Partitioning Graphs with Costs and Weights on Vertices: Algorithms and Applications

We prove separator theorems in which the size of the separator is minimized with respect to non-negative vertex costs. We show that for any planar graph G there exists a vertex separator of total vertex cost at most c q P v2V (G) (cost(v)) 2 and that this bound is optimal within a constant factor. Moreover such a separator can be found in linear time. This theorem implies a variety of other separation results discussed in the paper. We describe application of our separator theorems to graph embedding problems, graph pebbling, and multi{ commodity ow problems.

[1]  Arnold L. Rosenberg,et al.  Optimal Simulations by Butterfly Networks (Preliminary Version) , 1988, STOC 1988.

[2]  Frank Thomson Leighton,et al.  An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[3]  John R Gilbert,et al.  A Separator Theorem for Graphs of Bounded Genus , 1984, J. Algorithms.

[4]  Fillia Makedon,et al.  Fast Approximation Algorithms for Multicommodity Flow Problems , 1995, J. Comput. Syst. Sci..

[5]  Michael T. Goodrich,et al.  Planar separators and parallel polygon triangulation (preliminary version) , 1992, STOC '92.

[6]  Arnold L. Rosenberg,et al.  Optimal Simulations of Tree Machines (Preliminary Version) , 1986, FOCS 1986.

[7]  Samir Khuller,et al.  Designing Multi-Commodity Flow Trees , 1994, Inf. Process. Lett..

[8]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

[9]  Gary L. Miller,et al.  A unified geometric approach to graph separators , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[10]  Robert E. Tarjan,et al.  Applications of a planar separator theorem , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[11]  Gary L. Miller,et al.  Separators in two and three dimensions , 1990, STOC '90.

[12]  D. Rose,et al.  Generalized nested dissection , 1977 .

[13]  Hristo Djidjev,et al.  Reduced constants for simple cycle graph separation , 1997, Acta Informatica.

[14]  Nicholas Pippenger,et al.  Advances in Pebbling (Preliminary Version) , 1982, ICALP.

[15]  Philip N. Klein,et al.  Faster Shortest-Path Algorithms for Planar Graphs , 1997, J. Comput. Syst. Sci..

[16]  S. Teng.  Points, spheres, and separators: a unified geometric approach to graph partitioning , 1992 .

[17]  Ivan Hal Sudborough,et al.  Comparing Interconnection Networks , 1988, MFCS.

[18]  Frank Thomson Leighton,et al.  A Framework for Solving VLSI Graph Layout Problems , 1983, J. Comput. Syst. Sci..

[19]  Gary L. Miller,et al.  Finding Small Simple Cycle Separators for 2-Connected Planar Graphs , 1986, J. Comput. Syst. Sci..

[20]  Krzysztof Diks,et al.  Edge Separators of Planar and Outerplanar Graphs with Applications , 1993, J. Algorithms.

[21]  Charles E. Leiserson,et al.  Area-Efficient VLSI Computation , 1983 .

[22]  D. Rose,et al.  A Separator Theorem for Chordal Graphs , 1982 .

[23]  Greg N. Frederickson,et al.  Fast Algorithms for Shortest Paths in Planar Graphs, with Applications , 1987, SIAM J. Comput..

[24]  R. Tarjan,et al.  The analysis of a nested dissection algorithm , 1987 .

[25]  Robin Thomas,et al.  A separator theorem for graphs with an excluded minor and its applications , 1990, STOC '90.

[26]  Gary L. Miller,et al.  Planar Separators and the Euclidean Norm , 1990, SIGAL International Symposium on Algorithms.

[27]  Robin Thomas,et al.  Call routing and the ratcatcher , 1994, Comb..

[28]  Martin Tompa,et al.  A New Pebble Game that Characterizes Parallel Complexity Classes , 1986, FOCS.