Polytopes and arrangements: Diameter and curvature

Transcription

1 Operations Research Letters Operations Research Letters wwwelsevierco/locate/orl Polytopes and arrangeents: Diaeter and curvature Antoine Deza, Taás Terlaky, Yuriy Zinchenko McMaster University, 280 Main Street West, Hailton, Ont, Canada L8S4K Received 9 Septeber 2006; accepted 29 June 2007 Available online 2 Septeber 2007 Abstract We introduce a continuous analogue of the Hirsch conjecture and a discrete analogue of the result of Dedieu, Malajovich and Shub We prove a continuous analogue of the result of Holt and Klee, naely, we construct a faily of polytopes which attain the conjectured order of the largest total curvature 2007 Elsevier BV All rights reserved Keywords: Polytopes; Arrangeents; Diaeter; Central path; Total curvature Continuous analogue of the conjecture of Hirsch By analogy with the conjecture of Hirsch, we conjecture that the order of the largest total curvature of the central path associated to a polytope is the nuber of inequalities defining the polytope By analogy with a result of Dedieu, Malajovich and Shub, we conjecture that the average diaeter of a bounded cell of a siple arrangeent is less than the diension We prove a continuous analogue of the result of Holt Klee, naely, we construct a faily of polytopes which attain the conjectured order of the largest total curvature We substantiate the conjectures in low diensions and highlight additional links Let P be a full diensional convex polyhedron defined by inequalities in diension n The diaeter δp is the sallest nuber such that any two vertices of the polyhedron P can be connected by a path with at ost δp edges The conjecture of Hirsch, forulated in 97 and reported in [3], states that the diaeter of a polyhedron defined by inequalities in diension n is not greater than n The conjecture does not hold for unbounded polyhedra A polytope is a bounded polyhedron No polynoial bound is known for the diaeter of a polytope Conjecture Conjecture of Hirsch for polytopes The diaeter of a polytope defined by inequalities in diension n is not greater than n Intuitively, the total curvature [6] is a easure of how far off a certain curve is fro being a straight line Let ψ :[α, β] R n be a C 2 α ε, β ε ap for soe ε > 0 with a non-zero derivative in [α, β] Denote its arc length by lt = t α ψτ dτ, its paraetrization by the arc length by ψ arc = ψ l :[0,lβ] R n, and its curvature at the point t by κt = ψ arc tthe total curvature is defined as lβ 0 κt dt The requireent ψ = 0 insures that any given segent of the curve is traversed only once and allows to define a curvature at any point on the curve Fro now on we consider only polytopes, ie, bounded polyhedra, and denote those by P For a polytope P ={x : Ax b} with A R n, denote λp the largest total curvature of the prial central path corresponding to the standard logarithic barrier function, i= lna i x b i, of the linear prograing proble in{c T x : x P } over all possible c Following the analogy with the diaeter, let Λ, n be the largest total curvature λp of the prial central path over all polytopes P defined by inequalities in diension n Corresponding author Tel: x2370 E-ail address: deza@casterca A Deza /$ - see front atter 2007 Elsevier BV All rights reserved doi:006/jorl

2 26 A Deza et al / Operations Research Letters Conjecture 2 Continuous analogue of the conjecture of Hirsch The order of the largest total curvature of the prial central path over all polytopes defined by inequalities in diension n is the nuber of inequalities defining the polytopes, ie, Λ, n = O Reark In [6] the authors showed that a redundant Klee Minty n-cube C satisfies λc 3 2 n, providing a counterexaple to the conjecture of Dedieu and Shub [] that Λ, n = On For polytopes and arrangeents, respectively central path and linear prograing, we refer to the books of Grünbau [9] and Ziegler [7], respectively Renegar [3] and Roos et al [4] 2 Discrete analogue of the result of Dedieu, Malajovich, Shub Let A be a siple arrangeent fored by hyperplanes in diension n We recall that an arrangeent is called siple if n and any n hyperplanes intersect at a unique distinct point Since A is siple, the nuber of bounded cells, ie, bounded connected coponents of the copleent to the hyperplanes, of A is I = Let λ c P denote the total curvature of the prial central path corresponding to in{c T x : x P } Following the approach of Dedieu et al [4], let λ c A denote the average value of λ c P i over the bounded cells P i of A; that is i=i λ c i= A = λc P i I Note that each bounded cell P i is defined by the sae nuber of inequalities, soe being potentially redundant Given an arrangeent A, the average total curvature of a bounded cell λa is the largest value of λ c A over all possible c Siilarly, Λ A, n is the largest possible average total curvature of a bounded cell of a siple arrangeent defined by inequalities in diension n Proposition 2 Dedieu et al [4] The average total curvature of a bounded cell of a siple arrangeent defined by inequalities in diension n is not greater than 2πn n By analogy, let δa denote the average diaeter of a bounded cell of A; that is i=i i= δa = δp i I Siilarly, let Δ A, n denote the largest possible average diaeter of a bounded cell of a siple arrangeent defined by inequalities in diension n Conjecture 2 Discrete analogue of the result of Dedieu, Malajovich and Shub The average diaeter of a bounded cell of a siple arrangeent defined by inequalities in diension n is not greater than n Haiovich s probabilistic analysis of the shadow-vertex siplex algorith, see [2, Section 07], shows that the expected nuber of pivots is bounded by n While the result and Conjecture 2 are siilar in nature, they differ in soe aspects: Haiovich considers the average over bounded and unbounded cells, and the nuber of pivots could be saller than the diaeter for soe cells 3 Additional links and low diensions 3 Additional links Proposition 3 If the conjecture of Hirsch holds, then Δ A, nn 2n Proof Let i denote the nuber of hyperplanes of A which are non-redundant for the description of a bounded cell P i Ifthe conjecture of Hirsch holds, we have δp i i n It iplies δa i=i i= i n = I i=i i= i n I Since a facet belongs to at ost 2 cells, the su of i for i =,,I is less than twice the nuber of bounded facets of AAs a bounded facet induced by a hyperplane H of A corresponds to a bounded cell of the n -diensional siple arrangeent

3 A Deza et al / Operations Research Letters A H, the su of i is less than 2 n 2 n Fig The arrangeent A 6,2 Therefore, we have, for any siple arrangeent A, δa 2 2 n n= 2n n= n Reark 3 In the proof of Proposition 3, we overestiate the su of i for i =,,I as soe bounded facets belong to exactly bounded cell Let us call such bounded facets external We hypothesize that any siple arrangeent has at least n external facets, in turn, this would strengthen Proposition 3 to: If the conjecture of Hirsch holds, then Δ A, n n n Siilarly to Proposition 3, the results of Kalai and Kleitan [] and Barnette [] which bounds the diaeter of a polytope by, respectively, 2: logn and 2n 2 3 n 2, directly yield Proposition 32 Δ A, n 4n 2 2 log n n and Δ A, nn 2n 2n 2 3 The special case of = 2n of the conjecture of Hirsch is known as the d-step conjecture as the diension is often denoted by d in polyhedral theory In particular, it has been shown by Klee and Walkup [2] that the special case = 2n for all n is equivalent to the conjecture of Hirsch A continuous analogue would be: if Λ2n, n = On for all n, then Λ, n = O Reark 32 In contrast with Proposition 3, Λ, n = O does not iply that Λ A, n = On since all the inequalities count for each λp i while it is enough to consider the i non-redundant inequalities for each δp i 32 Low diensions 2 n In diensions 2 and 3 we have, respectively, δp 2 and δp 2 3, iplying: Proposition 33 Δ A, 22 2 and Δ A, 33 4 In diension 2, let S 2 be a unit sphere centered at, and consider the arrangeent P A,2 ade of the 2 lines foring the nonnegative orthant and additional 2 lines tangent to S 2 and separating the origin fro the center of the sphere See Fig for an illustration of A 6,2 Besides 2 triangles, the bounded cells of A,2 are ade of gons We have δa,2 = 2 2, and thus, Proposition Δ A, 22 2 Reark 33 The arrangeent A,2 was generalized in [7] to an arrangeent with n n cubical cells yielding that the diension n is an asyptotic lower bound for Δ A, n for fixed n In diension 2, for 4, consider the polytope P,2 defined by the following inequalities: y, x y 0 2, x y 3 3 and i x 0i 2 y 0 4 i for i = 4,, See Fig 2 for an illustration of P 6,2 and Fig 3 for the central path over P 34,2 with c = 0, Proposition 3 The total curvature of the central path of in{y : x, y P,2 } satisfies li inf λ 0, P,2 π

4 28 A Deza et al / Operations Research Letters y central path c Fig 2 The polytope P6,2 and its central path In y x Fig 3 The central path for P 34,2 Proof First we show that the central path P goes through a sequence of 2 points x j, 0 j for j =,, 2 with x j 0 for odd j and x j 0 4 for even j Fori = 2,,and j =,, 2, denote zj i the first coordinate of the intersection of the line y = 0 j and the facet of P,2 induced by the ith inequality defining P,2, that is, zj 2 = 0 j 2, zj 3 = 0 j 3, and z j i = i 0i j 0 4 i for i = 4,, As the central path ay be characterized as the set of iniizers of the barrier function over appropriate level sets of the objective function, the point x j, 0 j of P satisfies x j =arg ax x i=2 ln i z j i x Therefore, to show that x j 0 for odd j and that x j 0 4 for even j, it is enough to prove that gj 0>0 for odd j and g j 0 4 <0 for even j where gj x = d i=2 dx ln i z j i x For siplicity we assue that is even A siilar arguent applies for odd values of Since k > 0 for k j 4 and 0 4 x 0, we have x z j k x z j k i= 0, j odd, x= 0, i=j4 x z j i 0, j even, x= 0 4 This yields g 0 g j x = x z j 2 x z j 3 00 i<j2 772 = 639 > 0 For j 2, rewrite 0 4 x z j i i<j4 i=j2 x z j i i= i=j4 x z j i

5 Observe and x z j 2 i<j4 x z j = j 2 0 j 0 i=j2 x z j 0 i For odd j 3 and x = 0, we have i<j2 x z j 0 i 3 j 0 4 = = A Deza et al / Operations Research Letters j j 04 j j for x = 0,, j3 odd, x= 0, for x = 0 4, 2, j 4even,x= 0 4,, j = 2, x= j j j j 0 3 j j 0 j j j 2 j j/ j/ j j/ ,

6 220 A Deza et al / Operations Research Letters where the second inequality is based on v v v v2,v0 and the last equality is obtained by suing up the ters in three resulting geoetric series This, cobined with observations 3, gives, for odd j 3, g j = > Siilarly for even j 2 and x = 0 4 i<j2 x z j i Thus, for even j g j j 2 j 2 j 2 we have j/2 0 0 j/ j/ = <0 Therefore, the central path P goes through a sequence of 2 points x j,y j with y j = 0 j and x j 0 for odd j, x j 0 4 for even j One can easily check that x j,y j P for j =,, 2 by verifying that the analytic center χ is above the line y =

7 We have χ = χ, χ 2 = arg A Deza et al / Operations Research Letters ax ln y ln x y x,y P,2 0 ln x y ln i x 0i 2 y 0 4 i Therefore, to show that χ 2 >, it is enough to prove that the derivative with respect to y of the log-barrier function is negative for x, y P,2 and y, that is y 0x y 3x y which is iplied by y 00 x 00y i 2 i x 0 i 2 y > > 0, i = > 0 To show that li inf λ 0,T P,2 π, consider three consecutive points fro this sequence, say x j,y j, x j,y j, x j,y j, and observe that for any ε > 0 we can choose so that for all εj < 2wehave y j y j x j x j < ε, y j y j x j x j < ε Let be such a value and j ε Without loss off generality j ight be assued odd and let τ j, τ j, τ j R be such that P arc τ k = x k,y k, k=j,j,jwe show by contradiction that there is a t such that the first coordinate Ṗ arc t > ε 2 Suppose that for all t [τ j, τ j ] we have Ṗ arc t ε 2, then Ṗ arc t 2 ε since Ṗ arc t = and P arc t 2 is onotone-decreasing with respect to t By the Mean-Value Theore it follows that τ j τ j >x j x j, and thus, by the sae theore, we ust have Parc τ j 2 P arc τ j 2 =y j y j < εx j x j, a contradiction Siilarly, there is a t 2 such that Ṗ arc t 2 < ε 2 Since the total curvature K j of the segent of P arc connecting the points x j,y j, x j,y j, x j,y j corresponds to the length of the curve Ṗ arc connecting the corresponding derivative points on a unit 2-sphere, K j ay be bounded below by the length of the geodesic between the points Ṗ arc t and Ṗ arc t 2, that is, bounded below by a constant arbitrarily close to π Now siply add all K j for all εj < 2 Holt and Klee [0] showed that, for >n3, the conjecture of Hirsch is tight Fritzsche and Holt [8] extended the result to >n8 Since the polytope P,2 can be generalized to higher diensions by adding the box constraints 0x i for i 3, we have Λ,n Corollary 3 Continuous analogue of the result of Holt and Klee li inf π, that is, Λ, n is bounded below by a constant ties Acknowledgents Research supported by an NSERC Discovery grant, by a MITACS grant and by the Canada Research Chair progra References [] D Barnette, An upper bound for the diaeter of a polytope, Discrete Math [2] KH Borgwardt, The Siplex Method A Probabilistic Analysis, Springer, Berlin, 987 [3] G Dantzig, Linear Prograing and Extensions, Princeton University Press, Princeton, NJ, 963 [4] J-P Dedieu, G Malajovich, M Shub, On the curvature of the central path of linear prograing theory, Found Coput Math [] J-P Dedieu, M Shub, Newton flow and interior point ethods in linear prograing, International J Bifurcation Chaos [6] A Deza, T Terlaky, Y Zinchenko, Central path curvature and iteration-coplexity for redundant Klee-Minty cubes, In: D Gao, H Sherali Eds, Advances in Mechanics and Matheatics III, 2007, pp [7] A Deza, F Xie, Hyperplane arrangeents with large average diaeter, AdvOL-Report 2007/, McMaster University, 2007 [8] K Fritzsche, F Holt, More polytopes eeting the conjectured Hirsch bound, Discrete Math [9] B Grünbau, Convex Polytopes, Graduate Texts in Matheatics, vol 22, Springer, Berlin, 2003 [0] F Holt, V Klee, Many polytopes eeting the conjectured Hirsch bound, Discrete Coput Geoetry [] G Kalai, D Kleitan, A quasi-polynoial bound for the diaeter of graphs of polyhedra, Bull A Math Soc

8 222 A Deza et al / Operations Research Letters [2] V Klee, D Walkup, The d-step conjecture for polyhedra of diension d<6, Acta Math [3] J Renegar, A Matheatical View of Interior-Point Methods in Convex Optiization, SIAM, Philadelphia, PA, 200 [4] C Roos, T Terlaky, J-Ph Vial, Interior Point Methods for Linear Optiization, Springer, Berlin, 2006 [6] M Spivak, Coprehensive Introduction to Differential Geoetry, Publish or Perish, Berkeley, CA, 990 [7] G Ziegler, Lectures on Polytopes, Graduate Texts in Matheatics, vol 2, Springer, Berlin, 99