Menger-decomposition of a graph and its application to the structural analysis of a large-scale system of equations

Research Trends in Combinatorial Optimization

“…The concepts of principal structure and principal partition have been effectively applied to systems analysis and examined in details in matrix and matroidal frameworks in (Iwata 1996;Murota 1995, 1996;Murota 1987Murota , 1990Murota et al 1987) (see Murota's book, Murota 2000). The concepts of principal structure and principal partition have been effectively applied to systems analysis and examined in details in matrix and matroidal frameworks in (Iwata 1996;Murota 1995, 1996;Murota 1987Murota , 1990Murota et al 1987) (see Murota's book, Murota 2000).…”

Structural solvability of systems of equations —A mathematical formulation for distinguishing accurate and inaccurate numbers in structural analysis of systems—

“…In the graph-theoretic method [15], [16], [20], [21], as employed in JUSE-GIFS and DPS, for the structural analysis of a system of equations (3.1), the following strong "generality assumption" is postulated" GA1 : The nonvanishing elements of D are algebraically independent over the rational number filed Q.…”

“…The Menger-decomposition [20], [21], [23] of the representation graph brings about a decomposition of the whole system (3.1) into structurally solvable subsystems (and non-solvable systems if any) with a partial order among them, justas the Dulmage- In the analysis of an electric network, as in Example 1, the system of equations in question takes the form of (2.2), where the entries of the lower submatrix F are assumed to be algebraically independent over the rational number field Q [11], [13], [14], [29]. Then, the coefficient matrix of (2.2) can be recognized asa special type of mixed matrix with (olo/ Q= o_Lo 0f~R -V-…”

“…For example, when the structural aspects of a large-scale discrete system are in question, even the naive * Part of this paper is published in a Japanese journal [25]. classification of the quantities into zeros and nonzeros sometimes turns out to be fruitful [15], [16], [17], [18], [20], [21], [34].…”

“…The notion of structural solvability in its crude form seems to have been first proposed in [16] along with the graph-theoretic criterion for checking it, and has successfully been incorporated in the chemical process simulators such as JUSE-GIFS and DPS developed in Japan [7], [8], [31], [37]. (The mathematical basis for the techniques employed in JUSE-GIFS and DPS for testing the structural consistency of systems of linear/nonlinear equations, as well as for decomposing the whole system into hierarchical subsystems, is developed in [15], [16], [20], [21], [23] by means of graph theory under the assumption that the nonvanishing numbers appearing in the system are so general that they are mutually independent in a certain algebraic sense, as will briefly be mentioned in Section 5 of this paper. )…”