On the representability of totally unimodular matrices on bidirected graphs

We present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix B by finding node-edge incidence matrices Q and S such that QB=S. Seymour's famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called k-sums starting from network matrices and their transposes and two compact representation matrices B"1,B"2 of a certain ten element matroid. Given that B"1,B"2 are binet matrices we examine the k-sums of network and binet matrices. It is shown that thek-sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for k=2,3. A new class of matrices is introduced, the so-called tour matrices, which generalise network, binet and totally unimodular matrices. For any such matrix there exists a bidirected graph such that the columns represent a collection of closed tours in the graph. It is shown that tour matrices are closed under k-sums, as well as under pivoting and other elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the k-sum operation and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any TU matrix.

[1]  James G. Oxley,et al.  Matroid theory , 1992 .

[2]  Robert E. Bixby,et al.  Converting Linear Programs to Network Problems , 1980, Math. Oper. Res..

[3]  Thomas Zaslavsky,et al.  A simple algorithm that proves half-integrality of bidirected network programming , 2006, Networks.

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

[5]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[6]  W. T. Tutte.  An algorithm for determining whether a given binary matroid is graphic. , 1960 .

[7]  B. Kotnyek.  A generalization of totally unimodular and network matrices , 2002 .

[8]  Daniel C. Slilaty.  On cographic matroids and signed-graphic matroids , 2005, Discret. Math..

[9]  A. Musitelli,et al.  Recognition of generalized network matrices , 2008, 0807.3541.

[10]  Klaus Truemper,et al.  Matroid decomposition , 1992 .

[11]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[12]  Thomas Zaslavsky.  Signed graphs , 1982, Discret. Appl. Math..

[13]  Alan J. Hoffman,et al.  Integral Boundary Points of Convex Polyhedra , 2010, 50 Years of Integer Programming.

[14]  C. Kuratowski.  Sur le problème des courbes gauches en Topologie , 1930 .

[15]  W. T. Tutte.  Matroids and graphs , 1959 .

[16]  Balázs Kotnyek,et al.  A bidirected generalisation of network matrices , 2002 .

[17]  Balázs Kotnyek,et al.  A bidirected generalization of network matrices , 2006 .

[18]  W. T. Tutte.  Lectures on matroids , 1965 .

[19]  F. Harary.  On the notion of balance of a signed graph. , 1953 .

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

[21]  Paul D. Seymour,et al.  Decomposition of regular matroids , 1980, J. Comb. Theory, Ser. B.