Hyperplane arrangements with large average diameter Antoine Deza, McMaster Tamás Terlaky, Lehigh Yuriy Zinchenko, Calgary Feng Xie, McMaster P4 P6 P5 P3

Polytopes & Arrangements : Diameter & Curvature Motivations and algorithmic issues Diameter (of a polytope) : lower bound for the number of iterations for the simplex method (pivoting methods) Curvature (of the central path associated to a polytope) : large curvature indicates large number of iterations for (path following) interior point methods

Polytopes & Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension Polytope P defined by n inequalities in dimension d polytope : bounded polyhedron

Polytopes & Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension Polytope P defined by n inequalities in dimension d

Polytopes & Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension P Polytope P defined by n inequalities in dimension d

Polytopes & Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension (P) = 2 : diameter vertex c1 P vertex c2 Diameter (P): smallest number such that any two vertices (c1,c2) can be connected by a path with at most (P) edges

Polytopes & Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension (P) = 2 : diameter P Diameter (P): smallest number such that any two vertices can be connected by a path with at most (P) edges Hirsch Conjecture (1957): (P) ≤ n - d

Polytopes & Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension c(P): total curvature of the primal central path of max{cTx : xP}

Polytopes & Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension P c(P): total curvature of the primal central path of max{cTx : xP} c(P): redundant inequalities count

Polytopes & Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension P c(P): total curvature of the primal central path of max{cTx : xP} (P): largest total curvature c(P) over of all possible c

Polytopes & Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension P c(P): total curvature of the primal central path of max{cTx : xP} (P): largest total curvature c(P) over of all possible c Continuous analogue of Hirsch Conjecture: (P) = O(n)

Polytopes & Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension P c(P): total curvature of the primal central path of max{cTx : xP} (P): largest total curvature c(P) over of all possible c Continuous analogue of Hirsch Conjecture: (P) = O(n) Dedieu-Shub hypothesis (2005): (P) = O(d)

Polytopes & Arrangements : Diameter & Curvature n = 5 : inequalities d = 2 : dimension P c(P): total curvature of the primal central path of max{cTx : xP} (P): largest total curvature c(P) over of all possible c Continuous analogue of Hirsch Conjecture: (P) = O(n) Dedieu-Shub hypothesis (2005): (P) = O(d) D.-T.-Z. (2008) polytope C such that: (C) ≥ (1.5)d

Polytopes & Arrangements : Diameter & Curvature Central path following interior point methods start from the analytic center follow the central path converge to an optimal solution polynomial time algorithms for linear optimization : number of iterations n : number of inequalities L : input-data bit-length c analytic center central path optimal solution  : central path parameter xP : Ax ≥ b

Polytopes & Arrangements : Diameter & Curvature total curvature Polytopes & Arrangements : Diameter & Curvature total curvature C2 curve  : [a, b]  Rd parameterized by its arc length t (note: ║(t)║ = 1) curvature at t : y = sin (x)

Polytopes & Arrangements : Diameter & Curvature total curvature Polytopes & Arrangements : Diameter & Curvature total curvature C2 curve  : [a, b]  Rn parameterized by its arc length t (note: ║(t)║ = 1) curvature at t : total curvature : y = sin (x)

Polytopes & Arrangements : Diameter & Curvature n = 5 : hyperplanes d = 2 : dimension Arrangement A defined by n hyperplanes in dimension d

Polytopes & Arrangements : Diameter & Curvature n = 5 : hyperplanes d = 2 : dimension Simple arrangement: n  d and any d hyperplanes intersect at a unique distinct point

Polytopes & Arrangements : Diameter & Curvature n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells For a simple arrangement, the number of bounded cells I =

Polytopes & Arrangements : Diameter & Curvature n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells P1 P4 P6 P5 P3 c(A) : average value of c(Pi) over the bounded cells Pi of A: c(A) = with I = c(Pi): redundant inequalities count

Polytopes & Arrangements : Diameter & Curvature n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells P1 P4 P6 P5 P3 c(A) : average value of c(Pi) over the bounded cells Pi of A: (A) : largest value of c(A) over all possible c

Polytopes & Arrangements : Diameter & Curvature n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells P1 P4 P6 P5 P3 c(A) : average value of c(Pi) over the bounded cells Pi of A: (A) : largest value of c(A) over all possible c Dedieu-Malajovich-Shub (2005): (A) ≤ 2 d

Polytopes & Arrangements : Diameter & Curvature n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells P1 P4 P6 P5 P3 (A) : average diameter of a bounded cell of A: A : simple arrangement

Polytopes & Arrangements : Diameter & Curvature n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells P1 P4 P6 P5 P3 (A) : average diameter of a bounded cell of A: (A) = with I = (A): average diameter ≠ diameter of A ex: (A)= 1.333…

Polytopes & Arrangements : Diameter & Curvature n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells P1 P4 P6 P5 P3 (A) : average diameter of a bounded cell of A: (A) = with I = (Pi): only active inequalities count

Polytopes & Arrangements : Diameter & Curvature n = 5 : hyperplanes d = 2 : dimension I = 6 : bounded cells P1 P4 P6 P5 P3 (A) : average diameter of a bounded cell of A: Conjecture [D.-T.-Z.]: (A) ≤ d (discrete analogue of Dedieu-Malajovich-Shub result)

Polytopes & Arrangements : Diameter & Curvature (P) ≤ n - d (Hirsch conjecture 1957) (P) ≤ 2 n (conjecture D.-T.-Z. 2008) (A) ≤ d (conjecture D.-T.-Z. 2008) (A) ≤ 2 d (Dedieu-Malajovich-Shub 2005)

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation for unbounded polyhedra Hirsch conjecture is not valid central path may not be defined

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation for unbounded polyhedra Hirsch conjecture is not valid central path may not be defined Hirsch conjecture (P) ≤ n - d implies that (A) ≤ d Hirsch conjecture holds for d = 2, (A) ≤ 2 Hirsch conjecture holds for d = 3, (A) ≤ 3 Barnette (1974) and Kalai-Kleitman (1992) bounds imply (A) ≤

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation n = 6 d = 2 Remember that according the conjecture … it is natural to construct an arrangement this way … bounded cells of A* : ( n – 2) (n – 1) / 2 4-gons and n - 2 triangles 30

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation dimension 2 (A) ≥ 2 - 31

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation dimension 2 (A) = 2 - maximize(diameter) amounts to minimize( #external edge + #odd-gons) 32

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation dimension 3 (A) ≤ 3 + and A* satisfies (A*) = 3 - A* 33

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation dimension d A* satisfies (A*) ≥ d A* cyclic arrangement A* mainly consists of cubical cells 34

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation dimension d A* satisfies (A*) ≥ n Haimovich’s probabilistic analysis of shadow-vertex simplex method - Borgwardt (1987) Ziegler (1992) combinatorics of (pseudo) hyperplane addition to A* (Bruhat order) Forge – Ramírez Alfonsín (2001) counting k-face cells of A* 35

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation dimension 2 (A) = dimension 3 (A) asympotically equal to 3 dimension d d ≤ (A) ≤ d ? [D.-Xie 2008] Haimovich’s probabilistic analysis of shadow-vertex simplex method - Borgwardt (1987) Ziegler (1992) combinatorics of (pseudo) hyperplane addition to A* (Bruhat order) Forge – Ramírez Alfonsín (2001) counting k-face cells of A* 36

Polytopes & Arrangements : Diameter & Curvature Is it the right setting? (forget optimization and the vector c) bounded / unbounded cells (sphere arrangements)? simple / degenerate arrangements? pseudo-hyperplanes / underlying oriented matroid ? 37

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ 7 P such that (P) ≥ n - d Holt-Klee (1998) Fritzsche-Holt (1999) Holt (2004)

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ 7 P such that (P) ≥ n - d Holt-Klee (1998) Fritzsche-Holt (1999) Holt (2004) Is the continuous analogue true?

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ 7 P such that (P) ≥ n - d Holt-Klee (1998) Fritzsche-Holt (1999) Holt (2004) Is the continuous analogue true? yes

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ 7 P such that (P) ≥ n - d Holt-Klee (1998) Fritzsche-Holt (1999) Holt (2004) continuous analogue of tightness: for n ≥ d ≥ 2 D.-T.-Z. (2008)

Polytopes & Arrangements : Diameter & Curvature slope: careful and geometric decrease

Polytopes & Arrangements : Diameter & Curvature slope: careful and geometric decrease

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ 7 P such that (P) ≥ n - d continuous analogue of tightness: for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008)

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ 7 P such that (P) ≥ n - d continuous analogue of tightness: for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008) Hirsch special case n = 2d (for all d) is equivalent to the general case (d-step conjecture) Klee-Walkup (1967)

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ 7 P such that (P) ≥ n - d continuous analogue of tightness: for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008) Hirsch special case n = 2d (for all d) is equivalent to the general case (d-step conjecture) Klee-Walkup (1967) Is the continuous analogue true?

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ 7 P such that (P) ≥ n - d continuous analogue of tightness: for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008) Hirsch special case n = 2d (for all d) is equivalent to the general case (d-step conjecture) Klee-Walkup (1967) Is the continuous analogue true? yes

Polytopes & Arrangements : Diameter & Curvature Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ 7 P such that (P) ≥ n - d continuous analogue of tightness: for n ≥ d ≥ 2 P such that (P) = Ω(n) D.-T.-Z. (2008) Hirsch special case n = 2d (for all d) is equivalent to the general case (d-step conjecture) Klee-Walkup (1967) continuous analogue of d-step equivalence: (P) = O(n) (P) = O(d) for n = 2d D.-T.-Z. (2008)

Polytopes & Arrangements : Diameter & Curvature holds for n = 2 and n = 3 (P) ≤ n - d Hirsch conjecture (1957) (P) < n – d Holt-Klee (1998) (P) ≤ 2 n conjecture D.-T.-Z. (2008) (P)=Ω(n) D.-T.-Z (2008) holds for n = 2 and n = 3 redundant inequalities counts for (P) (A) ≤ d conjecture D.-T.-Z. (2008) (A) ≤ 2 d Dedieu-Malajovich-Shub (2005) (P) ≤ n – d (P) ≤ d for n = 2d Klee-Walkup (1967) (P) = O(n) (P) = O(d) for n = 2d D.-T.-Z. (2008)

Polytopes & Arrangements : Diameter & Curvature holds for n = 2 and n = 3 (P) ≤ n - d Hirsch conjecture (1957) (P) < n – d Holt-Klee (1998) (P) ≤ 2 n conjecture D.-T.-Z. (2008) (P)=Ω(n) D.-T.-Z (2008) holds for n = 2 and n = 3 redundant inequalities counts for (P) (A) ≤ d conjecture D.-T.-Z. (2008) (A) ≤ 2 d Dedieu-Malajovich-Shub (2005) (P) ≤ n – d (P) ≤ d for n = 2d Klee-Walkup (1967) (P) = O(n) (P) = O(d) for n = 2d D.-T.-Z. (2008)

Polytopes & Arrangements : Diameter & Curvature Proof of a continuous analogue of Klee-Walkup result (d-step conjecture) Want: {(P) = O(n), n}  {(P) = O(d) for n = 2d, d} Clearly {(P)  K n, n}  {(P)  K n for n = 2d, d} Need {(P)  K n, n}  {(P)   n for n = 2d, d} Case (i) d < n < 2d e.g., P with n=3, d=2 P  P with n=4, d=2 c(P)  c(P)  2 d< 2 n c   A4

Polytopes & Arrangements : Diameter & Curvature Proof of a continuous analogue of Klee-Walkup result (d-step conjecture) Want: {(P) = O(n), n}  {(P) = O(d) for n = 2d, d} Need {(P)  K n, n}  {(P)   n for n = 2d, d} Case (ii) 2d < n e.g., P with n=5, d=2 P  P with n=6, d=3 c(P)  (P)  2 (d + 1) … < 2 (n - d) < 2 n   c  c

Polytopes & Arrangements : Diameter & Curvature Motivations, algorithmic issues and analogies Diameter (of a polytope) : lower bound for the number of iterations for the simplex method (pivoting methods) Curvature (of the central path associated to a polytope) : large curvature indicates large number of iteration for (central path following) interior point methods we believe: polynomial upper bound for (P) ≤ d (n – d) / 2 53

Polytopes & Arrangements : Diameter & Curvature Is it the right setting? (forget optimization and the vector c) bounded / unbounded cells (sphere arrangements)? simple / degenerate arrangements? pseudo-hyperplanes / underlying oriented matroid ? thank you 54

Polytopes & Arrangements : Diameter & Curvature Motivations, algorithmic issues and analogies Diameter (of a polytope) : lower bound for the number of iterations for the simplex method (pivoting methods) Curvature (of the central path associated to a polytope) : large curvature indicates large number of iteration for (central path following) interior point methods we believe: polynomial upper bound for (P) ≤ d (n – d) / 2 thank you 55