Linear programming and the efficiency of the simplex algorithm for transportation polytopes
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1 Linear programming and the efficiency of the simplex algorithm for transportation polytopes Edward D. Kim University of Wisconsin-La Crosse February 20, 2015 Loras College Department of Mathematics Colloquium
2 2 What is Linear Programming? Maximize the value of a given linear function c : R n R subject to a finite set of linear inequalities.
3 2 What is Linear Programming? Maximize the value of a given linear function c : R n R subject to a finite set of linear inequalities. Example Maximize c(x, y, z) = 2x + 3y + 4z subject to 3x + 2y + z 10 2x + 5y + 3z 15 x, y, z 0.
4 2 What is Linear Programming? Maximize the value of a given linear function c : R n R subject to a finite set of linear inequalities. Example Maximize c(x, y, z) = 2x + 3y + 4z subject to 3x + 2y + z 10 2x + 5y + 3z 15 x, y, z 0. It is not enough to state what is the maximal value of c(x, y, z).
5 What is Linear Programming? Maximize the value of a given linear function c : R n R subject to a finite set of linear inequalities. Example Maximize c(x, y, z) = 2x + 3y + 4z subject to 3x + 2y + z 10 2x + 5y + 3z 15 x, y, z 0. It is not enough to state what is the maximal value of c(x, y, z). α = β iff α β and β α Constraints can also include linear equations
6 3 Polyhedra A polyhedron is any subset of R d formed by the intersection of finitely many closed half-spaces. In other words... A polyhedron is the set of solutions to a finite system of linear inequalities. If a polyhedron is bounded, it is also called a polytope.
7 4 Polytopes and Faces Face = P a supporting hyperplane
8 4 Polytopes and Faces Face = P a supporting hyperplane
9 4 Polytopes and Faces Face = P a supporting hyperplane vertex: faces of dimension zero (pl. vertices)
10 4 Polytopes and Faces Face = P a supporting hyperplane vertex: faces of dimension zero (pl. vertices)
11 4 Polytopes and Faces Face = P a supporting hyperplane vertex: faces of dimension zero (pl. vertices) edge: faces of dimension one
12 4 Polytopes and Faces Face = P a supporting hyperplane vertex: faces of dimension zero (pl. vertices) edge: faces of dimension one
13 4 Polytopes and Faces Face = P a supporting hyperplane vertex: faces of dimension zero (pl. vertices) edge: faces of dimension one facet: faces of dimension d 1
14 5 Polytopes and Graphs Given a polytope P, define the graph of P with: vertices: one vertex for each 0-dimensional face of P edges: an edge connects two vertices if there is 1-dimensional face of P containing the corresponding 0-dimensional faces
15 5 Polytopes and Graphs Given a polytope P, define the graph of P with: vertices: one vertex for each 0-dimensional face of P edges: an edge connects two vertices if there is 1-dimensional face of P containing the corresponding 0-dimensional faces
16 5 Polytopes and Graphs Given a polytope P, define the graph of P with: vertices: one vertex for each 0-dimensional face of P edges: an edge connects two vertices if there is 1-dimensional face of P containing the corresponding 0-dimensional faces Graph Distance dist(u, v) = 2
17 5 Polytopes and Graphs Given a polytope P, define the graph of P with: vertices: one vertex for each 0-dimensional face of P edges: an edge connects two vertices if there is 1-dimensional face of P containing the corresponding 0-dimensional faces Graph Distance dist(u, v) = 2 Graph Diameter diam(g) = 3
18 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
19 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
20 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
21 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
22 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
23 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
24 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
25 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
26 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
27 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
28 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
29 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
30 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
31 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P.
32 6 Klee-Minty Cube Geometrically, the simplex method is a path from vertex to vertex on the polytope P. How fast can the simplex algorithm run?
33 7 The Hirsch Conjecture (d, f ) = maximum diameter among d-polytope with f facets
34 7 The Hirsch Conjecture (d, f ) = maximum diameter among d-polytope with f facets Conjecture [Hirsch 1957] (d, f ) f d
35 The Hirsch Conjecture (d, f ) = maximum diameter among d-polytope with f facets Ex-Conjecture [Hirsch 1957] (d, f ) f d Theorem: Matschke, Weibel, Santos [2011] (d, f ) (f d) for d 20
36 8 Few solved cases The Hirsch Conjecture is known to be true when: d 3 [Klee 1966] f d 6 [Klee, Walkup 1967; Bremner, Schewe 2008] P is a dual transportation polytope [Balinski 1984] P is a 0-1 polytope [Naddef 1989] Polynomial diameter bounds exist for special cases: Integer polytopes in a bounded box [Kleinschmidt, Onn 1992] Totally unimodular constraints [Dyer, Frieze 2001] Transportation polytopes [Brightwell, van den Heuvel, Stougie 2004; Hurkens 2010] Multi-way transportation polytopes [De Loera, K., Onn, Santos 2009]
37 9 Classical Transportation Polytopes Let u R m and v R n with positive entries. Definition The m n transportation polytope determined by margins u and v is the set P of non-negative matrices X = (x i, j ) satisfying n x i, j = u i i and j=1 m x i, j = v j i=1 j.
38 9 Classical Transportation Polytopes Let u R m and v R n with positive entries. Definition The m n transportation polytope determined by margins u and v is the set P of non-negative matrices X = (x i, j ) satisfying n x i, j = u i i and j=1 m x i, j = v j i=1 j.
39 9 Classical Transportation Polytopes Let u R m and v R n with positive entries. Definition The m n transportation polytope determined by margins u and v is the set P of non-negative matrices X = (x i, j ) satisfying n x i, j = u i i and j=1 m x i, j = v j i=1 j.
40 9 Classical Transportation Polytopes Let u R m and v R n with positive entries. Definition The m n transportation polytope determined by margins u and v is the set P of non-negative matrices X = (x i, j ) satisfying n x i, j = u i i and j=1 m x i, j = v j i=1 j.
41 9 Classical Transportation Polytopes Let u R m and v R n with positive entries. Definition The m n transportation polytope determined by margins u and v is the set P of non-negative matrices X = (x i, j ) satisfying n x i, j = u i i and j=1 m x i, j = v j i=1 j.
42 9 Classical Transportation Polytopes Let u R m and v R n with positive entries. Definition The m n transportation polytope determined by margins u and v is the set P of non-negative matrices X = (x i, j ) satisfying n x i, j = u i i and j=1 m x i, j = v j i=1 j.
43 9 Classical Transportation Polytopes Let u R m and v R n with positive entries. Definition The m n transportation polytope determined by margins u and v is the set P of non-negative matrices X = (x i, j ) satisfying n x i, j = u i i and j=1 m x i, j = v j i=1 j.
44 9 Classical Transportation Polytopes Let u R m and v R n with positive entries. Definition The m n transportation polytope determined by margins u and v is the set P of non-negative matrices X = (x i, j ) satisfying n x i, j = u i i and j=1 m x i, j = v j i=1 j.
45 10 Hirsch-implied Bound The m n transportation polytope P defined by n x i, j = u i i j=1 has and m x i, j = v j j and x i,j 0 i, j i=1
46 10 Hirsch-implied Bound The m n transportation polytope P defined by n x i, j = u i i j=1 and m x i, j = v j j and x i,j 0 i, j i=1 has dimension d = (m 1)(n 1) = mn (m + n 1)
47 10 Hirsch-implied Bound The m n transportation polytope P defined by n x i, j = u i i j=1 and m x i, j = v j j and x i,j 0 i, j i=1 has dimension d = (m 1)(n 1) = mn (m + n 1) number of facets: f mn
48 10 Hirsch-implied Bound The m n transportation polytope P defined by n x i, j = u i i j=1 and m x i, j = v j j and x i,j 0 i, j i=1 has dimension d = (m 1)(n 1) = mn (m + n 1) number of facets: f mn If the Hirsch Conjecture is true for transportation polytopes, then diam(p) f d
49 10 Hirsch-implied Bound The m n transportation polytope P defined by n x i, j = u i i j=1 and m x i, j = v j j and x i,j 0 i, j i=1 has dimension d = (m 1)(n 1) = mn (m + n 1) number of facets: f mn If the Hirsch Conjecture is true for transportation polytopes, then diam(p) f d m + n 1.
50 11 Classical Transportation Polytopes Example: m = 3, n = 4 If u = (8, 8, 6) R m and v = (9, 1, 3, 9) R n,
51 Classical Transportation Polytopes Example: m = 3, n = 4 If u = (8, 8, 6) R m and v = (9, 1, 3, 9) R n, then the polytope P contains (among others), the following points:
52 Classical Transportation Polytopes Example: m = 3, n = 4 If u = (8, 8, 6) R m and v = (9, 1, 3, 9) R n, then the polytope P contains (among others), the following points:
53 Classical Transportation Polytopes Example: m = 3, n = 4 If u = (8, 8, 6) R m and v = (9, 1, 3, 9) R n, then the polytope P contains (among others), the following points:
54 Classical Transportation Polytopes Example: m = 3, n = 4 If u = (8, 8, 6) R m and v = (9, 1, 3, 9) R n, then the polytope P contains (among others), the following points:
55 Classical Transportation Polytopes Example: m = 3, n = 4 If u = (8, 8, 6) R m and v = (9, 1, 3, 9) R n, then the polytope P contains (among others), the following points:
56 Classical Transportation Polytopes Example: m = 3, n = 4 If u = (8, 8, 6) R m and v = (9, 1, 3, 9) R n, then the polytope P contains (among others), the following points: Both the point on the left and on the right happen to be vertices due to a Theorem of Klee and Witzgall.
57 12 Vertices with slice condition Same slice contains the same single support entry Vertices X and Y have X Y
58 12 Vertices with slice condition Same slice contains the same single support entry Vertices X and Y have the same slice X Y
59 12 Vertices with slice condition Same slice contains the same single support entry X Y Vertices X and Y have the same slice which has a single support entry
60 12 Vertices with slice condition Same slice contains the same single support entry X Y Vertices X and Y have the same slice which has a single support entry which is the same support entry
61 13 Quadratic Bound Theorem: van den Heuvel, Stougie [2002] Every m n transportation polytope has diameter at most (m + n) 2.
62 13 Quadratic Bound Theorem: van den Heuvel, Stougie [2002] Every m n transportation polytope has diameter at most (m + n) 2. Lemma Given two arbitrary vertices X and Y of an m n transportation polytope P, there are vertices X and Y of P such that: The same slice in X and Y contains the same single support entry dist P (X, X ) + dist P (Y, Y ) 2(m + n 2).
63 Lemma: Proof Sketch 14
64 Lemma: Proof Sketch 14
65 Lemma: Proof Sketch 14
66 Lemma: Proof Sketch 14
67 Lemma: Proof Sketch 14
68 Lemma: Proof Sketch 14
69 Lemma: Proof Sketch 14
70 15 Linear Bound Theorem: Brightwell, van den Heuvel, Stougie [2006] Every m n transportation polytope has diameter at most 8(m + n 1).
71 15 Linear Bound Theorem: Brightwell, van den Heuvel, Stougie [2006] Every m n transportation polytope has diameter at most 8(m + n 1). Lemma Given two arbitrary vertices X and Y of an m n transportation polytope P, there are vertices X and Y of P such that: The same slice in X and Y contains the same single support entry dist P (X, X ) + dist P (Y, Y ) 8.
72 Lemma: Proof Sketch 16
73 Lemma: Proof Sketch 16
74 Better Linear Bound Theorem: Hurkens [2009] Every m n transportation polytope has diameter at most 4(m + n 1).
75 Better Linear Bound Theorem: Hurkens [2009] Every m n transportation polytope has diameter at most 4(m + n 1). Lemma Given two arbitrary vertices X and Y of an m n transportation polytope P, there is an integer k > 0 such that X and Y are vertices of P The same k slices in X and Y each contain the same single support entry dist P (X, X ) + dist P (Y, Y ) 4k.
76 18 2 n Transportation Polytopes Theorem: De Loera, K. [2014] Every 2 n transportation polytope satisfies the Hirsch Conjecture.
77 18 2 n Transportation Polytopes Theorem: De Loera, K. [2014] Every 2 n transportation polytope satisfies the Hirsch Conjecture. Lemma Every vertex X of a non-degenerate 2 n transportation polytope contains a unique column with two strictly positive coordinates.
78 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p,
79 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations
80 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k
81 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k
82 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k
83 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k
84 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k
85 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k x i,j,k = v j i,k j,
86 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k x i,j,k = v j i,k j,
87 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k x i,j,k = v j i,k j,
88 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k x i,j,k = v j i,k x i,j,k = w k i,j j, k.
89 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k x i,j,k = v j i,k x i,j,k = w k i,j j, k.
90 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k x i,j,k = v j i,k x i,j,k = w k i,j j, k.
91 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k x i,j,k = v j i,k x i,j,k = w k i,j j, k.
92 19 3-way Transportation Polytopes Definition Given u R m, v R n and w R p, the 3-way transportation polytope given by 1-marginals is defined in mnp non-negative variables x i,j,k R 0 with the m + n + p equations x i,j,k = u i i, j,k x i,j,k = v j i,k x i,j,k = w k i,j j, k.
93 20 Significance of 3-way Transportation Polytopes Theorem: De Loera, Onn [2006] Given any rational polytope P,
94 20 Significance of 3-way Transportation Polytopes Theorem: De Loera, Onn [2006] Given any rational polytope P, There is a 3-way transportation polytope Q given by 1-marginals with a face that is isomorphic to P.
95 20 Significance of 3-way Transportation Polytopes Theorem: De Loera, Onn [2006] Given any rational polytope P, There is a 3-way transportation polytope Q given by 1-marginals with a face that is isomorphic to P. Moreover, the polytope Q can be computed in polynomial time (in the description of P).
96 Same slice with same single support entry 21
97 Same slice with same single support entry 21
98 Same slice with same single support entry 21
99 Same slice with same single support entry 21
100 Quadratic Bound for Axial Transportation Polytopes Theorem: De Loera, K., Onn, Santos [2009] Every 3-way axial m n p transportation polytope has diameter at most 2(m + n + p) 2.
101 22 Quadratic Bound for Axial Transportation Polytopes Theorem: De Loera, K., Onn, Santos [2009] Every 3-way axial m n p transportation polytope has diameter at most 2(m + n + p) 2. Lemma Given two arbitrary vertices X and Y of an m n p axial transportation polytope P, there are vertices X and Y of P such that: The same slice in X and Y contains the same single support entry dist P (X, X ) + dist P (Y, Y ) 4(m + n + p 1).
102 Lemma: Proof Sketch Case 1 23
103 Lemma: Proof Sketch Case 1 23
104 Lemma: Proof Sketch Case 1 23
105 Lemma: Proof Sketch Case 1 23
106 Lemma: Proof Sketch Case 2 24
107 Lemma: Proof Sketch Case 2 24
108 Lemma: Proof Sketch Case 2 24
109 Lemma: Proof Sketch Case 2 24
110 25 Thank you! N. Bonifas, M. Di Summa, F. Eisenbrand, N. Hähnle, M. Niemeier. On sub-determinants and the diameter of polyhedra. Discrete Comput. Geom., G. Brightwell, J. van den Heuvel, L. Stougie. A linear bound on the diameter of the transportation polytope. Combinatorica, 26(2): , J. A. De Loera, E. D. Kim. Combinatorics and Geometry of Transportation Polytopes: An Update. arxiv: , 2014 J. A. De Loera, E. D. Kim, S. Onn, F. Santos. Graphs of transportation polytopes. J. Combin. Theory Ser. A, 116(8): , J. A. De Loera, S. Onn. All linear and integer programs are slim 3-way transportation programs. SIAM J. Optim., 17: , J. van den Heuvel, L. Stougie. A quadratic bound on the diameter of the transportation polytope. SPOR-report , TU Eindhoven, and CDAM Research Report , LSE, V. Klee, C. Witzgall. Facets and Vertices of the Transportation Polytopes, Lecture Notes in Applied Mathematics, 11: , AMS, 1968.
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