# Finding solvable subsets of constraint graphs

Session 7a

First Online:

## Abstract

We present a network flow based, degree of freedom analysis for graphs that arise in geometric constraint systems. For a vertex and edge weighted constraint graph with m edges and n vertices, we give an *O*(*n*(*m* + *n*)) time max-flow based algorithm to isolate a subgraph that can be solved separately. Such a subgraph is called dense. If the constraint problem is not overconstrained, the subgraph will be minimal.

For certain overconstrained problems, finding minimal dense subgraphs may require up to *O*(*n*^{2} (*m* + *n*)) steps. Finding a minimum dense subgraph is NP-hard. The algorithm has been implemented and consistently outperforms a simple but fast, greedy algorithm.

## Keywords

Extremal subgraph dense graph network flow combinatorial optimization constraint solving geometric constraint graph geometric modeling## Preview

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## References

- 1.S. Ait-Aoudia, R. Jegou, and D. Michelucci. Reduction of constraint systems. In
*Compugraphics*, pages 83–92, 1993.Google Scholar - 2.W. Bouma, I. Fudos, C. Hoffmann, J. Cai, and R. Paige. A geometric constraint solver.
*Computer Aided Design*, 27:487–501, 1995.Google Scholar - 3.S. C. Chou, X. S. Gao, and J. Z. Zhang. A method of solving geometric constraints. Technical report, Wichita State University, Dept. of Computer Sci., 1996.Google Scholar
- 4.G. Crippen and T. Havel.
*Distance Geometry and Molecular Conformation*. John Wiley & Sons, 1988.Google Scholar - 5.S. Even and R. Tarjan. Network flow and testing graph connectivity.
*SIAM journal on computing*, 3:507–518, 1975.Google Scholar - 6.L.R. Ford and D.R. Fulkerson.
*Flows in Networks*. Princeton Univ. Press, 1962.Google Scholar - 7.I. Fudos.
*Geometric Constraint Solving*. PhD thesis, Purdue University, Dept of Computer Science, 1995.Google Scholar - 8.I. Fudos and C. M. Hoffmann. Constraint-based parametric conics for CAD.
*Computer Aided Design*, 28:91–100, 1996.Google Scholar - 9.I. Fudos and C. M. Hoffmann. Correctness proof of a geometric constraint solver.
*Intl. J. of Computational Geometry and Applications*, 6:405–420, 1996.Google Scholar - 10.I. Fudos and C. M. Hoffmann. A graph-constructive approach to solving systems of geometric constraints.
*ACM Trans on Graphics*, page in press, 1997.Google Scholar - 11.T. Havel. Some examples of the use of distances as coordinates for Euclidean geometry.
*J. of Symbolic Computation*, 11:579–594, 1991.Google Scholar - 12.C. M. Hoffmann. Solid modeling. In J. E. Goodman and J. O'Rourke, editors,
*CRC Handbook on Discrete and Computational Geometry*. CRC Press, Boca Raton, FL, 1997.Google Scholar - 13.C. M. Hoffmann and J. Peters. Geometric constraints for CAGD. In M. Daehlen, T. Lyche, and L. Schumaker, editors,
*Mathematical Methods fot Curves and Surfaces*, pages 237–254. Vanderbilt University Press, 1995.Google Scholar - 14.C. M. Hoffmann and J. Rossignac. A road map to solid modeling.
*IEEE Trans. Visualization and Comp. Graphics*, 2:3–10, 1996.Google Scholar - 15.Christoph M. Hoffmann and Pamela J. Vermeer. Geometric constraint solving in
*R*^{2}and*R*^{3}. In D. Z. Du and F. Hwang, editors,*Computing in Euclidean Geometry*. World Scientific Publishing, 1994. second edition.Google Scholar - 16.Christoph M. Hoffmann and Pamela J. Vermeer. A spatial constraint problem. In
*Workshop on Computational Kinematics*, France, 1995. INRIA Sophia-Antipolis.Google Scholar - 17.Ching-Yao Hsu.
*Graph-based approach for solving geometric constraint problems*. PhD thesis, University of Utah, Dept. of Comp. Sci., 1996.Google Scholar - 18.R. Latham and A. Middleditch. Connectivity analysis: a tool for processing geometric constraints.
*Computer Aided Design*, 28:917–928, 1996.Google Scholar - 19.E. Lawler.
*Combinatorial optimization, networks and Matroids*. Holt, Rinehart and Winston, 1976.Google Scholar - 20.J. Owen. Algebraic solution for geometry from dimensional constraints. In
*ACM Symp. Found. of Solid Modeling*, pages 397–407, Austin, Tex, 1991.Google Scholar - 21.J. Owen. Constraints on simple geometry in two and three dimensions. In
*Third SIAM Conference on Geometric Design*. SIAM, November 1993. To appear in Int J of Computational Geometry and Applications.Google Scholar - 22.T.L. Magnanti, R.K. Ahuja and J.B. Orlin.
*Network Flows*. Prentice-Hall, 1993.Google Scholar - 23.Dieter Roller. Dimension-Driven geometry in CAD: a Survey. In
*Theory and Practice of Geometric Modeling*, pages 509–523. Springer Verlag, 1989.Google Scholar - 24.O. E. Ruiz and P. M. Ferreira. Algebraic geometry and group theory in geometric constraint satisfaction for computer-aided design and assembly planning.
*IIE Transactions on Design and Manufacturing*, 28:281–294, 1996.Google Scholar - 25.P. Vermeer. Assembling objects through parts correlation. In
*Proc. 13th Symp on Comp Geometry*, Nice, France, 1997.Google Scholar - 26.W. Wunderlich. Starre, kippende, wackelige and bewegliche Achtflache.
*Elemente der Mathematik*, 20:25–48, 1965.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1997