {a, b, c, d} and B = {a , b , c , d } are two boxes such that dist(a, a ) = dist(b, b ) = dist(c, c ) = dist(d, d ) = 1. A 3D-Box is given by the eigth codewords differing in three fixed positions. For example, we may take T = {xyz00 : x, y, z ∈ {0, 1}}. In this case, we have a = 00000, b = 11000, c = 10000, and d = 01000 as in the standard box structure and, furthermore, a = 00100, b = 11100, c = 10100, and d = 01100. For x ∈ T , define NT (x) = N (x)\T and, for C ⊆ T , define NT (C) = ∪x∈C NT (x). Definition 4.1 [3D-Box 1-2 inequalities] If T = {a, b, c, d} ∪ {a , b , c , d } is a 3D-Box, we define (4)
2(xa + xb + xd + xd ) + (xa + xb + xc + xc ) + xNT (C) ≥ 3
to be the 3D-Box 1-2 inequality associated with T and the 6-cycle C = {a, a , c , b , b, d}. Theorem 4.2 The 3D-Box 1-2 inequalities (4) are valid and facet-inducing for Fpp. Definition 4.3 [3D-Box 2-3 inequalities] If T = {a, b, c, d} ∪ {a , b , c , d } is a 3D-Box, we define (5)
3(xa + xb + xd + xd ) + 2(xa + xb + xc + xc ) + xNT (C) ≥ 5
J.L. Marenco, P.A. Rey / Electronic Notes in Discrete Mathematics 30 (2008) 75–80
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to be the 3D-Box 2-3 inequality associated with T and the 6-cycle C = {a, a , c , b , b, d}. Theorem 4.4 The 3D-Box 2-3 inequalities (5) are valid and facet-inducing for Fpp. Note that the 3D-Box 1-2 inequality (4) and the 3D-Box 2-3 inequality (5) are defined over the same supporting codewords, the only difference between them being the assignment of coefficients 1 and 2 resp. 2 and 3 within the boxes B and B . As in the previous section, we can define a valid inequality to be contained in a 3D-Box structure. Again, it is possible to show that the only facetinducing valid inequalities contained in any 3D-Box structure are the point inequalities (1), the box and reinforced box inequalities (2) and (3), and the 3D-Box inequalities (4) and (5).
5
Diamond inequalities
Let B = {a, b, d, c} ⊆ A be a box. Let N (a) ∩ N (c) = {e}, N (c) ∩ N (b) = {f }, N (a)∩N (d) = {g}, and N (b)∩N (d) = {h}. We call the set D = B∪{e, f, g, h} a diamond of A. For x ∈ D, we define ND (x) = N (x)\D and, for C ⊆ D, we define ND (C) = ∪x∈C ND (x). Note that ND (x) and ND (y) are disjoint for any x, y ∈ B with x = y. The codewords in B are called the inner codewords of the diamond, and the codewords in D\B are called the outer codewords of the diamond. The diamond structure generates two further classes of facet-inducing inequalities for Fpp, based on the codewords required for covering different subsets of a diamond. The first family is based on the covering of one inner codeword and two outer codewords, and the second family is based on the covering of three outer codewords. Definition 5.1 [inner diamond inequalities] If D = {a, . . . , h} is a diamond, we define (6) xz + xND ({a,f,h}) ≥ 2 z∈D
to be the inner diamond inequality associated with D. Theorem 5.2 The inner diamond inequalities (6) are valid and facet-inducing for Fpp.
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Definition 5.3 [outer diamond inequalities] If D = {a, . . . , h} is a diamond, we define (7) xz + xND ({f,g,h}) ≥ 2 z∈D
to be the outer diamond inequality associated with D. Theorem 5.4 The outer diamond inequalities (7) are valid and facet-inducing for Fpp.
References [1] G. Cohen, I. Honkala, S. Litsyn, and A. Lobstein. Covering Codes. NorthHolland, 1997. [2] E. Balas and S. Ng, On the set covering polytope I: All the facets with coefficients in {0, 1, 2}. Mathematical Programming 43 (1989) 57–69. [3] H. Kamps and J. van Lint, The football pool problem for 5 matches. Journal of Combinatorial Theory 2 (1967) 315–325. [4] J. van Lint, Introduction to coding theory. Springer-Verlag, 1982. ¨ [5] P. Ostergard, A combinatorial proof of the football pool problem for six matches. Journal of Combinatorial Theory A-76 (1996) 160–163. ´ nchez-Garc´ıa, M. Sobro ´ n and B. Vitoriano, On the set covering [6] M. Sa polytope: Facets with coefficients in {0, 1, 2, 3}. Annals of Operations Research 81 (1998) 343–356.
The Football Pool Polytope Javier L. Marenco 1 Computer Science Dept., FCEN, University of Buenos Aires, Argentina Sciences Institute, National University of General Sarmiento, Argentina
Pablo A. Rey 2 Departamento de Ingenier´ıa Industrial, FCFM, Universidad de Chile, Chile
Abstract The football pool problem asks for the minimun number of bets on the result on n football matches ensuring that some bet correctly predicts the outcome of at least n − 1 of them. This combinatorial problem has proven to be extremely difficult, and is open for n ≥ 6. Integer programming techniques have been applied to this problem in the past but, in order to tackle the open cases, a deep knowledge of the polytopes associated with the integer programs modeling this problem is required. In this work we address this issue, by defining and studying the football pool polytope in connection with a natural integer programming formulation of the football pool problem. We explore the basic properties of this polytope and present several classes of facet-inducing valid inequalities over natural combinatorial structures in the original problem. Keywords: football pool, polyhedral combinatorics
1
Partially supported by Ubacyt Grant X212, Foncyt Grant 11-09112 (Argentina), and CNPq under Prosul project Proc. 490333/2004-4 (Brazil). Email: [email protected] 2 Partially supported by Fondecyt Grant 1040727, Millennium Science Nucleus Complex Engineering Systems F-06-04 (Chile), and CNPq under Prosul project Proc. 490333/2004-4 (Brazil). Email: [email protected] 1571-0653/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2008.01.014
76
1
J.L. Marenco, P.A. Rey / Electronic Notes in Discrete Mathematics 30 (2008) 75–80
Introduction
Let S = {0, 1, 2} be a set of symbols and, given a positive integer n, define An to be the set of all strings of length n over the alphabet S. The elements of An are called codewords. If a, b ∈ An , we say that a covers b if they differ in at most one symbol, i.e., if the Hamming distance between them is 0 or 1. The football pool problem asks for the minimum-cardinality subset C ⊂ An such that every codeword in An is covered by some codeword in C [1]. The football pool problem has been solved for n ≤ 5 [3], and for n = (3k − 1)/2 for any natural number k [4] (i.e., when there exists a ternary Hamming code of length n). The football poll problem for n ≥ 6 is an important open combinatorial problem [5]. Integer programming techniques have been applied to this problem in the past but, in order to tackle the open cases, a deep knowledge of the polytopes associated with the integer programs modeling this problem is required. In this work we address this issue, by defining and studying the football pool polytope in connection with a natural formulation of the football pool problem as an integer program. If the codeword length n is not explicitly needed, we write A instead of An . If a, b ∈ A, we denote by dist(a, b) the number of symbols in which a and b differ (i.e, the Hamming distance between a and b). For a ∈ A, we define the neighborhood of a to be N (a) = {x ∈ A : dist(a, x) = 1}. Moreover, if C ⊆ A, we define the neighborhood of C to be N (C) = {x ∈ A\C : dist(a, x) = 1 for some a ∈ C}. Finally, for a ∈ A we define the closed neighborhood of a to be N [a] = N (a) ∪ {a} and, similarly, for C ⊆ A we define N [C] = N (C) ∪ C.
2
The football pool polytope
If C ⊆ A is a set of codewords such that every codeword in A is covered by some codeword in C, we say that C is a feasible set. The incidence vector associated with a feasible set C is the vector xC ∈ {0, 1}|A| such that xC a = 1 if and only if a ∈ C, for every a ∈ A. Definition 2.1 [football pool polytope] We define the football pool polytope |A | Fppn ⊆ R n to be the convex hull of the incidence vectors associated to all the feasible sets of An . When the codeword length n is not explicitly needed, we write Fpp instead of Fppn . With this definition, we can state the football pool problem as min xa : x ∈ Fpp . a∈A
J.L. Marenco, P.A. Rey / Electronic Notes in Discrete Mathematics 30 (2008) 75–80
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The football poll polytope Fpp is a special set covering polytope, hence we can explore the basic properties and facets of Fpp by resorting to known results on set covering polytopes [2,6]. It is not difficult to verify that the polytope Fpp is full-dimensional [2]. For every a ∈ A, the binary bounds 0 ≤ xa and xa ≤ 1 are trivial valid inequalities of Fpp and, moreover, a straightforward argument shows that these inequalities define facets of Fpp. Definition 2.2 [point inequalities] If a ∈ A, we define (1) xz ≥ 1 z∈N [a]
to be the point inequality associated with the codeword a. Theorem 2.3 The point inequalities (1) are valid and facet-inducing for Fpp. Note that the point inequalities (1) together with the binary constraints xa ∈ {0, 1} for every a ∈ A define an integer programming formulation for Fpp, i.e., this polytope equals the convex hull of all points x ∈ {0, 1}|A| satisfying the point inequalities.
3
Box inequalities
We define a set B = {a, b, c, d} ⊆ A to be a box if dist(a, b) = 2, and c and d are the only two codewords with dist(a, c) = dist(c, b) = 1 and dist(a, d) = dist(d, b) = 1 (i.e., N (a) ∩ N (b) = {c, d}). For example, if a = 00000 and b = 11000, then c = 10000 and d = 01000. Note that c and d are the only two codewords which differ from a in only one symbol and differ from b in the other symbol separating a from b. Definition 3.1 [box inequalities] If B = {a, b, c, d} is a box, we define (2) xz + xz ≥ 2 z∈B
z∈N (B)
to be the box inequality associated with B. Theorem 3.2 The box inequalities (2) are valid and facet-inducing for Fpp. For x ∈ B, define NB (x) = N (x)\B and, for C ⊆ B, define NB (C) = ∪x∈C NB (x). Definition 3.3 [reinforced box inequalities] If B = {a, b, c, d} is a box, we define 2xd + (3) xz + xNB ({a,b,d}) ≥ 2 z∈B\{d}
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to be the reinforced box inequality associated with B and d. Theorem 3.4 The reinforced box inequalities (3) are valid and facet-inducing for Fpp. If πx ≤ π0 is a valid inequality for Fpp, we define its support to be supp(π) = {z ∈ A : πz = 0}, i.e., the set of codewords with nonzero coefficients in the inequality. Let B ⊆ A be a box, and let πx ≤ π0 be a valid inequality for Fpp. We say that πx ≤ π0 is contained in the box structure N [B] = B ∪ N (B) if supp(π) ⊆ N [B] and for every w ∈ B there exists αw ≥ 0 s.t. every z ∈ NB (B) has πz = max{αw : z ∈ N (w)}. Theorem 3.5 The only facet-inducing inequalities of Fpp contained in a box structure are the point inequalities (1), the box inequalities (2), and the reinforced box inequalities (3).
4
3D-Box inequalities
We define a 3D-Box to be a set T = B ∪ B , where B = {a, b, c, d} and B = {a , b , c , d } are two boxes such that dist(a, a ) = dist(b, b ) = dist(c, c ) = dist(d, d ) = 1. A 3D-Box is given by the eigth codewords differing in three fixed positions. For example, we may take T = {xyz00 : x, y, z ∈ {0, 1}}. In this case, we have a = 00000, b = 11000, c = 10000, and d = 01000 as in the standard box structure and, furthermore, a = 00100, b = 11100, c = 10100, and d = 01100. For x ∈ T , define NT (x) = N (x)\T and, for C ⊆ T , define NT (C) = ∪x∈C NT (x). Definition 4.1 [3D-Box 1-2 inequalities] If T = {a, b, c, d} ∪ {a , b , c , d } is a 3D-Box, we define (4)
2(xa + xb + xd + xd ) + (xa + xb + xc + xc ) + xNT (C) ≥ 3
to be the 3D-Box 1-2 inequality associated with T and the 6-cycle C = {a, a , c , b , b, d}. Theorem 4.2 The 3D-Box 1-2 inequalities (4) are valid and facet-inducing for Fpp. Definition 4.3 [3D-Box 2-3 inequalities] If T = {a, b, c, d} ∪ {a , b , c , d } is a 3D-Box, we define (5)
3(xa + xb + xd + xd ) + 2(xa + xb + xc + xc ) + xNT (C) ≥ 5
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to be the 3D-Box 2-3 inequality associated with T and the 6-cycle C = {a, a , c , b , b, d}. Theorem 4.4 The 3D-Box 2-3 inequalities (5) are valid and facet-inducing for Fpp. Note that the 3D-Box 1-2 inequality (4) and the 3D-Box 2-3 inequality (5) are defined over the same supporting codewords, the only difference between them being the assignment of coefficients 1 and 2 resp. 2 and 3 within the boxes B and B . As in the previous section, we can define a valid inequality to be contained in a 3D-Box structure. Again, it is possible to show that the only facetinducing valid inequalities contained in any 3D-Box structure are the point inequalities (1), the box and reinforced box inequalities (2) and (3), and the 3D-Box inequalities (4) and (5).
5
Diamond inequalities
Let B = {a, b, d, c} ⊆ A be a box. Let N (a) ∩ N (c) = {e}, N (c) ∩ N (b) = {f }, N (a)∩N (d) = {g}, and N (b)∩N (d) = {h}. We call the set D = B∪{e, f, g, h} a diamond of A. For x ∈ D, we define ND (x) = N (x)\D and, for C ⊆ D, we define ND (C) = ∪x∈C ND (x). Note that ND (x) and ND (y) are disjoint for any x, y ∈ B with x = y. The codewords in B are called the inner codewords of the diamond, and the codewords in D\B are called the outer codewords of the diamond. The diamond structure generates two further classes of facet-inducing inequalities for Fpp, based on the codewords required for covering different subsets of a diamond. The first family is based on the covering of one inner codeword and two outer codewords, and the second family is based on the covering of three outer codewords. Definition 5.1 [inner diamond inequalities] If D = {a, . . . , h} is a diamond, we define (6) xz + xND ({a,f,h}) ≥ 2 z∈D
to be the inner diamond inequality associated with D. Theorem 5.2 The inner diamond inequalities (6) are valid and facet-inducing for Fpp.
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Definition 5.3 [outer diamond inequalities] If D = {a, . . . , h} is a diamond, we define (7) xz + xND ({f,g,h}) ≥ 2 z∈D
to be the outer diamond inequality associated with D. Theorem 5.4 The outer diamond inequalities (7) are valid and facet-inducing for Fpp.
References [1] G. Cohen, I. Honkala, S. Litsyn, and A. Lobstein. Covering Codes. NorthHolland, 1997. [2] E. Balas and S. Ng, On the set covering polytope I: All the facets with coefficients in {0, 1, 2}. Mathematical Programming 43 (1989) 57–69. [3] H. Kamps and J. van Lint, The football pool problem for 5 matches. Journal of Combinatorial Theory 2 (1967) 315–325. [4] J. van Lint, Introduction to coding theory. Springer-Verlag, 1982. ¨ [5] P. Ostergard, A combinatorial proof of the football pool problem for six matches. Journal of Combinatorial Theory A-76 (1996) 160–163. ´ nchez-Garc´ıa, M. Sobro ´ n and B. Vitoriano, On the set covering [6] M. Sa polytope: Facets with coefficients in {0, 1, 2, 3}. Annals of Operations Research 81 (1998) 343–356.