András Frank

An application of simultaneous Diophantine approximation in combinatorial optimization

We present a preprocessing algorithm to make certain polynomial time algorithms strongly polynomial time. The running time of some of the known combinatorial optimization algorithms depends on the size of the objective functionw. Our preprocessing algorithm replacesw by an integral valued-w whose size is polynomially bounded in the size of the combinatorial structure and which yields the same set of optimal solutions asw.As applications we show how existing polynomial time algorithms for finding the maximum weight clique in a perfect graph and for the minimum cost submodular flow problem can be made strongly polynomial.Further we apply the preprocessing technique to make H. W. Lenstra’s and R. Kannan’s Integer Linear Programming algorithms run in polynomial space. This also reduces the number of arithmetic operations used.The method relies on simultaneous Diophantine approximation.

Generalized polymatroids and submodular flows

Polyhedra related to matroids and sub- or supermodular functions play a central role in combinatorial optimization. The purpose of this paper is to present a unified treatment of the subject. The structure of generalized polymatroids and submodular flow systems is discussed in detail along with their close interrelation. In addition to providing several applications, we summarize many known results within this general framework.

An Algorithm for Submodular Functions on Graphs

A constructive method is described for proving the Edmonds-Giles theorem which yields a good algorithm provided that a fast subroutine is available for minimizing a submodular set function. The algorithm can be used for finding a maximum weight common independent set of two matroids, for finding a minimum weight covering of directed cuts of a digraph, and, as a new application, for finding a minimum cost k strongly connected orientation of an undirected graph. As a theoretical consequence of the algorithm, we prove a combinatorial feasibility theorem for Edmonds-Giles polyhedron and then we derive a discrete separation theorem which says, roughly, an integer valued submodular function B and an integer valued supermodular function R can be separated by an integer valued modular function provided that R ≤ B .

An application of submodular flows

Extending theorems of Rado and Lovasz, we introduce a new framework for problems concerning supermodular functions and graphs. Among the application is an optimization problem for finding a minimum-cost subgraph H of a digraph G=(V, E) such that H contains k disjoint paths from a fixed paths from a node of G to any other node. Another consequence is a characterization for graphs having a branching that meets all directed cuts. A theorem of Vidyasankar on optimal covering by arborescences and a matroid intersection theorem of Groflin and Hoffman are also shown to be special cases.

On chain and antichain families of a partially ordered set

Abstract A common generalization of the theorems of Greene and Greene and Kleitman is presented. This yields some insight into the relation of optimal chain and antichain families of a partially ordered set. The fundamental device is the minimal cost flow algorithm of Ford and Fulkerson.

Edge-disjoint paths in planar graphs

Given a planar graph G = (V, E), find k edge-disjoint paths in G connecting k pairs of terminals specified on the outer face of G. Generalizing earlier results of Okamura and Seymour (J. Combin. Theory Ser. B 31 (1981), 75–81) and of the author (Combinatorica 2, No. 4 (1982), 361–371), we solve this problem when each node of G not on the outer face has even degree. The solution involves a good characterization for the solvability and the proof gives rise to an algorithm of complexity O(|V|3log|V|). In particular, the integral multicommodity flow problem is proved to belong to the problem class P when the underlying graph is outerplanar.

Disjoint paths in a rectilinear grid

We give a good characterization and a good algorithm for a special case of the integral multicommodity flow problem when the graph is defined by a rectangle on a rectilinear grid. The problem was raised by engineers motivated by some basic questions of constructing printed circuit boards.

On decomposing a hypergraph into k connected sub-hypergraphs

By applying the matroid partition theorem of J. Edmonds (J. Res. Nat. Bur. Standards Sect. B 69 (1965) 67) to a hypergraphic generalization of graphic matroids, due to Lorea (Cahiers Centre Etudes Rech. Oper. 17 (1975) 289), we obtain a generalization of Tuttes disjoint trees theorem for hypergraphs. As a corollary, we prove for positive integers k and q that every (kq)-edge-connected hypergraph of rank q can be decomposed into k connected sub-hypergraphs, a well-known result for q = 2. Another by-product is a connectivity-type sufficient condition for the existence of k edge-disjoint Steiner trees in a bipartite graph.

On the Orientation of Graphs

Let G(V, E) be a finite, undirected graph, and let l(X) be a set function on 2V. When can the edges of G be oriented so that the indegree of every subset X is at least l(X)? A necessary and sufficient condition is given for the existence of such an orientation when l(X) is “convex”.

Algorithms for routing around a rectangle

Abstract Simple efficient algorithms are given for three routing problems around a rectangle. The algorithms find routing in two or three layers for two-terminal nets specified on the sides of a rectangle. All algorithms run in linear time. One of the three routing problems is the minimum area routing previously considered by LaPaugh and Gonzalez and Lee. The algorithms they developed run in time O(n3) and O(n) respectively. Our simple linear time algorithm is based on a theorem of Okamura and Seymour and on a data structure developed by Suzuki, Ishiguro and Nishizeki.