Carl W. Lee

The Associahedron and Triangulations of the n-gon

Let Pn be a convex n-gon in the plane, n ⩾ 3. Consider Σn, the collection of all sets of mutually non-crossing diagonals of Pn. Then Σn is a simplicial complex of dimension n − 4. We prove that Σn is isomorphic to the boundary complex of some (n − 3)-dimensional simplicial convex polytope, and that this polytope can be geometrically realized to have the dihedral group Dn as its group of symmetries. Formulas for the f-vector and h-vector of this polytope and some implications for related combinatorial problems are discussed.

Combinatorial Aspects of Convex Polytopes

This chapter discusses some techniques that have been successful in analyzing the combinatorial aspects of convex polytopes. A convex polyhedron is a subset of ℝ d that is the intersection of a finite number of closed half-spaces. A bounded convex polyhedron is called a convex polytope. The following can be regarded as the fundamental theorem of convex polytopes: P ⊂ ℝ d is a polytope if and only if it is the convex hull of a finite set of points in ℝ d . This theorem and related results are foundational to the theory of linear programming duality, and one of the central themes of combinatorial optimization is to make this conversion for special polytopes related to specific programming problems. The problem of developing algorithms to convert from one description of a polytope to the other arises in mathematical programming and computational geometry. The second theorem states that the collection of all the faces of a polyhedron P , ordered by inclusion, is a lattice. This lattice is called the face lattice or boundary complex of P , and two polytopes are (combinatorially) equivalent if their face lattices are isomorphic. The third theorem states that the face lattices of P and P * are anti-isomorphic. Two polytopes with anti-isomorphic face lattices are said to be dual. Two important dual classes of d -polytopes are the class of simplicial d -polytopes and the class of simple d -polytopes.

Transportation problems which can be solved by the use of Hirsch-paths for the dual problems

Balinski uses his signature method for the proof of the Hirsch-conjecture for dual transportation polyhedra to obtain an efficient algorithm for the assignment problem. We will show how to extend this method to other primal transportation problems, including transportation problems with unit demands. We then prove that Balinskis assignment algorithm is equivalent, cycle by cycle, to that of Hung and Rom. We demonstrate that, under some assumptions for our probability model, a modification of the latter algorithm has an average complexity of O(n2logn) and present some computational results confirming this. We also present results that indicate that this modification compares favorably with Balinskis algorithm and other codes.

Generalized Stress and Motions

In 1987 Kalai presented a new proof of the Lower Bound Theorem for simplicial convex d-polytopes by linking the problem to results in rigidity and stress. He suggested that if higher-dimensional analogues of stress and rigidity were developed, they might lead to other combinatorial results on polytopes, and in particular another proof of the g-Theorem. Here we discuss such a generalization of stress and its relationship to face rings, h-vectors, shellings, bistellar operations, spheres, and simplicial polytopes. In particular, stress plays a role in McMullen’s recent new geometric proof of the g-Theorem using his polytope algebra.

On the dominating set polytope

In this paper, we study the dominating set polytope in the class of graphs that decompose by one-node cutsets where the pieces are cycles. We describe some classes of facets and procedures to construct facets of the polytope in that class of graphs, and establish some structural properties. As a consequence we obtain a complete description of the polytope by a system of inequalities when the graph is a cycle. We also show that the separation problem related to that system can be solved in polynomial time. This yields a polynomial time cutting plane algorithm for the minimum weight dominating set problem in that case. We further discuss the applications for the class of cactus graphs.

P.L.-Spheres, convex polytopes, and stress

We describe here the notion of generalized stress on simplicial complexes, which serves several purposes: it establishes a link between two proofs of the Lower Bound Theorem for simplicial convex polytopes; elucidates some connections between the algebraic tools and the geometric properties of polytopes; leads to an associated natural generalization of infinitesimal motions; behaves well with respect to bistellar operations in the same way that the face ring of a simplicial complex coordinates well with shelling operations, giving rise to a new proof that p.l.-spheres are Cohen-Macaulay; and is dual to the notion of McMullens weights on simple polytopes which he used to give a simpler, more geometric proof of theg-theorem.

Kalai's Squeezed Spheres Are Shellable

Abstract. Kalai constructs a large collection of simplicial spheres to show that there is a substantial gap between the numbers of simplicial polytopes and of simplicial spheres. We prove here that Kalais spheres are shellable.

On k-stacked polytopes

It is proved that equality in the Generalized Simplicial Lower Bound Conjecture can always be obtained by k-stacked polytopes.

Bounding the Numbers of Faces of Polytope Pairs and Simple Polyhedra

Let P be a simplicial d -polytope with v vertices and Σ( P ) be the simplicial ( d – 1)-complex associated with the boundary of P . Suppose, for a given face F of P , that we know the numbers of faces of various dimensions of lk Σ( p ) F . Then we are able to determine upper and lower bounds for the possible numbers of faces of all dimensions of P and of Σ( P ) F . As a consequence, we can bound the numbers of faces of a simple d -polyhedron P if the numbers of bounded and unbounded facets of P and the dimension of the recession cone of P are specified.

On the cone of nonnegative circuits

We discuss three equivalent formulations of a theorem of Seymour on nonnegative sums of circuits of a graph, and present a different (but not shorter) proof of Seymours resut.