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Minimum-Congestion Hypergraph Embedding in a Cycle
May 1997 (vol. 46 no. 5)
pp. 600-602

Abstract—The minimum-congestion hypergraph embedding in a cycle (MCHEC) problem is to embed the n edges in an m-vertex hypergraph as paths in a cycle on the same number of vertices, such that congestion—the maximum number of paths that use any single edge in the cycle—is minimized. The MCHEC problem has applications in electronic design automation and parallel computing. In this paper, it is proven that the MCHEC problem is NP-complete. An O((nm)k+1) algorithm is described that computes an embedding with congestion k or determines that such an embedding does not exist. Finally, a linear-time approximation algorithm for arbitrary instances is presented that computes an embedding whose congestion is at most three times optimal.

[1] A. Frank, T. Nishizeki, N. Saito, H. Suzuki, and É. Tardos, “Algorithms for Routing around a Rectangle,” Discrete Applied Math., vol. 40, pp. 363-378, 1992.
[2] T.F. Gonzalez and S.L. Lee, "A 1.6 Approximation Algorithm for Routing Multiterminal Nets," SIAM J. Computing, vol. 16, pp. 669-704, 1987.
[3] T.F. Gonzalez and S.L. Lee, "A Linear Time Algorithm for Optimal Routing Around a Rectangle," J. ACM, vol. 35, pp. 810-831, 1988.
[4] A.S. LaPaugh, "A Polynomial Time Algorithm for Optimal Routing Around a Rectangle," Proc. 21st Symp. Foundations of Computer Science, pp. 282-293, 1980.
[5] M. Sarrafzadeh and F.P. Preparata, "A Bottom-Up Layout Technique Based on Two-Rectangle Routing," Integration: The VLSI Journal, vol. 5, pp. 231-246, 1987.
[6] B.S. Baker and R.Y. Pinter, "An Algorithm for the Optimal Placement and Routing of a Circuit within a Ring of Pads," Proc. 24th Symp. Foundations of Computer Science, pp. 360-370, 1983.
[7] J.L. Ganley and J.P. Cohoon, "A Provably Good Moat Routing Algorithm," Proc. Sixth Great Lakes Symp. VLSI, pp. 86-91, 1996.
[8] R.K. McGehee, "A Practical Moat Router," Proc. 24th Design Automation Conf., pp. 216-222, 1987.
[9] D.C. Wang, "Pad Placement and Ring Routing for Custom Chip Layout," Proc. 27th Design Automation Conf., pp. 193-199, 1990.
[10] F.T. Leighton,Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes.San Mateo, Calif.: Morgan Kaufmann, 1992.
[11] H. Okamura and P.D. Seymour, "Multicommodity Flows in Planar Graphs," J. Combinatorial Theory, Series B, vol. 31, pp. 75-81, 1981.
[12] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness.New York: W.H. Freeman, 1979.
[13] J.L. Ganley, "Geometric Interconnection and Placement Algorithms," PhD thesis, Dept. of Computer Science, Univ. of Virginia, Charlottesville, Va., 1995.

Index Terms:
Hypergraph embedding in a cycle, congestion, NP-completeness, approximation algorithms, routing around a rectangle, moat routing, ring routing.
Joseph L. Ganley, James P. Cohoon, "Minimum-Congestion Hypergraph Embedding in a Cycle," IEEE Transactions on Computers, vol. 46, no. 5, pp. 600-602, May 1997, doi:10.1109/12.589233
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