aa r X i v : . [ m a t h . C O ] A ug COMPLEXITY YARDSTICKS FOR f -VECTORS OFPOLYTOPES AND SPHERES ERAN NEVO
Dedicated to the memory of Branko Gr¨unbaum
Abstract.
We consider geometric and computational measuresof complexity for sets of integer vectors, asking for a qualitativedifference between f -vectors of simplicial and general d -polytopes,as well as flag f -vectors of d -polytopes and regular CW ( d − d ≥ Introduction
The face numbers of simplicial d -polytopes are characterized by thecelebrated g -theorem, conjectured by McMullen [13] and proved byStanley [21] and Billera-Lee [5]. In contrast, the f -vector, and the finerflag f -vector, of general d -polytopes of dimension d ≥ fatness , see e.g. Ziegler’s ICM pa-per [25], and e.g. [27, 20] for general d .1.1. Geometric complexity.
Let F be a family of graded posets ofrank d + 1 with a minimum and a maximum. For instance denote by F = P d (resp. P ds ) the face lattices of all (resp. simplicial) d -polytopes.Let f ( F ) be the set of f -vectors of elements in F , counting the numberof elements in each rank i , denoted f i , for 1 ≤ i ≤ d . (Note the shiftof index by 1 with respect to the dimension convention.)For a subset T of R d and t ∈ T let Conv( T ) (resp. Cone t ( T )) be theminimal closed convex set (resp. cone with apex t ) containing T . Let σ d denote the d -simplex.The following are geometric consequences of the g -theorem. Theorem 1.1. (1)
Convex hull : C d := Conv( f ( P ds )) = Cone f ( σ d ) ( f ( P ds )) is a simplicial cone of dimension ⌊ d/ ⌋ .(2) Density of rays : for any ǫ > and any x ∈ C d there exists asimplicial polytope P ∈ P ds such that the angle between x − f ( σ d ) and f ( P ) − f ( σ d ) is less than ǫ . (3) Density of points : for any x ∈ C d there exists a simplicial poly-tope P ∈ P ds such that in the l -norm || x − f ( P ) || = O ( || x || − ⌊ d/ ⌋ ) = o ( || x || ) . (The O ( · ) estimate is tight; d ≥ .)(4) Boundary polytopes : the only polytopes P ∈ P ds with f ( P ) onthe boundary of C d are the k -stacked polytopes for some k ≤ d − ; onlythe -stacked polytopes have f ( P ) on an extremal ray, all are on thesame ray. When d ≥
4, all analogous statements for P d seem open. Explicitly: Problem 1.2. (1)
Convex hull.
Is Conv( f ( P d )) = Cone f ( σ d ) ( f ( P d ))?(1’) Finite generation.
Is Cone f ( σ d ) ( f ( P d )) finitely generated?(2) Ray density.
Are the rays from f ( σ d ) through f ( P ) for all P ∈ P d dense in Cone f ( σ d ) ( f ( P d ))?(3) Point density.
Is it true that for any x ∈ Conv( f ( P d )) thereexists P ∈ P d such that || x − f ( P ) || = o ( || x || )?(4) Boundary.
For which polytopes P ∈ P d does f ( P ) lie on theboundary of Conv( f ( P d ))? Of Cone f ( σ d ) ( f ( P d ))?For d = 4 Ziegler [27] showed that the limits of the rays spanned by f ( P d ) in Cone f ( σ d ) ( f ( P d )) form a convex set; this is open for d > f ( σ d ) ( f ( P d )) are limit rays, which is equivalentto a YES answer to (1,3); and just to (1) if restricting to the extremalrays.As for (1’) for d = 4, it is not known if the fatness parameter f + f f + f is bounded above by some constant C . If not, then Cone f ( σ ) ( f ( P ))would be determined, with exactly 5 facets [2, 7]. Ziegler [26] showedthat if C exists then C ≥ flag f -vectors of d -polytopes and again are open for d ≥ d ≤ f -vector is determinedby the f -vector, see the cd -index below).For F as above, let flag( F ) be the set of flag f -vectors of elements in F , counting the number of chains occupying each subset of ranks S ⊆ [ d ] (called S -chains). Billera and Ehrenborg [4] proved that the simplex σ d minimizes all components of the flag f -vector among d -polytopes,so we choose it as the apex and consider the cone Cone flag( σ d ) (flag( P d ))in the flag analog of Problem 1.2.For the larger family of regular CW ( d − W d the family of face posetsof regular CW ( d − D d ∈ W d be the dihedral ( d − d −
1; then D d minimizes the number of S -chains for any S ⊆ [ d ]. Combining a con-struction of Stanley [22] with the nonnegativity of the cd -index provedby Karu [11], gives the following known analog of Theorem 1.1(1). OMPLEXITY YARDSTICKS FOR f -VECTORS OF POLYTOPES AND SPHERES3 Proposition 1.3. W d := Conv(flag( W d )) = Cone flag(D d ) (flag( W d )) isa simplicial cone of dimension c d − , for c d the d th Fibonacci number(e.g. c = 5 ). The dimension c d − f ( σ d ) (flag( P d )); see also [10].The flag analogs of Theorem 1.1(2–4) are open for W d , to be dis-cussed in Sec. 3.2.1.2. Computational complexity.
Computational complexity gainsimportance in Enumerative Combinatorics in recent years, see Pak’sICM paper [18] for a recent survey. Yet, this perspective is still largelymissing in f -vector theory.Fix d and consider the following decision problems: given a vector v ∈ Z d ≥ (resp. v ∈ Z d ≥ ), does v = f ( P ) (resp. v = flag(P)) for some P ∈ F ?For F = P d this is decidable, by finding all combinatorial typesof d -polytopes with n vertices – see Gr¨unbaum’s book [8, Sec.5.5] fora proof using Tarski’s elimination of quantifiers theorem. Using theexistential theory of the reals, e.g. [6, 19], gives an algorithm that runsin time double exponential in size of the encoding of v (in binary, on adeterministic Turing machine).For F = P ds this is effectively decidable, namely: For a vector v =( v , . . . , v d ) ∈ Z d ≥ denote N ( v ) := P di =1 ⌈ lg ( v i ) ⌉ , the number of bits inits encoding in binary. Then, Theorem 1.4.
Deciding if v ∈ f ( P ds ) can be done in polynomial timein N ( v ) . Problem 1.5.
Can deciding whether v ∈ f ( P d ) be done in polynomialtime in N ( v )?Recognizing the cone Cone f ( σ d ) ( f ( P d )) may turn out undecidable: Problem 1.6.
Fix d ≥
4. Is the following problem decidable?: given ahyperplane H through f ( σ d ), does it support the cone Cone f ( σ d ) ( f ( P d )),or contain an interior ray of it?As mentioned, for d = 4, if fatness of 4-polytopes is unbounded thenthe decision problem is easy.The analogs of Problems 1.5 and 1.6 for flag- f vectors of d -polytopesare open; likewise for Problem 1.5 for regular CW ( d − Proposition 1.7.
Deciding if v is the flag f -vector of a Gorenstein*poset can be done in doubly exponential time in N ( v ) . Is there an effective decision algorithm? In the case d = 4 the flag f -vectors in W are characterized [16]; yet it is not clear whether thenumerical conditions given can be verified effectively; see Problem 3.2. ERAN NEVO Preliminaries g -numbers. For P ∈ P ds with the convention of the introduction, f i ( P ) denotes the number of rank i (i.e. ( i − P . Define the numbers h i ( P ) ( i = 0 , . . . , d ) by x d d X i =0 h i ( P )( 1 x ) i = ( x − d d X i =0 f i ( P )( 1 x − i . Note that the f -vector of P , f ( P ) = ( f , . . . , f d ), and its h -vector h ( P ) = ( h , h , . . . , h d ), are obtained one from the other by applyingan invertible linear transformation. Thus, the following theorem indeedcharacterizes the face numbers of simplicial polytopes. Theorem 2.1 ( g -theorem [5, 21]) . An integer vector h = ( h , . . . , h d ) is the h -vector of a simplicial d -polytope iff the following two conditionshold:(1) h i = h d − i for every ≤ i ≤ ⌊ d ⌋ , and(2) ( h = 1 , h − h , . . . , h ⌊ d ⌋ − h ⌊ d ⌋− ) is an M-sequence. Gorenstein* posets.
A poset P with minimum ˆ0 and maxi-mum ˆ1 is Gorenstein* if the reduced order complex O ( P ), consist-ing of all chains in P \ { ˆ0 , ˆ1 } , is a Gorenstein* simplicial complex.Namely, for any face F ∈ O ( P ) including the empty one, the linklk O ( P ) ( F ) has dimension dim( O ( P )) − | F | and is homologous to a ra-tional (dim( O ( P )) − | F | )-sphere. For example, all regular CW spheresare Gorenstein*, thus also all polytope face lattices.2.3. cd -index. For fixed d and P ∈ W d , or any Gorenstein* poset ofrank d + 1, we recall its cd -index, introduced by Fine. For a word w = w · · · w d over alphabet { a , b } , let S ( w ) := { i : w i = b } andfor a subset S ⊆ [ d ] let w ( S ) be the unique word w over { a , b } with d letters such that S ( w ) = S . Define polynomials in non-commutingvariables Γ P ( a , b ) := P S ⊆ [ d ] f S ( P ) w ( S ) and Ψ P ( a , b ) := Γ P ( a − b , b ).It turns out that for c = a + b of degree 1 and d = ab + ba ofdegree 2, Ψ P ( a , b ) = Φ P ( c , d ); this uniquely defined polynomial Φ P ofhomogenous degree d in non-commuting variables c and d is called the cd -index of P . Stanley [22] proved for P ∈ P d , and Karu [11] for anyGorenstein* poset, that: Theorem 2.2.
For any Gorenstein* poset P , all coefficients of its cd -index Φ P are nonnegative. For B the boolean lattice on two atoms, Q m the face poset of the m -gon, and any cd -word w = w · · · w k , Stanley [22] considered the joinposet P w,m = P ∗ . . . ∗ P k where P i = B if w i = c and P i = Q m if w i = d . It is a regular CW sphere as a join of such. As Φ P ∗ Q = Φ P Φ Q holds for any posets P, Q admitting a cd -index, Stanley concluded that OMPLEXITY YARDSTICKS FOR f -VECTORS OF POLYTOPES AND SPHERES5 when m approaches infinity the coefficient vector of Φ P w,m approachedthe ray spanned by the w th coordinate. This explains Proposition 1.3.3. Proofs and Discussion
Consequences of the g -theorem. Proof of Theorem 1.4.
Recall the g -theorem, Theorem 2.1. Denote g i = h i − h i − for 0 < i ≤ ⌊ d/ ⌋ . Checking whether (1 , g , . . . , g ⌊ d/ ⌋ )is an M-sequence can be done in polynomial time in the size of theencoding of g := ( g , . . . , g ⌊ d/ ⌋ ) in binary. Indeed, we recall the trivialalgorithm one needs to run: (i) for each i produce the i th Macaulayrepresentation (see e.g. [23] for a definition) of g i in poly(lg g i )-time, (ii) then check if the Macaulay inequalities g i ≥ g i +1 holdin poly(max(lg g i , lg g i +1 )-time. This verifies Theorem 1.4. (cid:3) Proof of Theorem 1.1.
The cone Cone f ( σ d ) ( f ( P ds )) is affinely equivalentto the g -cone with apex the origin Cone ( g ( P ) : P ∈ P ds ), which issimply the nonnegative orthant A ⌊ d/ ⌋ in R ⌊ d/ ⌋ . Thus we verify The-orem 1.1 by considering the analogous statements for g -vectors g ( P )and the cone A ⌊ d/ ⌋ rather than f -vectors f ( P ) and the cone C d .The McMullen-Walkup polytopes [14] approach the extremal rays of A ⌊ d/ ⌋ , verifying (1).For (2), first recall the connected sum construction (with respectto a given facet): for two d -polytopes P and P , after applying aprojective transformation to one of them, they can be glued along acommon facet (namely, ( d − σ to form a new convex d -polytope P = P σ P . Combinatorially, the face lattices are related by ∂P =( ∂P ∪ σ ∂P ) \{ σ } . Note that on the level of face lattices, the operations σ are associative and commutative, so we omit the order of summandsand of operations from the language.Now, take connected sum of an appropriate number of copies ofappropriate McMullen-Walkup polytopes to show that any ray in A ⌊ d/ ⌋ is a limit of a sequence of distinct rays spanned by the g ( P ) in A ⌊ d/ ⌋ .Indeed, the g -vectors sum up under connected sum: g ( P σ P ) = g ( P ) + g ( P ).More strongly, for (3) one requires the M-sequence inequalities in the g -theorem: consider the vector x ( a ) = (0 , , . . . , , a ) in A ⌊ d/ ⌋ for a >> M -sequence with respect to the reversed lexicographicorder such that its ⌊ d/ ⌋ th coordinate equals a , denoted M ( a ). TheMacaulay inequalities show that || x ( a ) − M ( a ) || = Θ( a ⌊ d/ ⌋− ⌊ d/ ⌋ ) (for all d ≥ x = ( x , x , . . . , x ⌊ d/ ⌋ ) ∈ A ⌊ d/ ⌋ there exists an M -sequence M ( x ) with || x − M ( x ) || = O ( || x || ⌊ d/ ⌋− ⌊ d/ ⌋ ),e.g. by repeating the above argument for the coordinate vectors x i e i and summing up. No better estimate is possible: if M = ( v , v . . . , v ⌊ d/ ⌋ ) ∈ ERAN NEVO A ⌊ d/ ⌋ is an M -sequence with || x ( a ) − M || = o ( a − ⌊ d/ ⌋ ) then v ⌊ d/ ⌋− = o ( a − ⌊ d/ ⌋ ) so by the Macaulay inequalities v ⌊ d/ ⌋ = o ( a ), and we getthe contradiction || M − x ( a ) || ≥ | a − o ( a ) | = Ω( || x ( a ) || ).For (4) consider the Macaulay conditions again. We see that the only g -vectors of simplicial d -polytopes g ( P ) on the boundary of A ⌊ d/ ⌋ arethose of the form ( a , . . . , a k , , . . . ,
0) for positive a i s, correspondingexactly to ( k − g ( P ) on anextremal ray are of the form ( a , , . . . , (cid:3) cd -index of regular CW spheres. For a flag analog of Theo-rem 1.1(2–4) for W d we first linearly transform to the cd -nonnegativeorthant A c d − in R c d − and consider cd -indices rather than flag f -vectors. As for (2), we lack the needed connected sum type construc-tions for complexes in W d . For example, is it possible to modify aposet P ∈ W d to another poset P ′ ∈ W d such that (i) P ′ has atop dimensional cell whose boundary is dihedral (i.e. isomorphic to D d − ), and (ii) the coefficients are close, namely, for any cd -word w , | [ w ] Φ P ′ − [ w ] Φ P | = o (Φ P (1 , cd -index by Murai and Yanagawa [17,Thm.1.4] shows that the only extremal rays in W d realized by posetsin W d are those corresponding to cd -words with a single d . For d = 4,thanks to a complete characterization of the possible cd -indices [16],(4) is answered. In particular, on the facet of A c − with coordinatecoefficient [ cdc ] = 0 the points are sparse. They in fact lie in “lowerdimension”, where the d coordinate is uniquely determined by thecoordinates c d and dc via the coefficient equation [ d ] = [ c d ][ dc ].Next we consider the characterization in [16] from the computationalcomplexity point of view; the part relevant for a potential computa-tional hardness result is: Theorem 3.1 (Murai-N. [16]) . Let Φ be a cd -polynomial of homoge-nous degree with nonnegative integer coefficients satisfying [ c ] = 1 and [ cdc ] = 1 . Then Φ = Φ( P ) for some Gorenstein* poset of rank (or even regular CW -sphere) iff there exist nonnegative integers x , x , x and y , y , y such that (1) x + x + x = [ c d ] , y + y + y = [ dc ] , x y + x y + x y = [ c d ][ dc ] − [ d ] . Problem 3.2.
Let N = ⌈ lg [ c d ] ⌉ + ⌈ lg [ dc ] ⌉ + ⌈ lg [ d ] ⌉ . Can it bedecided in poly( N )-time whether the diophantine system Eq. (1) hasa solution? OMPLEXITY YARDSTICKS FOR f -VECTORS OF POLYTOPES AND SPHERES7 Deciding in exp( N )-time is trivial. Recall that some binary diophan-tine quadratics are known to be NP-complete [12]. Next we considerarbitrary Gorenstein* posets. Proof of Proposition 1.7.
All Gorenstein* posets of fixed rank with givenflag f -vector of binary bit complexity N can be obtained in exp(exp( N ))-time. Indeed, the total number of possible chains of faces is Π i f { i } =exp( O ( N )), and each potential poset corresponds to a subset, so alltogether we have to consider exp(exp( O ( N ))) number of posets P . Foreach P we compute the cellular homology groups over say the field ofrationals, for all intervals in P ; they are exp( O ( N )) many. For eachinterval the computation is polynomial in the size of the interval, sotakes poly(exp( O ( N ))) time. Proposition 1.7 follows. (cid:3) Possibly a decision can be made in poly( N )-time. Acknowledgements.
I thank the anonymous referees for helpfulsuggestions on the presentation.
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