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Counter-examples to the Hirsch conjecture arXiv:1006.2814 Francisco - PowerPoint PPT Presentation

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The conjecture Motivation: LP Why n d ? The construction (I) The construction(s) (II) Its limitations Conclusion Counter-examples to the Hirsch conjecture arXiv:1006.2814 Francisco Santos http://personales.unican.es/santosf/Hirsch

  1. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion A quasi -polynomial bound, and a bound in fixed dimension Theorem [Kalai-Kleitman 1992] For every d -polytope with n facets: δ ( P ) ≤ n log 2 d + 2 . Theorem [Barnette 1967, Larman 1970] For every d -polytope with n facets: δ ( P ) ≤ n 2 d − 3 . 8

  2. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion A quasi -polynomial bound, and a bound in fixed dimension Theorem [Kalai-Kleitman 1992] For every d -polytope with n facets: δ ( P ) ≤ n log 2 d + 2 . Theorem [Barnette 1967, Larman 1970] For every d -polytope with n facets: δ ( P ) ≤ n 2 d − 3 . 8

  3. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Motivation: linear programming A linear program is the problem of maximization (or minimization) of a linear functional subject to linear inequality constraints. That is: finding max { c · x : x ∈ R d , Mx ≤ b } for given c ∈ R d , b ∈ R n , M ∈ R d × n . 9

  4. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Motivation: linear programming A linear program is the problem of maximization (or minimization) of a linear functional subject to linear inequality constraints. That is: finding max { c · x : x ∈ R d , Mx ≤ b } for given c ∈ R d , b ∈ R n , M ∈ R d × n . 9

  5. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Motivation: linear programming A linear program is the problem of maximization (or minimization) of a linear functional subject to linear inequality constraints. That is: finding max { c · x : x ∈ R d , Mx ≤ b } for given c ∈ R d , b ∈ R n , M ∈ R d × n . “If one would take statistics about which mathematical problem is using up most of the computer time in the world , then ( not including database handling problems like sorting and searching ) the answer would probably be linear programming.” (László Lovász, 1980) 9

  6. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Conection to the Hirsch conjecture The set of feasible solutions P = { x ∈ R d : Mx ≤ b } is a polyhedron P with (at most) n facets and d dimensions. The optimal solution (if it exists) is always attained at a vertex. The simplex method [Dantzig 1947] solves linear programming by starting at any feasible vertex and moving along the graph of P , in a monotone fashion, until the optimum is attained. In particular, the Hirsch conjecture is related to the question of what is the worst-case complexity of the simplex method. 10

  7. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Conection to the Hirsch conjecture The set of feasible solutions P = { x ∈ R d : Mx ≤ b } is a polyhedron P with (at most) n facets and d dimensions. The optimal solution (if it exists) is always attained at a vertex. The simplex method [Dantzig 1947] solves linear programming by starting at any feasible vertex and moving along the graph of P , in a monotone fashion, until the optimum is attained. In particular, the Hirsch conjecture is related to the question of what is the worst-case complexity of the simplex method. 10

  8. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Conection to the Hirsch conjecture The set of feasible solutions P = { x ∈ R d : Mx ≤ b } is a polyhedron P with (at most) n facets and d dimensions. The optimal solution (if it exists) is always attained at a vertex. The simplex method [Dantzig 1947] solves linear programming by starting at any feasible vertex and moving along the graph of P , in a monotone fashion, until the optimum is attained. In particular, the Hirsch conjecture is related to the question of what is the worst-case complexity of the simplex method. 10

  9. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Conection to the Hirsch conjecture The set of feasible solutions P = { x ∈ R d : Mx ≤ b } is a polyhedron P with (at most) n facets and d dimensions. The optimal solution (if it exists) is always attained at a vertex. The simplex method [Dantzig 1947] solves linear programming by starting at any feasible vertex and moving along the graph of P , in a monotone fashion, until the optimum is attained. In particular, the Hirsch conjecture is related to the question of what is the worst-case complexity of the simplex method. 10

  10. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Conection to the Hirsch conjecture The set of feasible solutions P = { x ∈ R d : Mx ≤ b } is a polyhedron P with (at most) n facets and d dimensions. The optimal solution (if it exists) is always attained at a vertex. The simplex method [Dantzig 1947] solves linear programming by starting at any feasible vertex and moving along the graph of P , in a monotone fashion, until the optimum is attained. In particular, the Hirsch conjecture is related to the question of what is the worst-case complexity of the simplex method. 10

  11. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming For most of the pivot rules devised so far there is an analogue of the Klee-Minty cube, which makes the simplex method take an exponential number of steps: 11

  12. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming For most of the pivot rules devised so far there is an analogue of the Klee-Minty cube, which makes the simplex method take an exponential number of steps: 11

  13. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming For most of the pivot rules devised so far there is an analogue of the Klee-Minty cube, which makes the simplex method take an exponential number of steps: 11

  14. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming For most of the pivot rules devised so far there is an analogue of the Klee-Minty cube, which makes the simplex method take an exponential number of steps: 11

  15. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming For most of the pivot rules devised so far there is an analogue of the Klee-Minty cube, which makes the simplex method take an exponential number of steps: 11

  16. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming For most of the pivot rules devised so far there is an analogue of the Klee-Minty cube, which makes the simplex method take an exponential number of steps: 11

  17. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming For most of the pivot rules devised so far there is an analogue of the Klee-Minty cube, which makes the simplex method take an exponential number of steps: 11

  18. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming For most of the pivot rules devised so far there is an analogue of the Klee-Minty cube, which makes the simplex method take an exponential number of steps: 11

  19. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming For most of the pivot rules devised so far there is an analogue of the Klee-Minty cube, which makes the simplex method take an exponential number of steps: 11

  20. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming For most of the pivot rules devised so far there is an analogue of the Klee-Minty cube, which makes the simplex method take an exponential number of steps: 11

  21. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming For most of the pivot rules devised so far there is an analogue of the Klee-Minty cube, which makes the simplex method take an exponential number of steps: 11

  23. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming For most of the pivot rules devised so far there is an analogue of the Klee-Minty cube, which makes the simplex method take an exponential number of steps: 11

  24. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming There are more recent algorithms for linear programming which are proved to be polynomial: (ellipsoid [1979], interior point [1984]). But: 12

  25. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming There are more recent algorithms for linear programming which are proved to be polynomial: (ellipsoid [1979], interior point [1984]). But: 12

  26. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming There are more recent algorithms for linear programming which are proved to be polynomial: (ellipsoid [1979], interior point [1984]). But: The number of pivot steps [that the simplex method takes] to solve a problem with m equality constraints in n nonnegative variables is almost always at most a small multiple of m, say 3 m. (M. Todd, 2011) 12

  27. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming There are more recent algorithms for linear programming which are proved to be polynomial: (ellipsoid [1979], interior point [1984]). But: The number of pivot steps [that the simplex method takes] to solve a problem with m equality constraints in n nonnegative variables is almost always at most a small multiple of m, say 3 m. The simplex method has remained, if not the method of choice, a method of choice, usually competitive with, and on some classes of problems superior to, the more modern approaches. (M. Todd, 2011) 12

  28. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming Besides, the known polynomial algorithms for linear programming known are not strongly polynomial : They are polynomial in the bit model of complexity (Turing machine) but not in the arithmetic model (real RAM machine). Finding strongly polynomial algorithms for linear programming is one of the “mathematical problems for the 21st century" according to [Smale 2000]. A polynomial pivot rule would solve this problem in the affirmative. 13

  29. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming Besides, the known polynomial algorithms for linear programming known are not strongly polynomial : They are polynomial in the bit model of complexity (Turing machine) but not in the arithmetic model (real RAM machine). Finding strongly polynomial algorithms for linear programming is one of the “mathematical problems for the 21st century" according to [Smale 2000]. A polynomial pivot rule would solve this problem in the affirmative. 13

  30. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Complexity of linear programming Besides, the known polynomial algorithms for linear programming known are not strongly polynomial : They are polynomial in the bit model of complexity (Turing machine) but not in the arithmetic model (real RAM machine). Finding strongly polynomial algorithms for linear programming is one of the “mathematical problems for the 21st century" according to [Smale 2000]. A polynomial pivot rule would solve this problem in the affirmative. 13

  31. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? It holds with equality in simplices ( n = d + 1, δ = 1) and cubes ( n = 2 d , δ = d ). It holds for all 0-1 polytopes [Naddef 1989] and for 3-polytopes [Klee 1966]. If P and Q satisfy it, then so does P × Q : δ ( P × Q ) = δ ( P ) + δ ( Q ) . In particular: For every n ≤ 2 d , there are polytopes in which the bound is tight (products of simplices). For every n > d , it is easy to construct unbounded polyhedra where the bound is tight. 14

  32. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? It holds with equality in simplices ( n = d + 1, δ = 1) and cubes ( n = 2 d , δ = d ). It holds for all 0-1 polytopes [Naddef 1989] and for 3-polytopes [Klee 1966]. If P and Q satisfy it, then so does P × Q : δ ( P × Q ) = δ ( P ) + δ ( Q ) . In particular: For every n ≤ 2 d , there are polytopes in which the bound is tight (products of simplices). For every n > d , it is easy to construct unbounded polyhedra where the bound is tight. 14

  33. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? It holds with equality in simplices ( n = d + 1, δ = 1) and cubes ( n = 2 d , δ = d ). It holds for all 0-1 polytopes [Naddef 1989] and for 3-polytopes [Klee 1966]. If P and Q satisfy it, then so does P × Q : δ ( P × Q ) = δ ( P ) + δ ( Q ) . In particular: For every n ≤ 2 d , there are polytopes in which the bound is tight (products of simplices). For every n > d , it is easy to construct unbounded polyhedra where the bound is tight. 14

  34. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? It holds with equality in simplices ( n = d + 1, δ = 1) and cubes ( n = 2 d , δ = d ). It holds for all 0-1 polytopes [Naddef 1989] and for 3-polytopes [Klee 1966]. If P and Q satisfy it, then so does P × Q : δ ( P × Q ) = δ ( P ) + δ ( Q ) . In particular: For every n ≤ 2 d , there are polytopes in which the bound is tight (products of simplices). For every n > d , it is easy to construct unbounded polyhedra where the bound is tight. 14

  35. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? It holds with equality in simplices ( n = d + 1, δ = 1) and cubes ( n = 2 d , δ = d ). It holds for all 0-1 polytopes [Naddef 1989] and for 3-polytopes [Klee 1966]. If P and Q satisfy it, then so does P × Q : δ ( P × Q ) = δ ( P ) + δ ( Q ) . In particular: For every n ≤ 2 d , there are polytopes in which the bound is tight (products of simplices). For every n > d , it is easy to construct unbounded polyhedra where the bound is tight. 14

  36. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? It holds with equality in simplices ( n = d + 1, δ = 1) and cubes ( n = 2 d , δ = d ). It holds for all 0-1 polytopes [Naddef 1989] and for 3-polytopes [Klee 1966]. If P and Q satisfy it, then so does P × Q : δ ( P × Q ) = δ ( P ) + δ ( Q ) . In particular: For every n ≤ 2 d , there are polytopes in which the bound is tight (products of simplices). For every n > d , it is easy to construct unbounded polyhedra where the bound is tight. 14

  37. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? It holds with equality in simplices ( n = d + 1, δ = 1) and cubes ( n = 2 d , δ = d ). It holds for all 0-1 polytopes [Naddef 1989] and for 3-polytopes [Klee 1966]. If P and Q satisfy it, then so does P × Q : δ ( P × Q ) = δ ( P ) + δ ( Q ) . In particular: For every n ≤ 2 d , there are polytopes in which the bound is tight (products of simplices). For every n > d , it is easy to construct unbounded polyhedra where the bound is tight. 14

  38. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? Hirsch conjecture has the following interpretations: 15

  39. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? Hirsch conjecture has the following interpretations: Assume n = 2 d , P a simple polytope, and let u and v be two complementary vertices of P (no common facet): 15

  40. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? Hirsch conjecture has the following interpretations: Assume n = 2 d , P a simple polytope, and let u and v be two complementary vertices of P (no common facet): d -step conjecture It is possible to go from u to v so that at each step we abandon a facet containing u and we enter a facet containing v . 15

  41. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? Hirsch conjecture has the following interpretations: Assume n = 2 d , P a simple polytope, and let u and v be two complementary vertices of P (no common facet): d -step conjecture It is possible to go from u to v so that at each step we abandon a facet containing u and we enter a facet containing v . 15

  42. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? Hirsch conjecture has the following interpretations: More generally, given any two vertices u and v of a polytope P : 15

  43. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? Hirsch conjecture has the following interpretations: More generally, given any two vertices u and v of a polytope P : non-revisiting path conjecture It is possible to go from u to v so that at each step we enter a new facet, one that we had not visited before. d -step conjecture ⇐ non-revisiting path ⇒ Hirsch 15

  44. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? Hirsch conjecture has the following interpretations: More generally, given any two vertices u and v of a polytope P : non-revisiting path conjecture It is possible to go from u to v so that at each step we enter a new facet, one that we had not visited before. d -step conjecture ⇐ non-revisiting path ⇒ Hirsch 15

  45. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · 16

  46. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · 16

  47. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · 16

  48. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · 16

  49. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · 16

  50. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · 16

  51. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · If n < 2 d , then H ( n − 1 , d − 1 ) ≥ H ( n , d ) : Every pair of vertices lie in a common facet F , which is a polytope with one less dimension and (at least) one less facet Use induction on n and n − d . 16

  52. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · If n < 2 d , then H ( n − 1 , d − 1 ) ≥ H ( n , d ) : Every pair of vertices lie in a common facet F , which is a polytope with one less dimension and (at least) one less facet Use induction on n and n − d . 16

  53. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · If n < 2 d , then H ( n − 1 , d − 1 ) ≥ H ( n , d ) : Every pair of vertices lie in a common facet F , which is a polytope with one less dimension and (at least) one less facet Use induction on n and n − d . 16

  54. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · If n < 2 d , then H ( n − 1 , d − 1 ) ≥ H ( n , d ) : Every pair of vertices lie in a common facet F , which is a polytope with one less dimension and (at least) one less facet Use induction on n and n − d . 16

  55. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · For every n , d , H ( n , d ) ≤ H ( n + 1 , d + 1 ) : Let P ′ be the wedge of Let u and v be two vertices of P . P over any facet F . Then, P ′ has vertices u ′ , v ′ such that d P ( u , v ) ≤ d P ′ ( u ′ , v ′ ) . 16

  56. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · For every n , d , H ( n , d ) ≤ H ( n + 1 , d + 1 ) : Let P ′ be the wedge of Let u and v be two vertices of P . P over any facet F . Then, P ′ has vertices u ′ , v ′ such that d P ( u , v ) ≤ d P ′ ( u ′ , v ′ ) . 16

  57. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · For every n , d , H ( n , d ) ≤ H ( n + 1 , d + 1 ) : Let P ′ be the wedge of Let u and v be two vertices of P . P over any facet F . Then, P ′ has vertices u ′ , v ′ such that d P ( u , v ) ≤ d P ′ ( u ′ , v ′ ) . 16

  58. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · For every n , d , H ( n , d ) ≤ H ( n + 1 , d + 1 ) : Let P ′ be the wedge of Let u and v be two vertices of P . P over any facet F . Then, P ′ has vertices u ′ , v ′ such that d P ( u , v ) ≤ d P ′ ( u ′ , v ′ ) . 16

  59. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Why was n − d a “reasonable” bound? d -step Theorem [Klee-Walkup 1967] Hirsch ⇔ d -step ⇔ non-revisiting path. Proof: Let H ( n , d ) = max { δ ( P ) : P is a d -polytope with n facets } . Then, for any fixed k = n − d we have: · · · ≤ H ( 2 k − 1 , k − 1 ) ≤ H ( 2 k , k ) = H ( 2 k + 1 , k + 1 ) = · · · For every n , d , H ( n , d ) ≤ H ( n + 1 , d + 1 ) : Let P ′ be the wedge of Let u and v be two vertices of P . P over any facet F . Then, P ′ has vertices u ′ , v ′ such that d P ( u , v ) ≤ d P ′ ( u ′ , v ′ ) . 16

  60. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Wedging, a.k.a. one-point-suspension f F P P’ 17

  61. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Wedging, a.k.a. one-point-suspension f F v P u d(u, v)=2 v’ u’ P’ d(u’, v’)=2 17

  62. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion So, the d -step Theorem is based in the following lemma: Lemma Let P be a polytope of dimension d, with n > 2 d facets and diameter λ . Then there is another polytope P ′ of dimension d + 1 , with n + 1 facets and diameter λ . That is: we can increase the dimension and number of facets of a polytope by one, preserving its diameter, until n = 2 d . 18

  63. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion So, the d -step Theorem is based in the following lemma: Lemma Let P be a polytope of dimension d, with n > 2 d facets and diameter λ . Then there is another polytope P ′ of dimension d + 1 , with n + 1 facets and diameter λ . That is: we can increase the dimension and number of facets of a polytope by one, preserving its diameter, until n = 2 d . 18

  64. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion So, the d -step Theorem is based in the following lemma: Lemma Let P be a polytope of dimension d, with n > 2 d facets and diameter λ . Then there is another polytope P ′ of dimension d + 1 , with n + 1 facets and diameter λ . That is: we can increase the dimension and number of facets of a polytope by one, preserving its diameter, until n = 2 d . 18

  65. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion The counter-example(s) Our construction of counter-examples has two ingredients: A strong d -step theorem for spindles/prismatoids. 1 The construction of prismatoids of dimension 5 and “width” 2 6. 19

  66. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion The counter-example(s) Our construction of counter-examples has two ingredients: A strong d -step theorem for spindles/prismatoids. 1 The construction of prismatoids of dimension 5 and “width” 2 6. 19

  67. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion The counter-example(s) Our construction of counter-examples has two ingredients: A strong d -step theorem for spindles/prismatoids. 1 The construction of prismatoids of dimension 5 and “width” 2 6. 19

  68. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Spindles and prismatoids Definition A spindle is a polytope P with two distinguished vertices u and v such that every facet contains either u or v . v v Definition The length of a spindle is the graph distance from u to v . u u 20

  69. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Spindles and prismatoids Definition A spindle is a polytope P with two distinguished vertices u and v such that every facet contains either u or v . v v Definition The length of a spindle is the graph distance from u to v . u u 20

  70. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Spindles and prismatoids Definition A prismatoid is a polytope Q with two facets Q + and Q − containing all vertices. Definition Q + The width of a Q prismatoid is the dual graph distance from Q + Q − to Q − . 21

  71. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion Spindles and prismatoids Definition A prismatoid is a polytope Q with two facets Q + and Q − containing all vertices. Definition Q + The width of a Q prismatoid is the dual graph distance from Q + Q − to Q − . 21

  72. The conjecture Motivation: LP Why n − d ? The construction (I) The construction(s) (II) Its limitations Conclusion The strong d -step Theorem Theorem (Strong d -step, spindle version) Let P be a spindle of dimension d, with n > 2 d facets, and with length δ . Then there is another spindle P ′ of dimension d + 1 , with n + 1 facets and with length δ + 1 . That is: we can increase the dimension, number of facets and length of a spindle, all by one, until n = 2 d . Corollary In particular, if a spindle P has length > d then there is another spindle P ′ (of dimension n − d, with 2 n − 2 d facets, and length ≥ δ + n − 2 d > n − d) that violates the Hirsch conjecture. 22

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