# Graph theoretical methods for the design of parallel algorithms

## Abstract

We give an overview on graph decomposition techniques to obtain fast algorithms for optimization problems on graphs.

Based on the observation that most algorithmical problems can be solved easily on trees these methods try to represent a given graph *G* as a tree of subgraphs of *G*, its components. One aims to keep the size of these components and the intersection between different components small. When using such a decomposition to solve an optimization problem on *G* partial solutions on different components have to be combined efficiently to obtain a global solution for *G*. Logical and algebraic characterization of optimization problems have been developed for this purpose. That way, algorithmic solution strategies on decomposed graphs can be derived in a uniform way for various optimization problems. They turn out to be very efficient for classes of graphs that are highly decomposable.

Furthermore extending this approach slightly, fast parallel algorithms can be designed for these problems in a very similar and general way. For this purpose we first have to discuss parallel procedures for graph decomposition. Based on such a decomposition local solutions are computed in parallel and combined to a global one. It turns out that a parallel strategy for the combining process does not have to exploit specific properties of the optimization problems to be solved. The parallel evaluation of the decomposition tree can be formalized by a transitive operation on an appropriate state space of components. Even unbalanced decomposition trees can be evaluated fast by parallel tree contraction techniques.

## Preview

Unable to display preview. Download preview PDF.

## 5 References

- [A85]S. Arnborg,
*Efficient Algorithms for Combinatorial Problems on Graphs with Bounded Decomposability — A Survey,*BIT 25, 1985, 2–23.Google Scholar - [ACP87]S. Arnborg, D. Corneil, A. Proskurowski,
*Complexity of Finding Embeddings in a k-Tree,*SIAM J. Alg. and Disc. Meth. 8, 1987, 277–284.Google Scholar - [ADKP89]K. Abrahamson, N. Dadoun, D. Kirkpatrick, T. Przytycka,
*A Simple Parallel Tree Contraction Algorithm,*J. Alg. 10, 1989, 287–302.Google Scholar - [ALS88]S. Arnborg, J. Lagergren, D. Seese,
*Problems Easy for Tree Decomposable Graphs,*Proc. 15. ICALP, 1988, 38–51.Google Scholar - [AP89]S. Arnborg, A. Proskurowski,
*Linear Time Algorithms for NP-hard Problems Restricted to Partial k-Trees,*Disc. Appl. Mathematics 23, 1989, 11–24.Google Scholar - [B83]B. Baker,
*Approximation Algorithms for NP-Complete Problems on Planar Graphs,*Proc. 24. FoCS, 1983, 265–273.Google Scholar - [B88a]H. Bodlaender,
*Dynamic Programming on Graphs with Bounded Treewidth,*Proc. 15. ICALP, 1988, 105–118.Google Scholar - [B88b]H. Bodlaender,
*NC-Algorithms for Graphs with Small Treewidth,*Proc. 14. Workshop on Graphtheoretical Concepts in Computer Science, 1988, Springer Lec. Notes, 1–10.Google Scholar - [BLW87]M. Bern, E. Lawler, A. Wong,
*Linear Time Computation of Optimal Subgraphs of Decomposable Graphs*, J. Algorithms 8, 1987, 216–235.Google Scholar - [C90]B. Courcelle,
*The Monadic Second-Order Logic of Graphs I: Recognizable Sets of Finite Graphs*, Information and Computation 85, 1990, 12–75.Google Scholar - [CM90]B. Courcelle, M. Moshab,
*Monadic Second-Order Evaluations on Tree-Decomposable Graphs*, Technical Report 90–110, Universite de Bordeaux I, 1990.Google Scholar - [H89]W. Hohberg,
*Zerlegung von Graphen — ein allgemeines, sequentielles und paralleles Lösungsverfahren für Graphenprobleme*, Dissertation, TH Darmstadt, 1989.Google Scholar - [H90]W. Hohberg,
*The Decomposition of Graphs into k-Connected Components for Arbitrary κ*, Technical Report, TH Darmstadt, 1990.Google Scholar - [HR90]W. Hohberg, R. Reischuk,
*A Framework to Design Algorithms for Optimization Problems on Graphs*, Technical Report, TH Darmstadt, 1990.Google Scholar - [HY88]X. He, Y. Yesha,
*Binary Tree Algebraic Computation and Parallel Algorithms for Simple Graphs*, J. Alg. 9, 1988, 92–113.Google Scholar - [J85]D. Johnson,
*The NP-Completeness Column*, J. Algorithms 6, 1985, 434–451.Google Scholar - [L88a]C. Lautemann,
*Decomposition Trees: Structured Graph Representation and Efficient Algorithms*, Proc. CAAP 1988, Springer Lec. Notes 299, 28–39.Google Scholar - [L88b]C. Lautemann,
*Efficient Algorithms on Context-Free Graph Languages*, Proc. 15. ICALP, 1988, 362–378.Google Scholar - [L90a]C. Lautemann,
*Tree Automata, Tree Decomposition and Hpyeredge Replacement*, Technical Report, Universität Mainz, 1990.Google Scholar - [L90b]J. Lagergren,
*Efficient Parallel Algorithms for Tree-Decomposition and Related Problems*, Proc. 31. FoCS, 1990, 173–182.Google Scholar - [LW88]T. Lengauer, E. Wahnke,
*Efficient Solution of Connectivity Problems on Hierarchically Defined Graphs*, SIAM J. Comput. 17, 1988, 1063–1080.Google Scholar - [MR85]G. Miller, J. Reif,
*Parallel Tree Contraction and Its Application*, Proc. 26. FoCS, 1985, 478–489.Google Scholar - [R90]R. Reischuk,
*An Algebraic Divide-and-Conquer Approach to Design Highly Parallel Solution Strategies for Optimization Problems on Graphs*, Technical Report, TH Darmstadt, 1990.Google Scholar - [RS86]N. Robertson, P. Seymour,
*Graph Minors II. Algorithmic Aspects of Tree-Width*, J. Alg. 7, 1986, 309–322.Google Scholar - [TNS82]K. Kakamizawa, T. Nishizeki, N. Saito,
*Linear-Time Computability of Combinatorial Problems on Series-Parallel Graphs*, J. ACM 29, 1982, 623–641.Google Scholar