Download PDF by Lawler E.L.: Combinatorial optimization: networks and matroids

By Lawler E.L.

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Each row of the node-arc incidence matrix is identified with a node and each column with an arc. If the arcs are numbered by the index k. then the incidence matrix 1: = (bik) is defined as follows: if node i is incident to arc k, b,, = 1 = 0 otherwise. 1 is 5\0 0 0 0 10 111 -----h-e P4rnbrilobvsv, Pi N” cc m’ 4 c*s< vvvvv Note that each column contains exactly two 1’s. In the case of a directed graph the arc (i, j). from i and incident to j. The arc-node incidence matrix B = (bik) is defined. as follows : bi, = + 1 if arc k is incident to node i =- 1 if arc k is incident from node i = otherwise.

In practice, degeneracy seldom results in circling. Moreover, there are several schemes for insuring that no basis is repeated, so that finite convergence is assured. Possibly the mose elegant of these involves a “lexicographic” condition which is incorporated into the ratio test. However, to describe this scheme would require more space than the issue deserves here. We should mention that nearly all of the linear programs formulated in later chapters are highly degenerate. Yet this creates no difliculty for the algorithms we shall describe.

Is planar, the? (G + e)D is obtained and the arc e*, dual to r. is by definition directed from s* to t*. Now note the relationship between GU and (G + e)” - e*. The addition of e to G simply subdivides IInto two parts some face F of G that has nodes s and t on its boundar,y. Hence, GD differs from (G + e)” - e* only in that the node in GD corresponding to F is split into two nodes s* and t*. 12. --b* t*. ‘12 (a) D graph G with terminals s, t. (b) Addition of (t, S) to G and dualization. (c) Dual digraph G” with terminals s*, 36 Mathematical Preliminaries By defining e* to be directed from :i* to t* rather than the opposite, we obtain the following results.

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