Tetsuya Fujie

An exact algorithm for the maximum leaf spanning tree problem

Given a connected graph, the Maximum Leaf Spanning Tree Problem (MLSTP) is to find a spanning tree whose number of leaves (degree-one vertices) is maximum. We propose a branch-and-bound algorithm for MLSTP, in which an upper bound is obtained by solving a minimum spanning tree problem. We report computational results for randomly generated graphs and grid graphs with up to 100 vertices.

The maximum‐leaf spanning tree problem: Formulations and facets

The Maximum Leaf Spanning Tree Problem (MLSTP) is to find a spanning tree in a given undirected graph, whose number of leaves (vertices of degree 1) is maximum. In this article, we consider an integer programming approach to the MLSTP. We provide two formulations of the MLSTP and study the facial structure of polytopes arising from the formulations. Moreover, several relaxation problems are compared.

A study of the quadratic semi-assignment polytope

We study a polytope which arises from a mixed integer programming formulation of the quadratic semi-assignment problem. We introduce an isomorphic projection and transform the polytope to a tractable full-dimensional polytope. As a result, some basic polyhedral properties, such as the dimension, the affine hull, and the trivial facets, are obtained. Further, we present valid inequalities called cut- and clique-inequalities and give complete characterizations for them to be facet-defining. We also discuss a simultaneous lifting of the clique-type facets. Finally, we show an application of the quadratic semi-assignment problem to hub location problems with some computational experiences.

A Dynamic Load Balancing Mechanism for New ParaLEX

ParaLEX, developed recently by the authors, is a parallel extension for the CPLEX mixed integer optimizer which is known as one of the fastest commercial solvers for the mixed integer programming problems. In our previous work, we showed that ParaLEX could efficiently perform 30 solver parallelizations. On the other hand, the simple load balancing mechanism of ParaLEX did not obviously have scalability. In this paper, we propose a load balancing mechanism for a new version of ParaLEX. Preliminary computational results show that the load balancing mechanism is quite efficient in solving a lot of classes of problem instances.

ParaLEX: a parallel extension for the CPLEX mixed integer optimizer

The ILOG CPLEX Mixed Integer Optimizer is a state-of-the-art solver for mixed integer programming. In this paper, we introduce ParaLEX which realizes a master-worker parallelization specialized for the solver on a PC cluster using MPI. To fully utilize the power of the solver, the implementation exploits almost all functionality available in it. Computational experiments are performed for MIPLIB instances on a PC cluster composed of fifteen 3.4GHz pentiumD 950 (with 2G bytes RAM) PCs (running a maximum of 30 CPLEX Mixed Integer Optimizers). The results show that ParaLEX is highly effective in accelerating the solver for hard problem instances.

Solving the maximum clique problem using PUBB

Given an (undirected) graph G=(V, E), a clique of G is a subset of vertices in which every pair is connected by an edge. The problem of finding a clique of maximum size is a classical NP-hard problem, and many algorithms, both heuristic and exact, have been proposed. While the philosophy behind the heuristic algorithms varies greatly, almost all of the exact algorithms are designed in the branch-and-bound framework. As is well known, branch-and-bound is well suited to parallelization, and PUBB is a software utility which implements a generic version of it. The authors show effectiveness of parallelization of branch-and-bound for the maximum clique problem. Especially, by using PUBB with good heuristics and branching techniques, they were able to solve five previously unsolved DIMACS benchmark problems to optimality.

A new heuristic signed-power of two term allocation approach for designing of FIR filters

In this paper, we consider design problems of linear phase FIR filters with CSD (or SP2) coefficients. When the total number of non-zero SP2 terms is given for the design problem, we have to determine the number of non-zero SP2 terms allocated for each filter coefficient respectively, while maintaining the total number. However, this is considered to be an NP-hard problem. Hence, Lim et al. (1998) developed an heuristic method for this allocation problem. In this paper, we propose a new heuristic method for the problem and compare it with Lims method through numerical experiments.

Effectiveness of Parallelizing the ILOG-CPLEX Mixed Integer Optimizer in the PUBB2 Framework

In this paper, we introduce a new method of parallelizing a MIP (Mixed Integer Programming) solver. This method is different from a standard implementation that constructs a parallel branch-and-cut algorithm from scratch (except using an LP solver). The MIP solver we use is ILOG-CPLEX MIP Optimizer (Version 8.0), which is one of the most efficient implementations of branch-and-cut algorithms. The parallelization of the solver is performed by using the software tool PUBB2 developed by the authors. We report a part of our computational experience using up to 24 processors. In addition, we point out some problems that should be resolved for a more efficient parallelization.

Application of Semidefinite Programming to Maximize the Spectral Gap Produced by Node Removal

The smallest positive eigenvalue of the Laplacian of a network is called the spectral gap and characterizes various dynamics on networks. We propose mathematical programming methods to maximize the spectral gap of a given network by removing a fixed number of nodes. We formulate relaxed versions of the original problem using semidefinite programming and apply them to example networks.

A Semidefinite Programming Relaxation for the Generalized Stable Set Problem

In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grotschel, Lovasz and Schrijver to the generalized stable set problem. We define a convex set which serves as a relaxation problem, and show that optimizing a linear function over the set can be done in polynomial time. This implies that the generalized stable set problem for perfect bidirected graphs is polynomial time solvable. Moreover, we prove that the convex set is a polytope if and only if the corresponding bidirected graph is perfect. The definition of the convex set is based on a semidefinite programming relaxation of Lovasz and Schrijver for the maximum weight stable set problem, and the equivalent representation using infinitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof.