Exam Logistics

Qualifying Exam Syllabus ProposalEdward D.KimDraft of November9,2007Exam CommitteeProf.Nina Amenta(Dept.of Computer Science) Prof.Eric Babson(Dept.of Mathematics) Prof.Jes´u s De Loera(Dept.of Mathematics) Prof.Francisco Santos(University of Cantabria) Prof.Roman Vershynin(Dept.of Mathematics) Prof.Roger Wets a(Dept.of Mathematics)a Committee Chairperson Exam Logistics Date:Feb/Mar2008 Time:To be determined Location:To be determinedThesis Proposal Presentation:Graphs of Convex Polytopes1IntroductionConvex polytopes are the sets of feasible solutions to linear programs.The combinatorics and geometry of polytopes are essential to understanding the efficiency of algorithms that solve these classical optimization problems.In particular,the graph(or1-skeleton)of a polytope is intimately connected to Dantzig’s simplex family of methods for solving linear programs.When the iterative method is implemented on a computer,ambiguities in the procedure are resolved by the specification of a pivot rule.Bounding the diameters of the graphs of polytopes is particularly interesting since the diameter of its graph is a lower bound on the number of iterations required for the simplex method using any pivot rule.Let n be afixed positive integer.A halfspace H is a set of the formH:={x∈R n| α,x ≤a}for some a∈R and a non-zero vectorα∈R n.The boundary∂H of a halfspace H forms a hyperplane,which can be described as follows:∂H:={x∈R n| α,x =a}A polytope P in R n is the bounded intersection offinitely-many halfspaces.For a subset K⊆R n,we define the convex hull conv K to be the smallest convex set containing K.Then,we can give an equivalent definition for a polytope:A subset P⊆R n is a polytope if1and only if it is the convex hull of afinite set K⊂R.The dimension d of P is the dimension d≤n of its affine hull.The faces of a polytope P are of special interest.These subsets of the boundary of P are the possible sets of optimal solutions for a linear program.We define a face as follows:Let H be a halfspace that contains P.Then,the face F of P associated to H is the intersectionF:=P∩∂Hof P with the associated hyperplane∂H.The dimension dim F of a face F is the dimension of its affine hull.Faces of dimension d−1are called facets.The faces of dimension0,which are single points,are called vertices.The faces of dimension1,which are always line segments,are called edges.The vertices and edges of a polytope P form an abstract,undirected,finite graph.This graph is called the1-skeleton G(P)(or simply the graph)of P.The distance between two vertices is the length(in number of edges)of the shortest path connecting them.The diameterδ(G)of a graph G is the distance between the two vertices which are furthest from each other.In1957,Hirsch conjectured that the diameterδ(G(P))of the graph of a polytope P is linearly bounded above in its dimension and the number of facets.Conjecture1.1(Hirsch Conjecture).Let f>d≥2.The graph of an d-dimensional polytope with f facets has diameter at most f−d.Remarkably,it is not even known if the diameter of the graph grows polynomially with respect to the dimension d and number f of facets.Specifically,the following problem remains open:Question1.2(Diameter Bound Conjecture).Is there afixed q∈N such that:The graph of a d-dimensional polytope with f facets has diameter at most(fd)q?The Hirsch Conjecture is known to hold only in a few cases:•When d≤3(See[15])•When f−d≤5(See[16])•0-1polytopes(See[17])•Dual transportation polyhedra(See[1])In[14],Kalai and Kleitman proved thefirst sub-exponential bound on the diameter of all convex polytopes.Theorem1.3.The graph of a d-dimensional polytope with f facets has diameter at most f log2d+2.22Transportation PolytopesOur discussion on convex polyhedra will focus on our results in[8]about transportation polytopes. These are the polyhedra whose values x i,j(or x i,j,k,in the3-way axial and planar cases)are organized in rectangular arrays satisfying specified sum conditions.2.1Classical Transportation PolytopesFix positive integers m and n.Let X=(X1,...,X m)∈R m≥0and Y=(Y1,...,Y n)∈R n≥0.The m×n classical transportation polytope P determined by the marginals X and Y is the polytope in the mn variablesx i,j∈R≥0,for i∈[m],j∈[n]satisfying the m+n equations:nj=1x i,j=X i,∀i∈[m]mi=1x i,j=Y j,∀j∈[n].In other words,if we organize the values x i,j≥0inside the tablex1,1x1,2···x1,nx2,1x2,2···x2,n.. ..........x m,1x m,2···x1,nthen we are describing a polytope where each column and each row add up to specified values.In[3],Brightwell et al.prove the following result on the diameters of classical transportation polytopes:Theorem2.1.The graph of an m×n classical transportation polytope has diameter at most 8(m+n−2).When the marginals are generic,we say that P is non-degenerate.In[8],we prove a relationship between the number f0of vertices of a non-degenerate classical transportation polytope and the greatest common divisor of the margin sizes:Theorem2.2.The number of vertices of a non-degenerate m×n classical transportation polytope is divisible by gcd(m,n).32.2Axial Transportation PolytopesWe consider the3-way transportation polytopes.Let l,m and n be three positive integers.We define the polyhedra whose points are l×m×n tables satisfying certain sum conditions.They come in two main varieties.First,we consider the3-way axial transportation polytopes.Let X=(X1,...,X l),Y=(Y1,...,Y m)and Z=(Z1,...,Z n)be three vectors of lengths l, m and n,respectively.The3-way l×m×n axial transportation polytope P determined by the 1-marginals X,Y and Z is the polytope in the lmn variablesx i,j,k∈R≥0,for i∈[l],j∈[m],k∈[n]satisfying the l+m+n equations:j,k x i,j,k=X i,∀i∈[l]i,k x i,j,k=Y j,∀j∈[m]i,jx i,j,k=Z k,∀k∈[n].The main theoretical result of[8]is the following quadratic bound on the diameter of3-way axial transportation polytopes:Theorem2.3.The graph of an l×m×n axial transportation polytope has diameter at most 4(l+m+n−2)2.Theorem2.3provides a quadratic bound for the diameters of axial3-way transportation poly-topes and,moreover,a sublinear one if we assume that the three parameters l,m and n are approximately the same.Bounding the diameter of3-way transportation polytopes is particularly interesting because of the following results proved in[9]:•Any rational convex polyhedron can be rewritten as a face F of an axial3-way transportation polytope.The sizes l,m and n,1-marginals X,Y and Z,and the entries x i,j,k that are prescribed to be zero in the face F can be computed in polynomial time on the size of the input.•Any convex rational polytope is isomorphically representable as a planar3-way transportation polytope.2.3Planar Transportation PolytopesNext,we consider the3-way planar transportation polytopes.Fix three matrices U∈R m×n≥0,V∈R l×n ≥0and W∈R l×m≥0.The3-way l×m×n planar transportation polytope P determined by the2-marginals U,V and W is the polytope in the lmn variablesx i,j,k∈R≥0,for i∈[l],j∈[m],k∈[n]4satisfying the lm+ln+mn equations:i x i,j,k=U j,k,for j∈[m]and k∈[n]j x i,j,k=V i,k,for i∈[l]and k∈[n]kx i,j,k=W i,j,for i∈[l]and j∈[m].In[8],we prove the following correspondence between families of planar transportation polytopes and classical transportation polytopes:Theorem2.4.The2×2×n planar transportation polytopes are in1-1correspondence with the 2×n classical transportation polytopes,with corresponding pairs being linearly isomorphic.3Future WorkFor my dissertation,I will study the following problems related to the graphs of convex polytopes.3.1Diameters of Convex PolytopesLet G=(N,A)be a directed network with n nodes and m arcs.Each arc(i,j)∈A also has a capacity upper bound u i,j and a lower bound l i,j.Each node i∈N is also assigned a number b(i) which indicates its supply or demand.The networkflow polytope P determined by G,u,l and b is the polytope in the m variablesx i,j∈[l i,j,u i,j]for(i,j)∈Asubject to the n equations{j|(i,j)∈A}x i,j−{j|(j,i)∈A}x j,i=b(i),∀i∈N.In[18],Orlin provesTheorem 3.1.The graph of a networkflow polytope with n nodes and m arcs has diameter O(mn log n).The class of unimodular polytopes includes networkflow polytopes.I will study the diameters of unimodular polytopes(those polytopes form P={x∈R n|Ax=b,x≥0}whose constraint matrix A forms a unimodular point configuration),as unimodular point configurations correspond to square-free monomial ideals(see[22]):5Lemma3.2.Let A={a1,...,a n}⊂Z d be a point configuration and let≺be a monomial order. The initial ideal in≺(I A)is square-free if and only if the corresponding regular triangulation∆≺of A is unimodular.There is a strong connection between Gr¨o bner bases and convex polytopes.The link is given by the notion of a state polytope whose vertices are in one-to-one correspondence with the set of all Gr¨o bner bases of a polynomial ideal.I will study the diameters of state polytopes.Secondary polytopes are polytopes whose vertices correspond to regular triangulations of a given convex polytope.I will study the diameters of secondary polytopes andfiber polytopes.I will study the diameters of pointed-cone polytopes(those polytopes of the form P={x∈R n|Ax=b,x≥0}whose constraint matrix A forms a pointed cone).As an initial step,I have shown that the graph of every pointed-cone polytope is isomorphic to the graph of some simple-cone polytope(that is,the cone generated by the columns of A forms a simple cone).3.2Hamiltonicity of Convex PolytopesLet P be a simple4-dimensional polytope.Barnette conjectured that the graph G(P)of P has a Hamiltonian cycle(a closed path in the graph G(P)visiting each vertex exactly once).Following the special families of polytopes given in Section3.1,I will study the4-polytopes in each special subfamily to resolve the question of whether or not some families of4-polytopes are Hamiltonian.6Topics(with book/paper references)putational Geometry•Convex Polyhedra(3-polytopes,Gale and Schlegel Diagrams).[25]•Convex hull Algorithms and vertex enumeration.[?]•Triangulations and Volume Computations.[10]•Fans,Polyhedral Simplicial Complexes,Manifolds and f-vectors.[12][20]•Homology Calculation,Betti Numbers.[12][20]•Basics of algorithm design and Complexity Theory.[5]binatorics•Basic Enumerative Combinatorics.[21]•Generating functions,Posets and Lattices.[21]•Graph Theory(matching theory,networkflows).[2][13]•Latin and Magic squares,0/1matrices.[?]•Transportation and Assignment problems.[24]3.Algebra•Matrix Analysis,Linear Algebra.[6]•Group Theory(Finitely-Generated Groups).[6]•Basics of Commutative Algebra.[6]•Gr¨o bner Bases,Hilbert Basis Theorem.[4]•Monomial Ideals,Hilbert functions.[20]•Stanley-Reisner Rings.[20]•Basics of Toric Ideals and Varieties.[20]•Combinatorial Representation theory(Symmetric groups,finite groups).[11]4.Optimization and Convex Analysis•Linear Programming(Simplex Method,Duality,Interior Point Methods).[19][23]7•Integer Programming(Lattice points,Cutting Planes).[7][23]•Barvinok’s Algorithm.[7]•Semidefinite programming.[?]•Basics of Convex Analysis.[23]References[1]M.L.Balinski:“The Hirsch conjecture for dual transportation polyhedra”.Math.Oper.Res.4,629–633,(1984).[2]B.Bollobas:“Graph Theory:an introductory course”.Springer,c1979,(1990).[3]G.Brightwell,J.van den Heuvel,L.Stougie:“A linear bound on the diameter of the trans-portation polytope”.Combinatorica26(2):133–139,(2006).[4]D.Cox,J.Little,D.O’Shea:“Ideals,Varieties and Algorithms:An Introduction to Compu-tational Algebraic Geometry and Commutative Algebra”.Springer,(1996).[5]T.Cormen,C.Leiserson,R.Rivest,C.Stein:“Introduction to Algorithms:Second Edition”.The MIT Press,(2001).[6]D.Dummit,R.Foote:“Abstract Algebra”.Wiley,3rd edition,(2003).[7]J.A.De Loera,R.Hemmecke:“Integer Optimization”.Pre-print,2007.[8]J.A.De Loera,E.D.Kim,S.Onn,F.Santos:“Graphs of Transportation Polytopes”.Pre-print.Submitted,(2007).[9]J.A.De Loera,S.Onn:“All linear and integer programs are slim3-way transportationprograms”.SIAM J.Optim.17806–821,(2006).[10]J.A.De Loera,J.Rambau,F.Santos:“Triangulations:Applications,Structures,Algorithms”.Pre-print,(2007).[11]W.Fulton,J.Harris:“Representation Theory:A First Course”.Springer,(1999).[12]A.Hatcher:“Algebraic Topology”.Cambridge University Press,(2002).[13]L.Lov´a sz,M.D.Plummer:“Matching Theory”.North-Holland,Amsterdam,(1986).[14]G.Kalai,D.Kleitman:“A quasi-polynomial bound on the diameter of convex polyhedra”.Bulletin Amer.Math.Soc.26315–316,(1992).[15]V.Klee:“Paths on polyhedra II”.Pacific J.Math.17,249–262,(1966).8[16]V.Klee,D.Walkup:“The d-step conjecture for polyhedra of dimension d<6”.Acta Math.11753–78,(1967).[17]D.Naddef:“The Hirsch conjecture is true for(0,1)-polytopes”.Mathematical Programming45,109–110,(1989).[18]J.B.Orlin:“A polynomial time primal network simplex algorithm for minimum costflows”.Mathematical Programming78,109–129,(1997).[19]A.Schrijver:“Theory of Linear and Integer Programming”.Wiley,(1998).[20]R.Stanley:“Combinatorics and Commutative Algebra”.Birkh¨a user,(1996).[21]R.Stanley:“Enumerative Combinatorics”.Cambridge University Press,(1997).[22]B.Sturmfels:“Gr¨o bner Bases and Convex Polytopes”.American Mathematical Society,(1995).[23]R.Wets:“An Optimization Primer:An Introduction to Linear,Nonlinear,Large Scale,Stochastic Programming and Variational Analysis”.Pre-print,lecture notes(2007).[24]V.A.Yemelichev,M.M.Kovalev,M.K.Kravtsov:“Polytopes,graphs and optimisation”.Cambridge University Press,(1984).[25]G.Ziegler:“Lectures on Polytopes”.Springer,(1995).9。