Yoshiko Wakabayashi

A cutting plane algorithm for a clustering problem

In this paper we consider a clustering problem that arises in qualitative data analysis. This problem can be transformed to a combinatorial optimization problem, the clique partitioning problem. We have studied the latter problem from a polyhedral point of view and determined large classes of facets of the associated polytope. These theoretical results are utilized in this paper. We describe a cutting plane algorithm that is based on the simplex method and uses exact and heuristic separation routines for some of the classes of facets mentioned before. We discuss some details of the implementation of our code and present our computational results. We mention applications from, e.g., zoology, economics, and the political sciences.

Facets of the clique partitioning polytope

A subsetA of the edge set of a graphG = (V, E) is called a clique partitioning ofG is there is a partition of the node setV into disjoint setsW1,⋯,Wk such that eachWi induces a clique, i.e., a complete (but not necessarily maximal) subgraph ofG, and such thatA = ∪i=1k1{uv|u, v ∈ Wi,u ≠ v}. Given weightswe∈ℝ for alle ∈ E, the clique partitioning problem is to find a clique partitioningA ofG such that ∑e∈Awe is as small as possible. This problem—known to be-hard, see Wakabayashi (1986)—comes up, for instance, in data analysis, and here, the underlying graphG is typically a complete graph. In this paper we study the clique partitioning polytope of the complete graphKn, i.e., is the convex hull of the incidence vectors of the clique partitionings ofKn. We show that triangles, 2-chorded odd cycles, 2-chorded even wheels and other subgraphs ofKn induce facets of. The theoretical results described here have been used to design an (empirically) efficient cutting plane algorithm with which large (real-world) instances of the clique partitioning problem could be solved. These computational results can be found in Grötschel and Wakabayashi (1989).

Algorithms for two-dimensional cutting stock and strip packing problems using dynamic programming and column generation

We investigate several two-dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed. We first present two dynamic programming based algorithms for the Rectangular Knapsack (RK) problem and its variants in which the patterns must be staged. The first algorithm solves the recurrence formula proposed by Beasley; the second algorithm - for staged patterns - also uses a recurrence formula. We show that if the items are not so small compared to the dimensions of the bin, then these algorithms require polynomial time. Using these algorithms we solved all instances of the RK problem found at the OR-LIBRARY, including one for which no optimal solution was known. We also consider the Two-dimensional Cutting Stock problem. We present a column generation based algorithm for this problem that uses the first algorithm above mentioned to generate the columns. We propose two strategies to tackle the residual instances. We also investigate a variant of this problem where the bins have different sizes. At last, we study the Two-dimensional Strip Packing problem. We also present a column generation based algorithm for this problem that uses the second algorithm above mentioned where staged patterns are imposed. In this case we solve instances for two-, three- and four-staged patterns. We report on some computational experiments with the various algorithms we propose in this paper. The results indicate that these algorithms seem to be suitable for solving real-world instances. We give a detailed description (a pseudo-code) of all the algorithms presented here, so that the reader may easily implement these algorithms.

An algorithm for the three-dimensional packing problem with asymptotic performance analysis

The three-dimensional packing problem can be stated as follows. Given a list of boxes, each with a given length, width, and height, the problem is to pack these boxes into a rectangular box of fixed-size bottom and unbounded height, so that the height of this packing is minimized. The boxes have to be packed orthogonally and oriented in all three dimensions. We present an approximation algorithm for this problem and show that its asymptotic performance bound is between 2.5 and 2.67. This result answers a question raised by Li and Cheng [5] about the existence of an algorithm for this problem with an asymptotic performance bound less than 2.89.

The maximum agreement forest problem: Approximation algorithms and computational experiments

There are various techniques for reconstructing phylogenetic trees from data, and in this context the problem of determining how distant two such trees are from each other arises naturally. Various metrics for measuring the distance between two phylogenies have been defined. Another way of comparing two trees T and U is to compute the so called maximum agreement forest of these trees. Informally, the number of components of an agreement forest tells how many edges from each of T and U need to be cut so that the resulting forests agree, after performing all forced edge contractions. This problem is NP-hard even when the input trees have maximum degree 2. Hein et al. [J. Hein, T. Jiang, L. Wang, K. Zhang, On the complexity of comparing evolutionary trees, Discrete Applied Mathematics 71 (1996) 153-169] presented an approximation algorithm for it, claimed to have performance ratio 3. We show that the performance ratio of the algorithm proposed by Hein et al. is 4, and we also present two new 3-approximation algorithms for this problem. We show how to modify one of the algorithms into a (d+1)-approximation algorithm for trees with bounded degree d, d>=2. Finally, we report on some computational experiments comparing the performance of the algorithms presented in this paper.

Algorithms for 3D guillotine cutting problems

We present algorithms for the following three-dimensional (3D) guillotine cutting problems: unbounded knapsack, cutting stock and strip packing. We consider the case where the items have fixed orientation and the case where orthogonal rotations around all axes are allowed. For the unbounded 3D knapsack problem, we extend the recurrence formula proposed by [1] for the rectangular knapsack problem and present a dynamic programming algorithm that uses reduced raster points. We also consider a variant of the unbounded knapsack problem in which the cuts must be staged. For the 3D cutting stock problem and its variants in which the bins have different sizes (and the cuts must be staged), we present column generation-based algorithms. Modified versions of the algorithms for the 3D cutting stock problems with stages are then used to build algorithms for the 3D strip packing problem and its variants. The computational tests performed with the algorithms described in this paper indicate that they are useful to solve instances of moderate size.

Approximation Algorithms for the Orthogonal Z -Oriented Three-Dimensional Packing Problem

We present approximation algorithms for the orthogonal z-oriented three-dimensional packing problem (TPPz) and analyze their asymptotic performance bound. This problem consists in packing a list of rectangular boxes L=(b1,b2,. . . ,bn) into a rectangular box B=(l,w,\infty)

Some Approximation Results for the Maximum Agreement Forest Problem

, orthogonally and oriented in the z-axis, in such a way that the height of the packing is minimized. We say that a packing is oriented in the z-axis when the boxes in L are allowed to be rotated (by ninety degrees) around the z-axis. This problem has some nice applications but has been less investigated than the well-known variant of it---denoted by TPP (three-dimensional orthogonal packing problem)---in which rotations of the boxes are not allowed. The problem TPP can be reduced to TPPz. Given an algorithm for TPPz, we can obtain an algorithm for TPP with the same asymptotic bound. We present an algorithm for TPPz, called R, and three other algorithms, called LS, BS, and SS, for special cases of this problem in which the instances are more restricted. The algorithm LS is for the case in which all boxes in L have square bottoms; BS is for the case in which the box B has a square bottom, and SS is for the case in which the box B and all boxes in L have square bottoms. For an algorithm

Packing Problems with Orthogonal Rotations

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On dominating sets of maximal outerplanar graphs

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