On Computing the Shadows and Slices of Polytopes

Complexit y of Computing the Pro jection of P olytop e s Hans Ra j Tiw ary hansra j@cs.uni-sb.de Abstract W e study the complexity of computing the p ro jection of an arbitrary d -p olytop e along k orthogonal vectors for v arious input and output forms. W e show that if d and k are part of the input ( i.e. not a constant) and w e are interes ted in output-sensitive algorithms, then in most forms the p roblem is equiv alent to en umerating vertices of p olytop es, except in tw o where it is NP-h ard. I n tw o o ther forms the problem is t rivial. W e also review the complexity of computing pro jections when the pro jection directions are in some sense non-degenerate. F or full-dimensional polytop es contai ning origin in the interi or, pro jection is an op eration dual to intersecting the p olytop e with a suitable linear sub space and so the results in t h is pap er can b e dualized by interc hanging vertices with facets and pro jection with inters ection. T o compare the comp lex ity of pro jection an d vertex enumeration, we define new complexit y classes based on the complexity of V ertex Enumeratio n. 1 In tro duction A po lytop e in R d is a closed con vex bo dy that ca n b e represented as either the conv ex h ull of a finite n um ber of p oints or as the in tersection of a finite num b er of halfspaces. W e will c a ll the for mer V -representation and the latter H -repr esentation. Accordingly a polytop e will be called a V -p olytop e, an H -p olytop e or an H V - po lytop e dep ending on whether the p olytop e is given by V , H or both representations. F or a ny polytop e each of these repre sentations is unique if no redundancies ar e allow ed and any of these repr esentations co mpletely determines the others. W e refer the reader to [5, 8] for a thorough trea tmen t of the sub ject. F or enumeration problems, it is often the cas e that the siz e of the output is not b o unded by a p o lynomial in the inpu t size, a nd hence a ny a lgorithm computing th e desired output can not ha ve p olyno mial running time. Therefore it is useful to talk ab out output-sensitive algor ithms who se r unning time is measured in terms o f bo th the size o f the input and the output. In this pap er w e consider only output-sensitive algor ithms and when w e talk ab out p oly nomiality of the algorithm we implicitly assume output-sensitivit y . The V er tex Enumeration problem (VE) a sks one to e numerate the vertices of a p olytop e given by its facets. Despite having b een studied for a long time by a num b er of r esearchers, the complexity s tatus of VE, for gener al dimension and for p olytop es that a re neither simple nor simplicial, is unk nown. It is neither known to b e NP-har d nor there exists a ny p o lynomial algor ithm for it. The dual pr oblem o f computing H -repr esentation from V -repres entation is known as Conv ex Hull pro blem (CH). These tw o problems a re equiv alent mo dulo so lving a Linear Progr am. Thus, for r ational input these tw o problems a re p oly no mial time equiv a lent and a p o lynomial output-sensitiv e a lgorithm for one can b e used to solve the other in output- sensitive p olynomia l time. F o r more details about the problems with curren t V er tex Enumeration methods, we re fer the reader t o [2 ]. Given a po lytop e P ⊂ R d the orthogona l pro jection π ( P ) of P, onto a k -dimensiona l subspace spa nned by the first k co ordinate directions, is obtained b y dropping the la st d − k co or dinates from every point of P . More for ma lly , the pro jection π : R d → R k is a map such that π ( P ) = { x ∈ R k |∃ y ∈ R ( d − k ) , ( x, y ) ∈ P } . In g eneral the pro jection need not b e orthogonal and the pro jection directions can be an arbitra ry orthogo nal set of vectors not necessarily aligned with the co ordina te axes . In such cases one can apply an affine transform 1 to align the pro jection directions along the coor dinate axes, changing the polytop e P in the pro ce ss to another po lytop e P ′ and t hen co nsidering an orthogonal pro jection of P ′ . In this paper w e consider the pr oblem of computing the pr o jection of a p olytop e. Apar t from the r elation of this pro blem to the vertex en umeration problem - which w e will establish in this pap er - a no ther motiv ation for considering the pro jection problem arises from the fact that one freq ue ntly needs to per form this op eration is many area s like Control Theo ry , Constr aint Logic P rogr amming Language s , Cons tr aint Query Languages etc ([6]). And although many har dness proo fs in this pap er ar e rather simple, they don’t appe a r an ywhere in the published literature. F urthermore, many pap ers seem to indicate that at least in some area s like control theory (See, for example, [7]) the quest for a polyno mial algorithm (ev en for the versions that w e prov e to be NP-hard) is still on. The pro jection pro blem has many v ariants dep ending on the input r epresentation of the p olytop e P and the desired output r epresentation of π ( P ). In this paper, we prove that for a rbitrar y pro jection directions, most versions o f this problem are either equiv alent to VE o r ar e NP-har d. V ariants that are neither o f these admit trivial p olynomial alg orithms. F or example computing the vertices of π ( P ) when P is a V -p olytop e can b e do ne simply by pro jecting ea ch vertex and removing redundancies using linear pro gramming. On the other hand, computing the facets or t he v er tices of π ( P ) when P is an H -polyto pe is NP-hard (Subsec tion 3.1). W e also review the complexit y o f the pro jection problem if the pro jection directions ar e in some sense non-degenera te and prov e that in man y cases one can en umera te the desired form of the output in polyno mial time. W e would like to note tha t ev en though the algorithm for computing non-degenerate pro jections w as conceived indep endently by the author, it app ear s to b e almos t similar to the o ne pre sented in [7]. W e nevertheless include it for completeness of discussion. In order to b e able to ta lk ab out the equiv alence o f V ertex Enumeration a nd pro jection, we will define a complexity c lass ba sed on V er tex En umera tion. Ke e ping in line with other notions of completeness, w e call an en umer ation problem Φ , VE-complete if an y output-sensitive polynomial algo r ithm for VE c a n be used to so lve Φ in output-sensitive po lynomial time a nd vice- versa. Similarly , we call a pr oblem VE- easy if it c a n be solved in output-sensitive po lynomial time using an o racle fo r VE, and w e call a pr oblem VE-hard if an oracle for this problem can b e used to solv e VE in output-sensitive p olyno mial time. The results in this pap er ar e summarize d in T able 1 and T able 2. T able 1 summar izes the complexity of computing pro jectio n a lo ng arbitra r y directions while T able 2 summar izes the complexity of computing pro jection a long pro jection directions tha t satisfy a non-degeneracy criteria . Input \ O utput V H HV V po ly VE-complete VE-complete H NP-hard NP-hard VE-complete HV po ly VE-complete VE-complete T able 1: Complexity of co mputing pro jection of a po lytop e onto an a rbitrar y s ubspace . Input \ O utput V H HV V po ly VE-complete VE-complete H VE-hard p oly VE-complete HV po ly po ly poly T able 2 : Co mplexity of computing pro jection of a p olytop e on to a non-degener a te subspa ce . Our res ults ab o ut the har dness, and the equiv alence of vertex enu meration and computing pro jection (in most forms) imply that in all for ms whe r e the pro jection can not b e computed by a tr iv ial alg orithm, finding an output-sensitiv e po lynomial alg orithm will b e a challenging task. Eq uiv a lently , an output-sensitiv e po lynomial algor ithm for vertex enumeration will hav e significant impac t in many fields outside algo r ith- 2 mic p oly to pe theor y , like Control Theor y , Cons traint Lo gic Pr ogra mming Languages , Constraint Query Languages etc, where one frequently needs to solve the pro jection problem ([6 ]). Since for b o unded po lytop es containing the origin in interior, pro jection is an op eration dual to inter- section with a suitably chosen linear subspace, all our results ca n b e dualiz e d by interc ha nging vertices and facets and replacing pro jection with in tersection. The r est o f the pa pe r is orga nized as follows. In the next section we briefly rev iew s ome related work ab out computing pro jection of a poly top e. Our result section is divided into tw o parts. In Subsection 3.1 we present the results about the complexity of computing the pr o jection o f a p oly to pe alo ng a r bitrary directions and in Subsection 3.2 w e formally state the notion of non-degeneracy of pro jection directions and describ e the complexity results for the case when the pr o jection directions a re non-degener ate with resp ect to the input p olytop e. 2 Related W ork Perhaps the best known algorithm for co mputing the facets of the pro jection of an H -p olytop e is the F ourier- Motzkin elimination discov er e d by F ourier in 1824 a nd then rediscov er ed by Motzkin in 1936. This method is ana logous to the metho d of Ga ussian elimination for equa tions and works by eliminating one v ar iable at a time. Since eliminating one v ariable from a system of m inequalities can result in j m 2 4 k facets, the algo rithm can hav e a terrible running time in bad cases wher e the in termedia te polyto pe s have very la r ge (exponential) nu m be r of facets but the final output has only a small num b er of facets. Many improv ements hav e b een made ov er the origina l algo rithm (See [6] for a survey) but ther e is no algorithm that has an o utput-sensitive p olynomial running time. The natural q uestion then is whether one can find a shortcut aro und the in ter mediate pro jection s teps in the F ourier-Motzkin elimination a nd obtain an output-sensitive p o lynomial algo rithm. As we will see in Section 3, the answer is no unless P = N P . Thu s the lac k o f a ny output-sensitive algorithm, for computing the facets of the pro jection of an H -p olytop e, is somewha t natural because t he pr oblem turns out b e NP-hard. F ark a’s Lemma pr ovides a w ay of generating a c one whose extreme rays corr esp ond to the facets of the pro jection of an H -p oly to pe ([3]), but unfortunately it do es not yield a bijectiv e mapping and many extreme rays of the resulting cone may corresp o nd to r edundant inequalities in the pro jection. Balas [3] found a way to get r id of these redundancies and provided a w ay to construct, given a n H - p o lytop e P , another polyhedra l cone W in p olynomial time who se pro jection W ′ yields a one-to -one corr esp ondence b etw een the extre me rays o f W ′ and t he fac e ts of the pro jection of P . Also , W has p olyno mially many facets compared to P . As noted earlier Jones, Kerrigan and Maciejo ws ki [7 ] describe a n algorithm for computing the fa c ets of the pro jection of an H -p olytop e for non-degenerate pro jection directions, that is similar to the one presented in this pap er. Also, in [1] Amenta a nd Zieg ler gav e a p o ly nomial algor ithm fo r the case when the input po lytop e is simple and the pro jectio n directions are no n-degenera te. 3 Results W e will denote the pro jection of a d - dimens ional p olytop e P onto a given k -dimensional s ubspace as π k ( P ). W e will mos tly o mit the subscript and simply refer to the pro jection as π ( P ) and the pro jection subspa ce will dep end on the c o ntext. F ormally , we ar e interested in the following problem: Giv en a p oly to pe P ∈ R d in H -, V - or HV - representation and a set of k orthogona l pro jection directions defining the pro jection space, w e w ant to compute the non-redundant H - , V - or H V -repres entation of π ( P ). 3.1 Pro jection onto arbitr ary subspaces Depending on the input and output form, we hav e nine v ar ia nts of the pr oblem. It is obvious that if the vertices ar e pa r t of the input and one w ants to compute the vertices of the pro jection, then eac h vertex ca n 3 be pro jected trivially and the v ertices that become re dundant in the pr o jection can be identified by solving one Linear Progra m p er v ertex. Hence, w e ha ve the fo llowing: Lemma 1. Given a p olytop e P ⊂ R d in V - or HV -r epr esentation and a set of arbitr ary pr oje ction dir e ctions, non-r e du n dant V -r epr esentation of π ( P ) c an b e c omput e d in p olynomial time. Also, it is easy to see that every p olytop e can be represented a s the pro jection of a suitable simplex. Assuming that the v ertices are num b er ed 0 thro ugh m − 1, simply app end e i to the i -th v e r tex, wher e e 0 is the zero vector and e i is the i -th unit vector in R m − 1 . Thus, given a p olytop e P by its vertices one ca n compute in p olynomia l time the vertices of this simplex ∆ and the pro jection dir ections such that P is the pro jection of ∆. Since it is trivial to compute the facets of a simplex given its vertices, V ertex Enumeration can b e solved in output-sensitive po lynomial time using a ny algorithm that computes the H - or HV -repr e sentation of pr o jection from V - or HV -re pr esentation of a p o lytop e. Clearly , one can also use any p olynomia l algorithm for V ertex En umeration to compute the H - or HV - representation of the pro jection of an y p olytop e given in V - o r H V -r epresentation in p oly no mial time. Hence, we have the follo wing lemma: Lemma 2. Computing the H - or H V -r epr esent ation of the pr oje ction of a p olytop e given in V - or HV - r epr esentation is VE-c omplete. In what follows, w e as sume the input p oly top e P is of the form { ( x, y ) | Ax + B y ≤ 1 } a nd we wan t to co mpute the pro jection π ( P ) = { x ∈ R k | ( x, y ) ∈ P for some y } , where A ∈ Q m × k , B ∈ Q m × ( d − k ) , x ∈ R k , y ∈ R ( d − k ). W e also assume P to b e full-dimensional and to contain the orig in in its interior. F or rational p o lytop es this assumption is justified b eca use o ne can always find a p oint in the interior of the po lytop e via Linear Pro gramming and mo ve the or igin to th is point. W e ar e now left with the thr ee cas es where the input po lytop e is given b y H - representation. As we will see now, co mputing either the facets or the vertices o f the pro jection in this case is hard while computing both facets and vertices o f the pro jection is equiv alent to V ertex Enumeration. Consider the following decision version o f the problem: Input: Polytopes P = { ( x, y ) | Ax + B y ≤ 1 } and Q = { x | A ′ x ≤ 1 } Output: YES if Q 6 = π ( P ), NO otherwise. W e will now prove that this decisio n problem is NP-co mplete th us pr oving the NP-hardness of the enum eration problem. Theorem 1. Given a p olytop e P = { ( x, y ) | Ax + B y ≤ 1 } and Q = { x | A ′ x ≤ 1 } it is NP-c omplete t o de cide if Q 6 = π ( P ) . Pr o of. It is ea sy to see that deciding whether a given set o f hyperplanes c ompletely define the pro jection of a giv en higher dimensional p o ly top e, is in NP , since if Q 6 = π ( P ) then ther e is a p oint that is either only in Q and not in π ( P ) or vice versa. F o r a given p o int x c hecking whether x ∈ Q is trivial when Q is defined by its facets, and c hecking whether x ∈ π ( P ) amounts to chec king th e feasibility of an LP . So it suffices to show tha t it is NP-hard as w ell. It is known ([4 ]) that given an H -p olytop e P 1 = { x | A ′ x ≤ 1 } a nd a V - po lytop e P 2 = C H ( V ), it is NP-complete to decide whether P 1 * P 2 . Clearly , P 1 ⊆ P 2 if and only if P 1 ∩ P 2 = P 1 . No w, P 1 ∩ P 2 has the following H -repres e n tation A ′ x ≤ 1 x − X v ∈ V λ v · v = 0 X v ∈ V λ v = 1 λ v ≥ 0 , ∀ v ∈ V 4 The v ar iables λ ensure that we co ns ider only those points in P that can be represented as a conv ex combination of v ertices of Q . O ne can further, in p oly nomial time, g et a full dimensional representation of P 1 ∩ P 2 by eliminating the d − 1 equatio ns. The resulting p o lytop e is a full-dimensional p olytop e in R | V |− 1 of the form { ( x, λ ) | Ax + B λ ≤ 1 } . Since we ar e interested in o nly the (vector) v ar iable x , the pro jection of this p olytop e along the a xes corres p o nding to the v ar iables λ v gives us the facets o f P 1 ∩ P 2 in the subspace of v ariables x . It follows that P 1 ∩ P 2 = P 1 if and only if the pr o jection of { ( x, y ) | Ax + B y ≤ 1 } onto the subspace of v aria bles x has the H -repr esentation same a s that of P 1 i.e. A ′ x ≤ 1. Thu s for arbitra ry p olytop es P = { ( x, y ) | Ax + B y ≤ 1 } and Q = { x | A ′ x ≤ 1 } , a n algorithm for deciding whether Q 6 = π ( P ) can b e use d to decide whether a n H -p olytop e is contained in a V -p oly top e or not, which is a NP-complete problem. Balas([3]) has shown that for a given H -po lytop e ( P ) and a set of pro jectio n directions, o ne can compute the facets of another po int ed polyhedr al cone W a nd another set of pro jection dir ections such that the facets of π ( P ) are in one-to-o ne corresp ondence with the e x treme ra y s π ( W ). No te that the pro jections of P and W ar e defined in differen t spaces and should not b e confused a s the sa me pro jectio n map despite the abuse of nota tio n here. The num b er of facets of W is p olynomia l in the num b er of facets of P . It is not difficult to mo dify the co nstruction in [3] so that W is b ounded i.e. a p o lytop e and the vertices in the pr o jection o f W , except the orig in, ar e in one-to-one corr e sp ondence with the facets of th e pro jection of P . F or completeness we state the res ult of Ba las a nd describ e the mo dification here. Lemma 3 ( Balas ) . Given an H p olytop e P and a set of pr oje ction dir e ctions, ther e exists a p olyhe dr al c one W and another set of pr oje ction dir e ctions such that t he fac ets of π ( P ) , ar e in one-to-one c orr esp ondenc e with the ex tr eme r ays of π ( W ) . F u rthermor e, W has p olynomial ly many f ac ets c omp ar e d to P and the fac ets of W c an b e c ompute d in p olynomial time. W e will use the notion of po la r duality to prov e that the c o ne W obta ined from the construction of Balas can b e turned into a b ounded polytop e. F or a p olyhedr al cone W in R n with facet inequalities Ax ≤ 0 and extreme rays the row vectors of V , where A and V a re matrices with each row a vector in R n , the p ola r dual W ∗ has the r oles of the e xtreme rays and facets reversed. In par ticular, the facet inequalities o f W ∗ are V x ≤ 0 and the extreme rays are the ro w vectors o f A . W e a g ain re fer the r e ader to [5, 8] for mor e de ta ils of the prop erties of polar dua lity . Lemma 4. Gi ven a p ointe d p olyhe dr al c one W ∈ R n and a s et of pr oje ct ion dir e ctions Γ one c an c onstru ct, in p olynomial time, a p olytop e P ∈ R n such that the extr eme r ays o f π ( W ) a r e in one-to-one c orr esp ondenc e with the vertic es of π ( P ) , ex c ept one vertex c orr esp onding to the ap ex of π ( W ) . F urt hermor e, b ot h the pr oje ctions ar e onto the same subsp ac e. Pr o of. Clearly , none of the pro jectio n directions lie in the interior of W , otherwise the pro jection spans th e whole subspace. Let W ∗ be the p olar dual of W . F o r any vector α in the interior of W ∗ the hyperplane α · x = 0 touches W only at the origin and hence W ∩ { x | α · x ≤ 1 } is a b ounded polyto pe . It is actually a pyramid with or igin as the apex. Now co nsider the pro jection π ( W ) which is a p ointed cone with orig in as ap ex. This cone is a full- dimensional cone in the subs pa ce co ntaining it and we can consider its p olar dual in that subspace. Let π ∗ ( W ) b e the p ola r dual of π ( W ). F o r any vector α ′ in the interior of π ∗ ( W ), π ( W ) ∩ { x | α ′ · x ≤ 1 } is a bo unded po ly top e. Mor eov er, fo r such an α ′ , γ i · α ′ = 0 for all γ i ∈ Γ. Since π ( W ) is a p ointed cone suc h an α ′ exists. Since π ∗ ( W ) can be o bta ined a s the in ter section of W ∗ with T γ i ∈ Γ { γ i · x = 0 } , a vector α ′ in the interior of π ∗ ( W ) c an be co mputed in polyno mial time if one knows either the vertices or face ts of W . Also, α ′ lies in the interior of W ∗ as well. Define Q = W ∩ { α ′ · x ≤ 1 } . Given the ex treme rays (facets resp ectively) of W and the pr o jection directions Γ one can compute the vertices (facets resp ectively) of Q in polynomial time. 5 Since α ′ is or thogonal to each of the pro jection directions , the v ertices and facets o f π ( P ) are in one-to - one corres p o ndence with the extreme r ays and the facets o f π ( W ). Note that one vertex of π ( P ) corresp onds to the a p ex of π ( W ). Theorem 1 t ogether with Lemma 3 and Lemma 4 g ives the follo wing: Theorem 2. Given a p olytop e P = { ( x, y ) | Ax + B y ≤ 1 } and Q = C H ( V ) it is N P- c omplete t o de cide if Q 6 = π ( P ) . Now w e consider the la s t v a r iant of the pro jection pr oblem where we are given an H -p oly top e a nd we wan t to compute the H V -repr esentation of the pr o jection. It turns out that altho ugh computing either the vertices or facets of the pro jection is NP-hard, computing both vertices a nd facets is VE-complete. Before w e prove this, we w ould lik e to remar k that the notion of output-sensitiveness can hav e v ario us meanings. An output-sensitive polynomial alg o rithm for an enumeration problem (lik e VE) could enumerate vertices such that a new v er tex is repo r ted within incremental p o lynomial delay i.e. each new rep o rting tak es time p olynomial in the input and the o utput pro duced so far. It is equally conceiv able that the algorithm takes tota l time p olynomial in the input and o utput but there is no guar antee that succ essive rep or tings take only incre mental p olynomial delay . W e will a ssume that if we hav e an o utput-se ns itive algo rithm o f the latter kind, then we a ctually know the co mplex it y of its running time. Under this assumption the tw o notions a r e same for VE. T o see why th is is true, consider the follo wing. Given an H -p olytop e P and a V - p olytop e Q , determinin g whether P = Q is polynomial time equiva len t to VE (See [2 ]). Also, sol ving this problem giv es an algorithm for VE that is not only output-sensitive p olynomial but als o has a polynomial dela y guarantee. If w e ha ve an enumeration algo rithm that has no guarantee of p olynomial dela y b etw een successive outpu ts, but for which we know the run n ing time, then w e can use this pro cedure to create a p olynomial algorithm for deciding the equiv alence of H - and V -p olytop es: Simply compute the time n eeded by the algorithm to enumerate all vertices of P assuming P = Q and run the enumeratio n algorithm for the time req u ired to output | v er t ( Q ) | + 1 vertices . If th e pro cedure stops th en w e can compare the list of vertices of P with th at of Q in p olynomial time. If, on t h e other hand, the proced u re doesn’t finish within the given time then P m ust have more vertices th an Q and h ence, P 6 = Q . This also implies that if w e hav e an alg orithm for VE that has outp ut-sensitive p olynomial running time, then w e can assume that the al gorithm prod uces successive vertices with only a dela y polynomial in the size of the input and the num b er of v ertices prod uced so far. So for pro ving VE-completeness in the next theorem, when w e ass ume the ex istence of an output-sensitive po lynomial algor ithm for VE, we also assume that this algorithm has a g uarantee of p olynomial delay b etw een successive outputs. Although w e will work with t he Con v ex H ull problem whic h is the du al v ersion of VE, with a sligh t abuse of la nguage we will refer to this dual problem as VE as w ell. Theorem 3. Given a p olytop e P = { ( x, y ) | Ax + B y ≤ 1 } it is VE-c omplete to c ompute the fac ets and vertic es of π ( P ) . Pr o of. Since every p olyto pe P ∈ R n given by m vertices can b e co nv erted to a ( m − 1 )-dimensional simplex ∆ such tha t P is a pro jection of ∆ it is clear that c o mputing HV -repres ent ation of the pr o jection o f an H -p olytop e is VE-har d. T o prov e that this problem is also VE-easy , w e give an a lgorithm that uses a routine for VE to en umera te the facets and v ertices of π ( P ). The algorithm pro ceeds as follows: A t any point w e hav e a list of v er tices V of π ( P ) and we want to v e r ify that V indeed contains all v ertices o f π ( P ). If the list is not complete, w e wan t to find another vertex o f π ( P ) that is not alre ady in V . T o do this, we sta rt e numerating facets of C H ( V ) and we verify that eac h generated facet is indeed a facet of π ( P ). Note that even though C H ( V ) can have many more fac e ts than π ( P ) (in fact it ca n hav e exp o ne ntially many facets co mpared to π ( P ) [2]), this is no t a problem s inc e we stop at the fir st facet of C H ( V ) that is no t a facet of π ( P ). Checking whether a given facet of C H ( V ) is a f acet of π ( P ) or not is easy because of the following: Suppo se h = { x | a · x = 1 } b e a hyperpla ne in the pro jection space. W e say that h int ersects P pr op erly if the intersection P ∩ { ( a, k times z }| { 0 , · · · , 0) · ( x, y ) = 1 } ha s some p oint in the int erior of P . W e will call such an 6 int ersection a pr op er in tersection. W e claim tha t the defining h yp er pla ne of every facet f o f C H ( V ), that is not a facet o f π ( P ), int ersects P prop erly . T o see this, pic k a p oint x 1 in the rela tive in ter ior of f . Such a p oint exists b ecause C H ( V ) ⊂ π ( P ) if some facet f of C H ( V ) is not a face t of π ( P ). This p oint also lie s in the r elative in terior o f π ( P ). Also, there is a p oint ( x 1 , y 1 ) that lies in the interior of P that pr o jects to x 1 . Clearly the hyperplane { ( a, k times z }| { 0 , · · · , 0) · ( x, y ) = 1 } contains ( x 1 , y 1 ) and hence the h yp erplane defining f intersects P prop erly . It follows t hat, if V do es not c ontain all v er tice s o f π ( P ) then there exis ts a fa c e t f = { x | a · x = 1 } of C H ( V ) intersecting P prope r ly . So if the enumeration pro cedure for facets of C H ( V ) s tops a nd none of the facets in tersect P pr op erly then V contains all the vertices o f π ( P ) , and we hav e all the v ertices a nd face ts of π ( P ). If so me intermediate facet { a · x = 1 } of C H ( V ) do es intersect P prop er ly then maximizing the ob jective function ( a, k times z }| { 0 , · · · , 0) · ( x, y ) ov er P pro duces a vertex of P that also gives a vertex v of π ( P ) up on pro jection. Mor eov er this vertex is not in the list V . This gives an output-sensitive po lynomial algo rithm for enumerating all facets and vertices of π ( P ). Hence, co mputing all vertices and facets of the pro jection π ( P ) of an H -p olytop e P is VE-easy as well. 3.2 Pro jection onto non-degenerate subspaces T o make the no tion of non-deg enerate pr o jection precise, no te that if P is the input p olytop e then every face of pro jection π ( P ) is the s hadow of some prop er face o f P . Call the maxima l dimensional face f ′ of P a pr e-image o f the face f of π ( P ) if f is obtained by pro jecting all vertices defining f ′ and taking their conv ex hull. In genera l, the dimensions of f and f ′ are not the same. This can happen if some pro jection directions lie in the affine hull o f f ′ . W e call a set of pro jection directio ns non-de gener ate with res pe c t to P if no directions lie in the affine h ull of any face of P . F act 1: F or non-degener ate pro jection directions a nd a face f of π ( P ), if f ′ is the pr e -image of f then dim ( f ) = dim ( f ′ ). This is easy to see b ecause for any face f ′ of P that app ear s in π ( P ) pro jection reduces the dimensio n if and only if the some pr o jection directions lie in the affine hull of f ′ . This is not p ossible f or no n-degenera te pro jection dir ections. Now, given a po lytop e P in H - representation and a set of non-degener ate pro jection directio ns Γ we wan t to compute the facets of the pr o jection π ( P ). Aga in, we assume that the facets of P are presen ted as inequalities of the for m Ax + B y ≤ 1, where A is an m × k ma trix, B is a n m × ( d − k ) matrix, and the pro jection has dimension k . Since, we will need to solve Linea r P r ogra ms w e a lso a ssume that the po lytop e is rational i.e. the entries in A and B ar e r a tional n um ber s. W e will assume that the pro jection directions ar e aligned along a subset of co-o rdinate a x es. If not, we can apply a suitable affine transform to P depending on the orthogonal pro jectio n dir ections. Our a lg orithm for enumerating the face ts of π ( P ) pro c e eds as follows: Giv en a partial list of facets o f π ( P ), for each facet f we identif y its pre- image f ′ in P . F or each o f these faces of P we identify their ( d − 2 )-dimensional faces and among all such ( d − 2)-faces of f ′ some give rise to r idges in π ( P ). W e identify which faces form the pre- ima ge of some ridge of π ( P ) and from the cor r esp onding r idg e, we ide ntify the tw o facets defining this ridge , th us fin ding a new facet of π ( P ) if the current list of facets is not complete. Lemma 5. Given an H - p olytop e P and a fac et f of its pr oje ction π ( P ) , one c an fin d the fac ets of P defining the pr e-image of f in p olynomial time. Pr o of. Let { x ∈ R d | a · x ≤ 1 , a ∈ Q d } b e the halfspace defining the facet f of π ( P ). Clear ly , the hype r plane h in R d + k with normal a ′ = ( a, k times z }| { 0 , · · · , 0) defines the supp orting hyperplane { x ∈ R d + k | a ′ · x = 1 } . Also, P ∩ h is a face of P and is exactly the pre-image o f f . A facet F of P contains t his face iff P ∩ h ∩ F has the same dimensions as P ∩ h . Thus, one can find all t he fac e ts o f P co nt aining the pre-image o f f in time po lynomial in the size of P . 7 The next lemma follows immediately from the non-degener acy of the pro jection directions, so w e mention it without the pro of (See F act 1). Lemma 6. Given P and a fac et f of its pr oje ction π ( P ) , if g is another fa c et of π ( P ) sharing a ridge with f then the pr e-images f ′ and g ′ shar e a fac e in P . F u rthermor e, dim ( f ′ ∩ g ′ ) = dim ( f ∩ g ) = dim ( f ′ ) − 1 = dim ( g ′ ) − 1 = d − 2 Since the face ts of P a re known, we can identify all ( d − 2)-faces of f ′ . The num b er of these faces is at most m for each pre-image f ′ and since f ′ is itse lf a po lytop e of dimension d − 1, we can compute the non-redundant ineq ualities defining the facets ( d − 2-dimensional faces) of f ′ . At this point, what remains is to iden tify these ridg e s and the face ts defining these r idges. The following tw o lemmas achiev e th is. Lemma 7. L et P = { ( x, y ) | Ax + B y ≤ 1 } b e a p olytop e in R d , wher e A ∈ Q m × k , B ∈ Q m × ( d − k ) x ∈ R k , y ∈ R ( d − k ) . Also, let f b e a ( d − 2) -fac e of P define d as f = { ( x, y ) | A ′ · ( x, y ) = 1 , ( x, y ) ∈ P } . Then, f defin es a ridge in the pr oje ct ion π ( P ) if and only if • ther e exists α ∈ R d such that ( α, k time s z }| { 0 , · · · , 0) ∈ C H ( A ′ ) , wher e e ach r ow of A ′ is tr e ate d as a p oint in R d + k . And, • The fe asible r e gion of al l such α is a line se gment. It is no t difficult to s e e that this lemma is just a rephr asing of the basic prop erties of supp orting hy- per planes of a p o lytop e. In other words, any hyperpla ne whos e normal is a conv ex co mbin ations of the normals of facets defining the face f , is a supp orting hyperplane of P and vice-versa. F ur ther more, if the normal lies in the subspace where the pro jection π ( P ) lives, then it is also a supp orting hyperplane of π ( P ). Also, th e no rmals of all h yp erplanes that supp ort a p o lytop e at some r idge, when treated a s points, form a 1-dimensional p olytop e i.e. a line se gment. This for mu lation a llows us to check in p o ly nomial time whether a ( d − 2 )-face of P forms a pr e-image of some ridge of π ( P ). Lemma 8. The end p oints of the fe asible r e gion of α in lemma 7 ar e the normals of t he fac ets of π ( P ) defining the ridge c orr esp onding to fac e f . As noted before, the norma ls of the hyper planes supp or ting a polyto pe at a ridge r form a line seg ment when view ed as points. The end p oints o f the segmen t re pr esent the normals of the t wo facets defining the ridge r . This lemma ensures that given a pre-image of some ridg e of π ( P ), o ne can compute the normals of the t w o facets of π ( P ) defining the ridge r b y solving a p oly nomial n um ber of linear programs eac h of size po lynomial in the size of input. Putting everything tog e ther we g e t the following theorem: Theorem 4. Given a p olytop e P define d by fac ets, and a set of non- de gener ate ortho gonal pr oje ction dir e c- tions Γ one c an enumer ate al l fac ets of π ( P ) in output- sensitive p olynomial time. Since randomly picked pro jection dir ections are non- degenerate with pr obability 1, one can also enumerate the facets of the pro jection of an H -p olytop e for s uch direc tio ns. Note, that this also g ives an output-sensitiv e po lynomial a lgorithm fo r the case when the input is an HV -p olytop e irresp ective o f the output form. Also, if the vertices of P are given then so me tests like those in Lemma 7 and 8 b ecome easier. W e leave the pro of of this to the reader since they do not affect our main argument ab o ut the existence of an o utput-sensitive po lynomial a lgorithm. Corollary 1. Given an HV -p olytop e P and a set of pr oje ction dir e ctions Γ that ar e n on-de gener ate with r esp e ct to P ther e is an alg orithm that c an enumer ate the vertic es and/or fac ets of π ( P ) in output-sensitive p olynomial time. 8 It is ea sy to s ee that c omputing the vertices o f the pro jection of an H -p olytop e along non- degenerate directions has VE as a sp ecial ca se - simply pick the se t of pr o jection directions to b e the empty s et. Hence, enum erating vertices of the pro jection of a n H -p olytop e a lo ng no n-degenera te pro jection directions remains VE-hard even though it is not clear if it remains NP- hard. Similarly one can ar g ue that the complexity status of computing the pro jection of a V -p olytop e along non-degenerate pro jectio n directions r emains the same as that of co mputing the pro jection along arbitrary dir ections. 4 Concluding Remarks Computing the pro jection of a po ly top e is a fundamen tal task that arises frequently in many different areas of co mputer science like Con trol Theory , Co nstraint Lo gic Prog ramming Languages, Constraint Query Languages among others [6]. The r esults in this pap er emphasize the imp ortance of obtaining a n efficient metho d for vertex enumeration for co mputing pro jections. W e also defined the notio ns o f VE-ha rdness, completeness etc in this pap er. Given the long histor y of rese arch on the vertex enumeration and lack of an y insight on its complexity , we believe t hat it is natural to try to relate the complexit y of new problems with that of vertex en umeration. W e hop e that the new complexity classes defined in this pap er will b e useful for future r esearch and new problems will be sho wn to be in these classes. References [1] N. Amen ta and G. Ziegler. Shadows and slices of po lytop es. In Pr o c e e dings of t he Twelf th Annual Symp osium On Computational Ge ometr y (ISG ’96 ) , pages 10– 19, New Y o r k, May 199 6. ACM Pr ess. [2] D. Avis, D. Bre mner , and R. Seidel. How g o o d are conv ex hull a lgorithms? Comput. G e om. , 7 :265–3 01, 1997. [3] E. Balas. Pro jection with a minimal system of inequalities. Comput. Optim. A ppl. , 1 0(2):189 –193, 1998 . [4] R. M. F reund and J. B. Orlin. On the complexity of four p olyhedra l set con ta inmen t problems. Mathe- matic al Pr o gr amming , 33(2):139– 145, 1 985. [5] B. Gr ¨ un baum. Convex Polytop es Se c ond Edition pr ep ar e d by V. Kaib el, V . L. K le e and G. M. Zie gler , volume 2 21 of Gr aduate T ex ts in Mathema tics . Springer, 2003. [6] J.-L. Im b ert. F our ier’s elimination: Which to c ho os e? In PPCP , pag es 11 7–12 9, 199 3. [7] C. Jones, E. Kerrigan, and J. Maciejowski. Equality set pro jection: A new algorithm for the pro jection of po lytop es in ha lfspace repre sentation. T echnical Rep ort CUED/F-INFENG/TR.46 3 , ETH Zurich, 2004. [8] G. M. Ziegler. L e ctur es on P olytop es , volume 152 of Gr aduate T exts in Mathema tics . Spring er-V erla g. 9