Jack Edmonds

Paths, Trees, and Flowers

A graph G for purposes here is a finite set of elements called vertices and a finite set of elements called edges such that each edge meets exactly two vertices, called the end-points of the edge. An edge is said to join its end-points.

MATCHING, EULER TOURS AND THE CHINESE POSTMAN

The solution of the Chinese postman problem using matching theory is given. The convex hull of integer solutions is described as a linear programming polyhedron. This polyhedron is used to show that a good algorithm gives an optimum solution. The algorithm is a specialization of the more generalb-matching blossom algorithm. Algorithms for finding Euler tours and related problems are also discussed.

Matroids and the greedy algorithm

Linear-algebra rank is the solution to an especially tractable optimization problem. This tractability is viewed abstractly, and extended to certain more general optimization problems which are linear programs relative to certain derived polyhedra.

Matching: a well-solved class of integer linear programs

A main purpose of this work is to give a good algorithm for a certain well-described class of integer linear programming problems, called matching problems (or the matching problem). Methods developed for simple matching [2]

Total Dual Integrality of Linear Inequality Systems

Let A be a rational m × n matrix and b be a rational m-vector. The linear system Ax ≤ b is said to be totally dual integral (TDI) if for all integer n-vectors c, the dual of the linear program max{cx : Ax≤b} has an integer-valued optimum solution if it has an optimum solution. We consider two topics: First we discuss the theoretical aspects of total dual integrality. Second, we survey several classes of TDI linear systems and show how the general properties of total dual integrality appy to each class.

Reductions to 1–matching polyhedra

The matching polyhedron theorem of Edmonds and Johnson, which gives the convex hull of capacitated perfect b-matchings of a bidirected graph, is proved by reducing this matching problem to the ordinary perfect 1–matching problem, for which there exists a short inductive proof of the corresponding polyhedral theorem. The proof method makes it possible to deduce nestedness and discreteness properties of optimal dual solutions to the general matching problem from analogous properties of optimal dual solutions to the perfect 1–matching problem. In particular, the total dual half-integrality of the inequality system for general matching is shown to follow from that for 1–matching. Applications considered include determining the convex hull of unions of disjoint circuits of a graph.

Theoretical improvements in algorithmic efficiency for network flow problems

This paper presents new algorithms for the maximum flow problem, the Hitchcock transportation problem and the general minimum-cost flow problem. Upper bounds on the number of steps in these algorithms are derived, and are shown to improve on the upper bounds of earlier algorithms.

Coflow polyhedra

Cameron, K. and J. Edmonds, Coflow polyhedra, Discrete Mathematics 101 (1992) 1-21. A system L of linear inequalities in the variables x is called totally dual integral (TDI) if for every linear function cx such that c is all integers, the dual of the linear program: maximize {cx: x satisfies L} has an integer-valued optimum solution or no optimum solution. A linear system L is called box TDI if L together with any inequalities b G x s a is TDI. The main result of this paper is the Coflow Polyhedron Theorem: For any digraph G with node-set V(G), and for any fixed rational-valued d = (d,: IJ E V(G)), the following system of inequalities in variables x = (1,: u E V(G)) is box TDI. V dicircuit C of G, c {x,: v E C} <c {d,: v E C}. By Edmonds and Giles’ basic theorem on TDI systems, the Coflow Polyhedron Theorem provides many combinatorial min-max relations. In particular, it implies min-max theorems for the maximum weight union of k antichains and the maximum weight union of k chains in a poset. It allows us to prove that a class of graphs which generalize comparability graphs are perfect, and to prove a theorem which for acyclic digraphs generalizes the Gallai-Milgram Theorem. The proof of dual integrality of the inequality system described above uses network flow techniques. The proof of primal integrality then follows by the Edmonds-Giles Theorem, or can be proved directly using the concept of projection of a polyhedron.

The poset scheduling problem

Let P and Q be two finite posets and for each p∈P and q∈Q let c(p, q) be a specified (real-valued) cost. The poset scheduling problem is to find a function s: P→Q such that Σp∈Pc(p,s(p)) is minimized, subject to the constraints that p<p′ in P implies s(p)<s(p′) in Q. We prove that the poset scheduling problem is NP-hard. This problem with a totally ordered poset Q is proved to be transformable to the closed set problem or the minimum cut problem in a network.

Polyhedral polarity defined by a general bilinear inequality

LetX andY be finite dimensional vector spaces over the real numbers. LetΩ be a binary relation betweenX andY given by a bilinear inequality. TheΩ-polar of a subsetP ofX is the set of all elements ofY which are related byΩ to all elements ofP. TheΩ-polar of a subset ofY is defined similarly. TheΩ-polar of theΩ-polar ofP is called theΩ-closure ofP andP is calledΩ-closed ifP equals itsΩ-closure. We describe theΩ-polar of any finitely generated setP as the solution set of a finite system of linear inequalities and describe theΩ-closure ofP as a finitely generated set. TheΩ-closed polyhedra are characterized in terms of defining systems of linear inequalities and also in terms of the relationship of the polyhedronP with its recessional cone and with certain subsets ofX andY determined by the relationΩ. Six classes of bilinear inequalities are distinguished in the characterization ofΩ-closed polyhedra.