R. Gary Parker

Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families

This paper describes a predicate calculus in which graph problems can be expressed. Any problem possessing such an expression can be solved in linear time on any recursively constructed graph, once its decomposition tree is known. Moreover, the linear-time algorithm can be generatedautomatically from the expression, because all our theorems are proved constructively. The calculus is founded upon a short list of particularly primitive predicates, which in turn are combined by fundamental logical operations. This framework is rich enough to include the vast majority of known linear-time solvable problems.We have obtained these results independently of similar results by Courcelle [11], [12], through utilization of the framework of Bernet al. [6]. We believe our formalism is more practical for programmers who would implement the automatic generation machinery, and more readily understood by many theorists.

Guaranteed performance heuristics for the bottleneck travelling salesman problem

We consider constant-performance, polynomial-time, nonexact algorithms for the minimax or bottleneck version of the Travelling Salesman Problem. It is first shown that no such algorithm can exist for problems with arbitrary costs unless P = NP. However, when costs are positive and satisfy the triangle inequality, we use results pertaining to the squares of biconnected graphs to produce a polynomial-time algorithm with worst-case bound 2 and show further that, unless P = NP, no polynomial alternative can improve on this value.

On multiple Steiner subgraph problems

Recently, much attention has been given to the solvability of (otherwise intractable) combinatorial optimization problems when instances are confined to series-parallel graphs. Substantially less is known, however, regarding those that remain hard on these particularly sparse structures. While a few ad hoc cases have been shown to be difficult, little appears to be known in terms of more generic settings. In this paper, we contend with this state of affairs by establishing that a class of Steiner subgraph-like problems are NP-complete on series-parallel graphs while remaining easy (even trivial) on trees.

Deterministic decomposition of recursive graph classes

The popular class of series-parallel graphs can be built recursively from single edges by combining smaller components via connections only at a fixed pair of vertices called terminals. This recursive construction property with a limited number of terminals is essential to the linear time solution of problems on these graphs. A second useful property of these graphs is that decomposition is deterministic with respect to the series-parallel rules. This implies that the parse-tree of decomposition (which is required by the algorithms) can be determined in a straightforward manner by repeatedly applying the decomposition rules. Subject to retaining these properties, we examine how far the series-parallel graphs can be generalized. Corollaries of our results yield the deterministic decomposition of the series-parallel and Halin graph classes.

Solving problems on recursively constructed graphs

Fast algorithms can be created for many graph problems when instances are confined to classes of graphs that are recursively constructed. This article first describes some basic conceptual notions regarding the design of such fast algorithms, and then the coverage proceeds through several recursive graph classes. Specific classes include trees, series-parallel graphs, <i>k</i>-terminal graphs, treewidth-<i>k</i> graphs, <i>k</i>-trees, partial <i>k</i>-trees, <i>k</i>-jackknife graphs, pathwidth-<i>k</i> graphs, bandwidth-<i>k</i> graphs, cutwidth-<i>k</i> graphs, branchwidth-<i>k</i> graphs, Halin graphs, cographs, cliquewidth-<i>k</i> graphs, <i>k</i>-NLC graphs, <i>k</i>-HB graphs, and rankwidth-<i>k</i> graphs. The definition of each class is provided. Typical algorithms are applied to solve problems on instances of most classes. Relationships between the classes are also discussed.

On completing latin squares

Abstract In 1984, Colbourn proved that completing a partially filled latin square is NP -complete. In this paper, we tighten the Colbourn result by showing that completing a partially filled square remains hard even if no more than three unfilled cells exist in any row or column of the square and where only three integers are available.

An Overview of Complexity Theory in Discrete Optimizations: Part I. Concepts

Abstract Over the past decade, complexity theory has emerged from a branch of computer science almost unknown to the operations research community into a topic of widespread interest and research. The goals of the theory are to broadly classify problems and algorithms according to their convenience for solution by digital computers. Very considerable progress has been achieved, but some of the concepts in the theory are so subtle that their implications are as often misunderstood as grasped correctly. In this and the succeeding paper, we will present an elementary tutorial review of the important concepts and results in complexity theory. Emphasis is placed on constructs and implications for persons interested in discrete optimization—especially scheduling theory. The present Part I develops background concepts and definitions. Part II (in the June 1982 TRANSACTIONS) will cover results and implications.

Minimum-maximal matching in series-parallel graphs

Abstract Given an undirected graph G = (V, E), we seek a minimum weight subset of E which is a matching and which is maximal in this regard. Known as minimum-maximal matching, this problem is hard in general but can be solved in linear time if G is series-parallel. After a brief background discussion on series-parallel graphs, wer provide an algorithm which is applicable for arbitrary integer edge weights. The paper concludes with a short description of relevant issues regarding general, series-parallel solvability.

An efficiently solvable case of the minimum weight equivalent subgraph problem

Given a directed graph G(,), the problem of finding a minimum cardinality subset υ ⊆ such that the subgraph, G(,), preserves the reachability properties of G(,) is well known to be difficult. In this article, we consider a generalization which seeks a minimum weight subset υ satisfying the stated conditions where the weights of arcs in are assigned arbitrary integer values. A polynomial-time algorithm is given for the case where the underlying, undirected graph is series-parallel. Naturally, the stated algorithm subsumes the cardinality case on such graphs as well.

An Overview of Complexity Theory in Discrete Optimization: Part II. Results and Implications

Abstract Over the past decade, complexity theory has emerged from a branch of computer science almost unknown in the operations research community into a topic of widespread interest and research. Part I of this tutorial overview of the subject (TRANSACTIONS, March 1982) developed important background concepts of the theory. This paper uses that background to define and investigate the implications of NP-Hardness, NP-Completeness, NP-Equivalency, the NP≠NP conjecture, and various approximations.