Hyperplane arrangements with large average diameter

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

n = 5 : inequalities d = 2 : dimension Polytope P defined by n inequalities in dimension d polytope : bounded polyhedron

n = 5 : inequalities d = 2 : dimension Polytope P defined by n inequalities in dimension d

n = 5 : inequalities d = 2 : dimension P Polytope P defined by n inequalities in dimension d

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

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

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

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

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

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)

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)

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

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

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)

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)

n = 5 : hyperplanes d = 2 : dimension Arrangement A defined by n hyperplanes in dimension d

n = 5 : hyperplanes d = 2 : dimension Simple arrangement: n  d and any d hyperplanes intersect at a unique distinct point

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

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

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

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

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

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…

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

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)

(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)

Links, low dimensions and substantiation for unbounded polyhedra Hirsch conjecture is not valid central path may not be defined

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) ≤

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) / gons and n - 2 triangles 30

Links, low dimensions and substantiation dimension 2 (A) ≥ 2 - 31

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

Links, low dimensions and substantiation dimension 3 (A) ≤ 3 + and A* satisfies (A*) = 3 - A* 33

Links, low dimensions and substantiation dimension d A* satisfies (A*) ≥ d A* cyclic arrangement A* mainly consists of cubical cells 34

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

Links, low dimensions and substantiation dimension (A) = dimension (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

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

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

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

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

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

slope: careful and geometric decrease

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

Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ P such that (P) ≥ n - d continuous analogue of tightness: for n ≥ d ≥ 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)

Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ P such that (P) ≥ n - d continuous analogue of tightness: for n ≥ d ≥ 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?

Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ P such that (P) ≥ n - d continuous analogue of tightness: for n ≥ d ≥ 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

Links, low dimensions and substantiation Hirsch conjecture is tight for n > d ≥ P such that (P) ≥ n - d continuous analogue of tightness: for n ≥ d ≥ 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)

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)

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

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

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

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

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

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