Fast Sequential and Randomised Parallel Algorithms for Rigidity and approximate Min k-cut

In this paper we use new techniques based on flows and matroid theory to produce fast sequential and randomised parallel algorithms for two important classes of problems. The first class arises in the study of rigidity of graphs (also in the study of graph realizations). The second class of problems may be called Principal Partition related problems. We take a representative of this class, viz, the min k-cut problem and produce an RNC algorithm which solves this NP — hard problem within twice the optimal.

[1]  Hassler Whitney,et al.  On the Classification of Graphs , 1933 .

[3]  Sachin B. Patkar,et al.  Principal Lattice of Partition of submodular functions on Graphs: Fast algorithms for Principal Partition and Generic Rigidity , 1992, ISAAC.

[4]  Sachin B. Patkar,et al.  A Fast Algorithm for the Principle Partition of a Graph , 1991, FSTTCS.

[5]  Y. Kajitani,et al.  Diagnosability of linear active networks , 1979 .

[6]  Masao Iri,et al.  Network theory and transversal matroids , 1980, Discret. Appl. Math..

[7]  K. Sugihara.  Machine interpretation of line drawings , 1986, MIT Press series in artificial intelligence.

[8]  杉原 厚吉,et al.  On Redundant Bracing in Plane Skeletal Structures , 1980 .

[9]  T. Idehara,et al.  MEMOIRS OF THE FACULTY OF ENGINEERING , 1977 .

[10]  András Frank,et al.  Generalized polymatroids and submodular flows , 1988, Math. Program..

[11]  H. Whitney.  On the Abstract Properties of Linear Dependence , 1935 .

[12]  H. Whitney.  Non-Separable and Planar Graphs. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[13]  Harold N. Gabow,et al.  Forests, frames, and games: algorithms for matroid sums and applications , 1988, STOC '88.

[14]  C. Nash-Williams.  Decomposition of Finite Graphs Into Forests , 1964 .

[15]  Kokichi Sugihara,et al.  An algebraic and combinatorial approach to the analysis of line drawings of polyhedra , 1984, Discret. Appl. Math..

[16]  H. Narayanan,et al.  The principal lattice of partitions of a submodular function , 1991 .

[17]  W. T. Tutte.  On the Problem of Decomposing a Graph into n Connected Factors , 1961 .

[18]  H. Imai.  NETWORK-FLOW ALGORITHMS FOR LOWER-TRUNCATED TRANSVERSAL POLYMATROIDS , 1983 .

[19]  Kokichi Sugihara.  A unifying approach to descriptive geometry and mechanisms , 1983, Discret. Appl. Math..

[20]  Vijay V. Vazirani,et al.  Finding k-cuts within twice the optimal , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[21]  Takao Ozawa,et al.  Topological conditions for the solvability of linear active networks , 1976 .

[22]  Vijay V. Vazirani,et al.  Randomized parallel algorithms for matroid union and intersection, with applications to arboresences and edge-disjoint spanning trees , 1992, SODA '92.

[23]  G. Laman.  On graphs and rigidity of plane skeletal structures , 1970 .

[24]  M. F.,et al.  Bibliography , 1985, Experimental Gerontology.

[25]  H. Narayanan,et al.  Fast loop matrix generation for hybrid analysis and a comparison of the sparsity of the loop impedance and MNA impedance submatrices , 1992, [Proceedings] 1992 IEEE International Symposium on Circuits and Systems.

[26]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[27]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[28]  R. Rado.  A THEOREM ON INDEPENDENCE RELATIONS , 1942 .

[29]  D. Sleator.  An 0 (nm log n) algorithm for maximum network flow , 1980 .

[30]  Bruce Hendrickson,et al.  Conditions for Unique Graph Realizations , 1992, SIAM J. Comput..

[31]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[32]  Hazel Perfect,et al.  Applications of Menger's graph theorem , 1968 .

[33]  C. Nash-Williams.  Edge-disjoint spanning trees of finite graphs , 1961 .

[34]  A. Recski.  Matroid theory and its applications in electric network theory and in statics , 1989 .

[35]  J. Stoer,et al.  Convexity and Optimization in Finite Dimensions I , 1970 .

[36]  R. V. Randow.  Introduction to the Theory of Matroids , 1975 .

[37]  Hazel Perfect,et al.  Independence Spaces and Combinatorial Problems , 1969 .

[38]  H. Narayanan,et al.  A new approach to the problem of PLA partitioning using the theory of the principal lattice of partitions of a submodular function , 1991, [1991] Proceedings Fourth Annual IEEE International ASIC Conference and Exhibit.

[39]  H. Watanabe,et al.  Topological degrees of freedom and mixed analysis of electrical networks , 1970 .

[40]  Masataka Nakamura.  ON THE REPRESENTATION OF THE RIGID SUB-SYSTEMS OF A PLANE LINK SYSTEM , 1986 .