Subexponential-Time Framework for Optimal Embeddings of Graphs in Integer Lattices

We present a general framework for computing various optimal embeddings of undirected and directed connected graphs in two and multi-dimensional integer lattices in time sub-exponential either in the minimum number n of lattice points used by such optimal embeddings or in the budget upper bound b on the number of lattice points that may be used in an embedding. The sub-exponential upper bounds in the two dimensional case and d-dimensional case are respectively of the form 2\(^{O(\sqrt{ln}log n)}\), 2\(^{O(\sqrt{lb}log b)}\) and 2\(^{O(dl^{1/d_n (d-1)/d}{\rm log} n)}\), 2\(^{O(dl^{1/d_b (d-1)/d}{\rm log} b)}\), where l stands for the degree of the allowed overlap. For the problem of minimum total edge length planar or multi-dimensional embedding or layout of a graph and the problem of an optimal protein folding in the so called HP model we obtain the upper bounds in terms of n. Note that in case of protein folding n is also the size of the input. The list of problems for which we can derive the upper bounds in terms of b includes among other things:

[1]  S. Vavasis,et al.  Geometric Separators for Finite-Element Meshes , 1998, SIAM J. Sci. Comput..

[2]  F. Leighton.  New lower bound techniques for VLSI , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[3]  Maurizio Patrignani,et al.  On the complexity of orthogonal compaction , 1999, Comput. Geom..

[4]  P. Raghavan.  Line and plane separators , 1993 .

[5]  Gary L. Miller,et al.  Separators for sphere-packings and nearest neighbor graphs , 1997, JACM.

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

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

[8]  Charles E. Leiserson,et al.  Area-efficient graph layouts , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[9]  Frank Wagner,et al.  The VLSI layout in various embedding models , 1990, WG.

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

[11]  Takao Nishizeki,et al.  Drawing Plane Graphs , 2003, ISAAC.

[12]  Mihalis Yannakakis,et al.  On the Complexity of Protein Folding , 1998, J. Comput. Biol..

[13]  Frank Thomson Leighton,et al.  Protein folding in the hydrophobic-hydrophilic (HP) is NP-complete , 1998, RECOMB '98.

[14]  K. Dill.  Theory for the folding and stability of globular proteins. , 1985, Biochemistry.

[15]  Mike Paterson,et al.  On the Complexity of String Folding , 1996, Discret. Appl. Math..

[16]  Charles E. Leiserson,et al.  Area-Efficient Graph Layouts (for VLSI) , 1980, FOCS.

[17]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[18]  Roberto Tamassia,et al.  On Embedding a Graph in the Grid with the Minimum Number of Bends , 1987, SIAM J. Comput..

[19]  Frank Thomson Leighton,et al.  Protein folding in the hydrophobic-hydrophilic (HP) is NP-complete , 1998, RECOMB '98.

[20]  Giorgio Gambosi,et al.  Complexity and Approximation , 1999, Springer Berlin Heidelberg.

[21]  Giorgio Gambosi,et al.  Complexity and approximation: combinatorial optimization problems and their approximability properties , 1999 .

[22]  David Eppstein,et al.  A deterministic linear time algorithm for geometric separators and its applications , 1993, SCG '93.

[23]  Jeffrey D Ullma.  Computational Aspects of VLSI , 1984 .

[24]  Uri Zwick,et al.  Spatial codes and the hardness of string folding problems , 1998, SODA '98.

[25]  James A. Storer,et al.  The node cost measure for embedding graphs on the planar grid (Extended Abstract) , 1980, STOC '80.

[26]  William E. Hart,et al.  Fast protein folding in the hydrophobic-hydrophilic model within three-eights of optimal , 1995, STOC '95.

[27]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.