Peh Ng - Academia.edu (original) (raw)

Papers by Peh Ng

Research paper thumbnail of Leontief flow problems: Integrality properties and strong extended formulations

This thesis has two main parts, both of which deal with Leontief directed hypergraphs (LDH\u27s),... more This thesis has two main parts, both of which deal with Leontief directed hypergraphs (LDH\u27s), a generalization of directed graphs where arcs have multiple (or no) tails and at most one head. Given an LDH with costs on its hyperarcs and with net demands on its vertices, the Leontief flow problem (LFP) is the problem of assigning flows on the hyperarcs at minimum cost and in such a way that net demands at all vertices are satisfied. In the first part of the thesis, we identify classes of Leontief flow problems, beyond ordinary network flow problems, that have integral optimal solutions. Specifically, we give necessary and sufficient conditions under which the associated vertex-hyperarc incidence matrices of classes of LDH\u27s are totally unimodular. We also characterize circuits of the underlying matroids for some subclasses of these LDH\u27s, one of which properly includes the class of graphic matroids. In the second part of the thesis, we consider the class of Uncapacitated Fixed Charge Network Flow problems, (UF)\u27s. These are single-source flow problems with fixed charge for setting up any arc in a specific subset, the usual unit flow costs for other arcs, and no capacities. As a mixed-integer linear programming problem, (UF) is difficult to solve; indeed, it is NP-Hard. This research develops a practical and concise way of modeling the Uncapacitated Fixed Charge Network Flow problem using (LFP)\u27s and exploiting the structures of Leontief directed hypergraphs. We show how an extended formulation, adding many new variables and constraints, yields far tighter LP-relaxations. Indeed, we give an axiomatic characterization of the best such extended formulations

Research paper thumbnail of Totally unimodular Leontief directed hypergraphs

Linear Algebra and its Applications, 1995

Research paper thumbnail of Commodity family extended formulations of uncapacitated fixed charge network flow problems

Networks, 1997

ABSTRACT Uncapacitated fixed charge network flow problems are single-commodity flow problems with... more ABSTRACT Uncapacitated fixed charge network flow problems are single-commodity flow problems with (positive) fixed charges for opening some arcs and no capacities. Previous research has shown that much improved linear programming relaxations can be obtained by reformulating these problems in terms of an extended variable set corresponding to flow commodities for each demand point. In this paper, we develop a theory of reformulations generalizing to families of commodities defined by arbitrary demand subsets. In particular, we show how to produce an extended formulation for any suitable commodity family and isolate simple axioms characterizing the families that yield the most useful reformulations. © 1997 John Wiley & Sons, Inc. Networks 30:57–71, 1997

Research paper thumbnail of A simplex algorithm for a class of leontief flow problems

Mathematical and Computer Modelling, 1996

A matrix N is hntief if it has exactly one positive entry per column and there exists a nonnegati... more A matrix N is hntief if it has exactly one positive entry per column and there exists a nonnegative x such that Nx > 0. A Lwntief flow problem is a linear-programming problem of the form min(cTx 1 Nx = b;x 2 0), where N is a certain type of Leontief matrix. It is shown that for b > 0 this problem can be solved in 0(n2U 1ognpU) pivots by the simplex method using Dantzig's rule for choosing the entering variable, where n is the number of variables, p is the largest entry of N in absolute value, and CT is a valid upper bound on any extreme-point solution. C~SSHS of problems where this is a strongly polynomial bound are identified.

Research paper thumbnail of A Decomposition of Copulas and Its Use

Communications in Statistics - Theory and Methods, 2005

ABSTRACT In this article, we create a decomposition that represents and describes the depen-dence... more ABSTRACT In this article, we create a decomposition that represents and describes the depen-dence structure between two variables. Since copulas provide a deep understanding of the dependence structure by eliminating the effects of the marginals, they play a key role in this study. We define a discretized copula density matrix and decompose it into a set of permutation matrices by using the Birkhoff–von Neumann theorem. This decomposition provides a way to effectively apply the concepts of copulas to solve problems in multivariate statistical data analysis.

Research paper thumbnail of Polyhedral study of the connected subgraph problem

Discrete Mathematics, 2015

ABSTRACT In this paper, we study the connected subgraph polytope which is the convex hull of the ... more ABSTRACT In this paper, we study the connected subgraph polytope which is the convex hull of the solutions to a related combinatorial optimization problem called the maximum weight connected subgraph problem. We strengthen a cut-based formulation by considering some new partition inequalities for which we give necessary and sufficient conditions to be facet defining. Based on the separation problem associated with these inequalities, we give a complete polyhedral characterization of the connected subgraph polytope on cycles and trees.

Research paper thumbnail of Polyhedral study of the connected subgraph problem

Discrete Mathematics, 2015

ABSTRACT In this paper, we study the connected subgraph polytope which is the convex hull of the ... more ABSTRACT In this paper, we study the connected subgraph polytope which is the convex hull of the solutions to a related combinatorial optimization problem called the maximum weight connected subgraph problem. We strengthen a cut-based formulation by considering some new partition inequalities for which we give necessary and sufficient conditions to be facet defining. Based on the separation problem associated with these inequalities, we give a complete polyhedral characterization of the connected subgraph polytope on cycles and trees.

Research paper thumbnail of Leontief flow problems: Integrality properties and strong extended formulations

This thesis has two main parts, both of which deal with Leontief directed hypergraphs (LDH\u27s),... more This thesis has two main parts, both of which deal with Leontief directed hypergraphs (LDH\u27s), a generalization of directed graphs where arcs have multiple (or no) tails and at most one head. Given an LDH with costs on its hyperarcs and with net demands on its vertices, the Leontief flow problem (LFP) is the problem of assigning flows on the hyperarcs at minimum cost and in such a way that net demands at all vertices are satisfied. In the first part of the thesis, we identify classes of Leontief flow problems, beyond ordinary network flow problems, that have integral optimal solutions. Specifically, we give necessary and sufficient conditions under which the associated vertex-hyperarc incidence matrices of classes of LDH\u27s are totally unimodular. We also characterize circuits of the underlying matroids for some subclasses of these LDH\u27s, one of which properly includes the class of graphic matroids. In the second part of the thesis, we consider the class of Uncapacitated Fixed Charge Network Flow problems, (UF)\u27s. These are single-source flow problems with fixed charge for setting up any arc in a specific subset, the usual unit flow costs for other arcs, and no capacities. As a mixed-integer linear programming problem, (UF) is difficult to solve; indeed, it is NP-Hard. This research develops a practical and concise way of modeling the Uncapacitated Fixed Charge Network Flow problem using (LFP)\u27s and exploiting the structures of Leontief directed hypergraphs. We show how an extended formulation, adding many new variables and constraints, yields far tighter LP-relaxations. Indeed, we give an axiomatic characterization of the best such extended formulations

Research paper thumbnail of Totally unimodular Leontief directed hypergraphs

Linear Algebra and its Applications, 1995

Research paper thumbnail of Commodity family extended formulations of uncapacitated fixed charge network flow problems

Networks, 1997

ABSTRACT Uncapacitated fixed charge network flow problems are single-commodity flow problems with... more ABSTRACT Uncapacitated fixed charge network flow problems are single-commodity flow problems with (positive) fixed charges for opening some arcs and no capacities. Previous research has shown that much improved linear programming relaxations can be obtained by reformulating these problems in terms of an extended variable set corresponding to flow commodities for each demand point. In this paper, we develop a theory of reformulations generalizing to families of commodities defined by arbitrary demand subsets. In particular, we show how to produce an extended formulation for any suitable commodity family and isolate simple axioms characterizing the families that yield the most useful reformulations. © 1997 John Wiley & Sons, Inc. Networks 30:57–71, 1997

Research paper thumbnail of A simplex algorithm for a class of leontief flow problems

Mathematical and Computer Modelling, 1996

A matrix N is hntief if it has exactly one positive entry per column and there exists a nonnegati... more A matrix N is hntief if it has exactly one positive entry per column and there exists a nonnegative x such that Nx > 0. A Lwntief flow problem is a linear-programming problem of the form min(cTx 1 Nx = b;x 2 0), where N is a certain type of Leontief matrix. It is shown that for b > 0 this problem can be solved in 0(n2U 1ognpU) pivots by the simplex method using Dantzig's rule for choosing the entering variable, where n is the number of variables, p is the largest entry of N in absolute value, and CT is a valid upper bound on any extreme-point solution. C~SSHS of problems where this is a strongly polynomial bound are identified.

Research paper thumbnail of A Decomposition of Copulas and Its Use

Communications in Statistics - Theory and Methods, 2005

ABSTRACT In this article, we create a decomposition that represents and describes the depen-dence... more ABSTRACT In this article, we create a decomposition that represents and describes the depen-dence structure between two variables. Since copulas provide a deep understanding of the dependence structure by eliminating the effects of the marginals, they play a key role in this study. We define a discretized copula density matrix and decompose it into a set of permutation matrices by using the Birkhoff–von Neumann theorem. This decomposition provides a way to effectively apply the concepts of copulas to solve problems in multivariate statistical data analysis.

Research paper thumbnail of Polyhedral study of the connected subgraph problem

Discrete Mathematics, 2015

ABSTRACT In this paper, we study the connected subgraph polytope which is the convex hull of the ... more ABSTRACT In this paper, we study the connected subgraph polytope which is the convex hull of the solutions to a related combinatorial optimization problem called the maximum weight connected subgraph problem. We strengthen a cut-based formulation by considering some new partition inequalities for which we give necessary and sufficient conditions to be facet defining. Based on the separation problem associated with these inequalities, we give a complete polyhedral characterization of the connected subgraph polytope on cycles and trees.

Research paper thumbnail of Polyhedral study of the connected subgraph problem

Discrete Mathematics, 2015

ABSTRACT In this paper, we study the connected subgraph polytope which is the convex hull of the ... more ABSTRACT In this paper, we study the connected subgraph polytope which is the convex hull of the solutions to a related combinatorial optimization problem called the maximum weight connected subgraph problem. We strengthen a cut-based formulation by considering some new partition inequalities for which we give necessary and sufficient conditions to be facet defining. Based on the separation problem associated with these inequalities, we give a complete polyhedral characterization of the connected subgraph polytope on cycles and trees.