# Polynomial time analysis of toroidal periodic graphs

## Abstract

A toroidal periodic graph *G*^{ α } is defined by a positive integer vector *α* and a directed graph *G* in which the edges are associated with integer vectors. *G*^{ α } has a vertex (*v, y*) for each vertex *v* of *G* and each integer vector \(\vec 0 \le y < \alpha \). *G*^{ α } has an edge from (*v, y*) to (*w, z*) if and only if *G* has an edge from *v* to *w* associated with *t*, and *z = y+t* mod *α.*

We show that path problems for toroidal periodic graphs *G*^{ α } can be solved in polynomial time if *G* has a constant number of strongly connected components. The general path problem in toroidal periodic graphs is shown to be NP-complete for all \(\alpha \ge \vec 2\). Additionally, we present a procedure for determining the number of strongly connected components in a toroidal periodic graph. This procedure takes polynomial time for all instances *G* and *α*.

The introduced methods are very general and can also be used to solve further graph problems in polynomial time on even more general toroidal periodic graphs.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]W. Backes, U. Schwiegelshohn, and L. Thiele. Analysis of free schedule in periodic graphs. In
*Proceedings of the Annual ACM Symposium on Parallel Algorithms and Architectures*, pages 333–343. ACM, 1992.Google Scholar - [2]I. Borosch and B. Treybig. Bounds on positive integral solutions of linear diophantine equations. In
*Proceedings of Amer. Math. Soc.*, volume 55, pages 299–304, 1976.Google Scholar - [3]E. Cohen and N. Megiddo. Recognizing properties of periodic graphs. In P. Gritzmann and B. Sturmfels, editors,
*Applied Geometry and Discrete Mathematics. The Victor Klee Festschrift*, volume 4, pages 135–146. ACM, 1991.Google Scholar - [4]E. Cohen and N. Megiddo. Strongly polynomial-time and NC algorithms for detecting cycles in dynamic graphs.
*Journal of the ACM*, 40(4):791–830, 1993.Google Scholar - [5]M.R. Garey and D.S. Johnson.
*Computers and Intractability, A Guide to the Theory of NP-Completeness*. W.H. Freeman and Company, San Francisco, 1979.Google Scholar - [6]F. Höfting and E. Wanke. Polynomial algorithms for minimum cost paths in periodic graphs. In
*Proceedings of the ACM-SIAM Symposium on Discrete Algorithms*, pages 493–499, 1993. To appear in SIAM Journal on Computing.Google Scholar - [7]K. Iwano and K. Steiglitz. Testing for cycles in infinite graphs with periodic structure. In
*Proceedings of the Annual ACM Symposium on Theory of Computing*, pages 46–55. ACM, 1987.Google Scholar - [8]R.M. Karp, R.E. Miller, and A. Winograd. The organization of computations for uniform recurrence equations.
*Journal of the ACM*, 14(3):563–590, 1967.Google Scholar - [9]M. Kodialam and J.B. Orlin. Recognizing strong connectivity in (dynamic) periodic graphs and its relation to integer programming. In
*Proceedings of the ACM-SIAM Symposium on Discrete Algorithms*, pages 131–135. ACM-SIAM, 1991.Google Scholar - [10]S.R. Kosaraju and G.F. Sullivan. Detecting cycles in dynamic graphs in polynomial time. In
*Proceedings of the Annual ACM Symposium on Theory of Computing*, pages 398–406, 1988.Google Scholar - [11]C. Lengauer. Loop parallelization in the polytope model. In
*Proceedings of the International Conference on Concurrency Theory*, volume 715 of*LNCS*, pages 398–416. Springer-Verlag, 1993.Google Scholar - [12]G.L. Nemhauser and L.A. Wolsey.
*Integer and Combinatorial Optimization*. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1989.Google Scholar - [13]J.B. Orlin. Some problems on dynamic/periodic graphs. In W.R. Pulleyblank, editor,
*Progress in Combinatorial Optimization*, pages 273–293, Orlando, Florida, 1984. Academic Press.Google Scholar - [14]A. Schrijver.
*Theory of Linear and Integer Programming*. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1986.Google Scholar - [15]E. Wanke. Paths and cycles in finite periodic graphs. In A.M. Borzyszkowski and S. Sokolowski, editors,
*Proceedings of Mathematical Foundations of Computer Science*, volume 711 of*LNCS*, pages 751–760. Springer-Verlag, 1993.Google Scholar