Polynomial time analysis of toroidal periodic graphs
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.
Unable to display preview. Download preview PDF.
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- A. Schrijver. Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1986.Google Scholar
- 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