Komei Fukuda

Reverse search for enumeration

Abstract The reverse search technique has been recently introduced by the authors for efficient enumeration of vertices of polyhedra and arrangements. In this paper, we develop this idea in a general framework and show its broader applications to various problems in operations research, combinatorics, and geometry. In particular, we propose new algorithms for listing 1. (i) all triangulations of a set of n points in the plane. 2. (ii) all cells in a hyperplane arrangement in Rd, 3. (iii) all spanning trees of a graph, 4. (iv) all Euclidean (noncrossing) trees spanning a set of n points in the plane. 5. (v) all connected induced subgraphs of a graph, and 6. (vi) all topological orderings of an acyclic graph. Finally, we propose a new algorithm for the 0 1 integer programming problem which can be considered as an alternative to the branch-and-bound algorithm.

A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra

We present a new pivot-based algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties:(a)Virtually no additional storage is required beyond the input data.(b)The output list produced is free of duplicates.(c)The algorithm is extremely simple, requires no data structures, and handles all degenerate cases.(d)The running time is output sensitive for nondegenerate inputs.(e)The algorithm is easy to parallelize efficiently. For example, the algorithm finds thev vertices of a polyhedron inRd defined by a nondegenerate system ofn inequalities (or, dually, thev facets of the convex hull ofn points inRd, where each facet contains exactlyd given points) in timeO(ndv) andO(nd) space. Thev vertices in a simple arrangement ofn hyperplanes inRd can be found inO(n2dv) time andO(nd) space complexity. The algorithm is based on inverting finite pivot algorithms for linear programming.

Double Description Method Revisited

The double description method is a simple and useful algorithm for enumerating all extreme rays of a general polyhedral cone in ℝd, despite the fact that we can hardly state any interesting theorems on its time and space complexities. In this paper, we reinvestigate this method, introduce some new ideas for efficient implementations, and show some empirical results indicating its practicality in solving highly degenerate problems.

From the Zonotope Construction to the Minkowski Addition of Convex Polytopes

A zonotope is the Minkowski addition of line segments in R d . The zonotope construction problem is to list all extreme points of a zonotope given by its line segments. By duality, it is equivalent to the arrangement construction problem that is to generate all regions of an arrangement of hyperplanes. By replacing line segments with convex V-polytopes, we obtain a natural generalization of the zonotope construction problem: the construction of the Minkowski addition of k polytopes. Gritzmann and Sturmfels studied this general problem in various aspects and presented polynomial algorithms for the problem when one of the parameters k or d is xed. The main objective of the present work is to introduce an ecien t algorithm for variable d and k. Here we call an algorithm ecient or polynomial if it runs in time bounded by a polynomial function of both the input size and the output size. The algorithm is a natural extension of a known algorithm for the zonotope construction, based on linear programming and reverse search. It is compact, highly parallelizable and very easy to implement. This work has been motivated by the use of polyhedral computation for optimal tolerance determination in mechanical engineering.

Primal—Dual Methods for Vertex and Facet Enumeration

Abstract. Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (resp. vertex) to the vertex (resp. halfspace) representation is called vertex enumeration (resp. facet enumeration ). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and the output size. In this paper we extend the known polynomially solvable classes of polytopes by looking at the dual problems. The dual problem of a vertex (resp. facet) enumeration problem is the facet (resp. vertex) enumeration problem for the same polytope where the input and output are simply interchanged. For a particular class of polytopes and a fixed algorithm, one transformation may be much easier than its dual. In this paper we propose a new class of algorithms that take advantage of this phenomenon. Loosely speaking, primal—dual algorithms use a solution to the easy direction as an oracle to help solve the seemingly hard direction.

Exact Volume Computation for Polytopes: A Practical Study

The invention relates to a process of manufacturing, in one and the same operation, a wall or a wall section having an opening therein, and a door or cover for the opening, the door or cover being larger than the free opening in the wall or wall section. The two components are formed in a common tool, the components being placed in staggered parallel planes and interconnected by means of a connecting flange which is directed substantially transversely of these planes and extends between the surrounding edge portion of the wall opening and the surrounding edge portion of the door or cover. After forming, the two components are severed by cutting off of the connecting flange, the overlap of the wall or wall section and the door or cover being determined by the thickness of the connecting flange.

Convexity recognition of the union of polyhedra

In this paper we consider the following basic problem in polyhedral computation: Given two polyhedra in R^d, P and Q, decide whether their union is convex, and, if so, compute it. We consider the three natural specializations of the problem: (1) when the polyhedra are given by halfspaces (H-polyhedra), (2) when they are given by vertices and extreme rays (V-polyhedra), and (3) when both H- and V-polyhedral representations are available. Both the bounded (polytopes) and the unbounded case are considered. We show that the first two problems are polynomially solvable, and that the third problem is strongly-polynomially solvable.

Mathematical Software - ICMS 2006

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Analysis of backtrack algorithms for listing all vertices and all faces of a convex polyhedron

In this paper, we investigate the applicability of backtrack technique to solve the vertex enumeration problem and the face enumeration problem for a convex polyhedron given by a system of linear inequalities. We show that there is a linear-time backtrack algorithm for the face enumeration problem whose space complexity is polynomial in the input size, but the vertex enumeration problem requires a backtrack algorithm to solve a decision problem, called the restricted vertex problem, for each output, which is shown to be NP-complete. Some related NP-complete problems associated with a system of linear inequalities are also discussed, including the optimal vertex problems for polyhedra and arrangements of hyperplanes.

A polynomial case of unconstrained zero-one quadratic optimization

Abstract.Unconstrained zero-one quadratic maximization problems can be solved in polynomial time when the symmetric matrix describing the objective function is positive semidefinite of fixed rank with known spectral decomposition.