Linear Programming the Simplex Algorithm and Simple Polytop es Gil Kalai Institute of Mathematics Hebrew University of Jerusalem Jerusalem Israel email kalaimathhujiacil May Abstract In the rst part of the pap er we survey some farreaching applications of the basic facts of linear programming to the combinatorial theory of simple p olytop es In the second part we discuss some recent developments concerning the simplex algorithm We describ e sub exp onential randomized pivot rules and upp er b ounds on the diameter of graphs of p olytop es Intro duction d A convex p olyhedron is the intersection P of a nite numb er of closed halfspaces in R P is a d ddimensional p olyhedron briey a dp olyhedron if the p oints in P anely span R A convex ddimensional p olytop e briey a dp olytop e is a bounded convex dp olyhedron Alternatively a d convex dp olytop e is the convex hull of a nite set of p oints which anely span R A nontrivial face F of a dp olyhedron P is the intersection of P with a supp orting hyp erplane F itself is a p olyhedron of some lower dimension If the dimension of F is k we call F a k face of P The empty set and P itself are regarded as trivial faces faces of P are called vertices faces are called edges and d faces are called facets For material on convex p olytop es and for many references see Zieglers recent b o ok The set of vertices and b ounded edges of P can b e regarded as an abstract graph called the graph of P and denoted by GP We will denote by f P the numb er of k faces of P The vector f P f P f P is k d called the f vector of P Eulers famous formula V E F gives a connection b etween the numb ers V E F of vertices edges and faces of every p olytop e A convex dp olytop e or p olyhedron is called simple if every vertex of P b elongs to precisely d edges Simple p olyhedra corresp ond to nondegenerate linear programming problems When you cut a simple p olytop e P near a vertex v with a hyp erplane H which intersect the interior of P the intersection P H is a d dimensional simplex S The vertices of S are the intersections of edges of P which contain v with H and the k dimensional faces of S are the intersection of k faces of P with H The following basic prop erty of simple p olytop es follows Let P b e a simple dp olytop e and let v b e a vertex of P Every set of k edges adjacent to v determines a k dimensional face of P which contains the vertex v d d In particular there are precisely k faces in P containing v and altogether faces of all k dimensions which contain v Linear programming and the simplex algorithm Linear programming is the problem of maximizing a linear ob jective function sub ject to a nite set of linear inequalities The relevance of convex p olyhedra to linear programming is clear The set P of feasible solutions for a linear programming problem is a p olyhedron There are two fundamental facts concerning linear programming the reader should keep in mind If is b ounded from ab ove on P then the maximum of on P is attained at a face of P in particular there is a vertex v for which the maximum is attained If is not b ounded from ab ove on P then there is an edge of P on which is not b ounded from ab ove A sucient condition for v to b e a vertex of P on which is maximal is that v is a local maximum namely v is bigger or equal than w for every vertex w which is a neighb or of v The simplex algorithm is a metho d to solve a linear programming problem by rep eatedly moving from one vertex v to an adjacent vertex w of the feasible p olyhedron so that in each step the value of the ob jective function is increased The sp ecic way to cho ose w given v is called the pivot rule The ddimensional simplex and the ddimensional cub e d The ddimensional simplex S is the convex hull of d anely indep endent p oints in R The d faces of S are themselves simplices In fact the convex hull of every subset of vertices of a simplex d d is a face and therefore f S The graph of S is the complete graph on d vertices k d d k d The ddimensional cub e C is the set of all p oints x x x in R such that for every i d d x The vertices of C are all the vectors of length d and two vertices are adjacent i d d dk in the graph of C if they agree in all but one co ordinates f C d k d k Applications of the fundamental prop erties of linear program ming to the combinatorial theory of simple p olytop es Let P b e a simple dp olytop e and let b e linear ob jective function which attains dierent values on dierent vertices of P Call such a linear ob jective function generic Actually it will b e enough to assume only that is not constant on any edge of P The fundamental fact concerning linear programming is that the maximum of on P is attained at a vertex v and that a sucient condition for v to b e the vertex of P on which is maximal is that v is a local maximum namely v is strictly bigger than w for every vertex w which is a neighb or of v Every face F of P is itself a p olytop e and attains dierent values on distinct vertices of F Among the vertices of F there is a vertex on which is maximal and again this vertex is the only vertex in F which is a lo cal maximum of in the face F These considerations have farreaching applications on the understanding of the combinatorial structure of simple p olytop es We refer the reader to Zieglers b o ok for historical notes and for references to the original pap ers Our presentation is also quite close to that in We hop e that the theory of hnumb ers describ ed b elow will reect back on linear programming but this is left to b e seen Degrees and hnumb ers Let P b e a simple dp olytop e and let b e a generic linear ob jective function For a vertex v of P dene the degree of v denoted by deg v to b e the numb er of its neighb oring vertices with smaller value of the ob jective function Clearly deg v d Dene now h P to b e the numb er of vertices of P of degree k This numb er as we dened it k dep ends on the ob jective function but we will so on see that it is actually indep endent from We can see one sign for this already no matter what is there will always b e precisely one vertex of degree d on which attains the maximum and one vertex of degree on which attains the minimum This follows at once from the fact that lo cal maximumglobal maximum To continue we will count pairs of the form F v where F is a k face of P and v is a vertex of F which is a lo cal maximum hence a global maximum of in F On the one hand the numb er of such pairs is precisely f P the numb er of k faces of P This k is b ecause every k face has a unique lo cal maximum On the other hand let us compute how many pairs contain a given vertex v of P This dep ends only on the degree of v Assume that deg v r and consider the set of edges of P T fv w v w g Thus jT j r As we mentioned ab ove every set S of k edges containing v determines a k face F S containing v In this face the set of edges containing v is precisely S In order for v to b e a lo cal maximum in this face it is necessary and sucient that for every edge v w in S v w This o ccurs if and only if S T Therefore the numb er of k faces containing v for which v is a r lo cal maximum is precisely the numb er of subsets of T of size k namely k Summing over all vertices v of P and recalling that h P denotes the numb er of vertices of k degree k we obtain d X r h P f P k d r k k r Note that this formula describ es the f vector of P f P f P f P as an upp er d triangular matrix with ones on the diagonal times the hvector of P h P h P h P d Therefore the hnumb ers are in fact linear combinations of the face numb ers and in particular they do not dep end on the linear ob jective function Put d X k F x f P x P k k and d X k H x h P x P k k Relation gives d X r H x h P x P r r d d d X X X r k k h P x f P x F x r P k k r k k Therefore H x F x and P P d X r r k h P f P r k k r In particular d h P f P f P f P f P d d h P f P f P f P df P d d d h P f P f P f P f P d h P f P d d h P f P d d d d h P f P d f P d d d For the simplex S h for every k The graph of S is the complete graph on d vertices d k d and for every generic ob jective function there will by precisely one vertex of degree k for k d d For the cub e C h To see this consider the ob jective function which is the sum of the d k k co ordinates This is not a generic ob jective function but it is not constant on edges of the p olytop e and this is sucient for our purp oses The vertices of degree k are precisely those having v k d and there are such vertices k Euler Formula and the DehnSommerville Relations For a generic linear ob jective function there is a unique maximal vertex and a unique minimal vertex Therefore h P h P and by the formulas ab ove we obtain f P f P f P d d f P which is Euler Formula usually written d d d f P f P f P f P d More generally if is a generic linear ob jective function then so is However if v is a vertex of a simple p olytop e P and v has degree k wrt then v has degree d k wrt This gives the DehnSommerville relations h P h P k dk The DehnSommerville relations are the only linear equalities among face numb ers of simple dp olytop es The cyclic p olytop es The cyclic dp olytop e with n vertices C d n is the convex hull of n distinct p oint on the moment