Leontief flow problems: Integrality properties and strong extended formulations

Peh Hoon Ng, Purdue University

Abstract

This thesis has two main parts, both of which deal with Leontief directed hypergraphs (LDH's), 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's are totally unimodular. We also characterize circuits of the underlying matroids for some subclasses of these LDH's, 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)'s. 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)'s 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.

Degree

Ph.D.

Advisors

Rardin, Purdue University.

Subject Area

Operations research|Industrial engineering

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