Connectivity augmentation of networks: structures and algorithms

Abstract

On the spring of 1994, while preparing a survey-paper on graph-connectivity augmentation problems, I read and heard of several exciting questions and new results in the area, and had the idea of organizing a small workshop on connectivity augmentation. Thanks to a generousfinancial help fromOTKA, theHungarianNational Foundation for Scientific Research, and from the Hungarian Ministry of Education, whose support is hereby gratefully acknowledged, the authors of this volume were able to meet in Budapest at Eotvos University in November, 1994. We spent a fruitful week working together which led to extensions of earlier results and initiated several new research directions. This volume includes the product of the work that was either presented or stemmed from discussions at the workshop. Connectivity is a classic topic of combinatorial optimization and graph theory. Typically, one is interested in finding an optimal subgraph or supergraph of a graph (or digraph, or hypergraph) which has a specified connectivity property. (The supergraph version, that is, when the existing graph is to be augmented, is called the connectivity augmentation problem.) Menger’s theorem (or Max-Flow Min-Cut) and the greedy algorithm for minimumweight trees are two basic results. Somemore difficult theorems of the sixties and seventies, such as Tutte’s theorem on disjoint trees, Edmonds’s disjoint arborescence theorem, the Lucchesi-Younger theorem on making digraphs strongly connected, or Fulkerson’s algorithm to compute a cheapest arborescence, showed that the area contains deep structural results and polynomial time algorithms. One characteristic of almost all of these tractable cases is that they can be embedded into a framework related to total dual integrality (TDI-ness) of linear inequality systems. Not surprisingly, however, the majority of connectivity problems turned out to be NP-complete; the Hamiltonian circuit and the Steiner tree problems are two prime examples, and in the last two decades the main stream of research turned to finding approximation algorithms for these NP-complete problems. An interesting feature of connectivity augmentation is that the area is rich in new problems (potentially) belonging to NP∩co-NP or even to P. All of the papers of the present volume include structural characterizations and (exact) polynomial-time solution algorithms. Another important phenomenon is that suband supermodular functions (and their relaxations, like T -intersecting, crossing, and skew submodular ones) play a fundamental role in almost all papers. A third point to observe is that TDI-ness has no role in these problems: it is typical that the cardinality version of an augmentation