networks - EE236A (Fall 2007-08) Lecture 8 Network...

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Unformatted text preview: EE236A (Fall 2007-08) Lecture 8 Network optimization network flows minimum cost network flow problem extreme flows shortest path problem bipartite matching 81 Networks network (directed graph): m nodes connected by n directed arcs arcs are ordered pairs ( i, j ) we assume there is at most one arc from node i to node j we assume there are no self-loops (arcs ( i, i ) ) arc-node incidence matrix A R m n : A ij = 1 arc j starts at node i- 1 arc j ends at node i otherwise column sums of A are zero: 1 T A = 0 reduced arc-node incidence matrix A R ( m- 1) n : the matrix formed by the first m- 1 rows of A Network optimization 82 example ( m = 6 , n = 8 ) 1 2 3 4 5 6 1 2 3 4 5 6 7 8 A = 1 1- 1- 1 1 1- 1- 1- 1 1 1 1- 1- 1 1- 1 Network optimization 83 Network flow flow vector x R n x j : flow (of material, traffic, charge, information, . . . ) through arc j positive if in direction of arc; negative otherwise total flow leaving node i : n summationdisplay j =1 A ij x j = ( Ax ) i i x j A ij =- 1 x k A ik = 1 Network optimization 84 External supply supply vector b R m b i : external supply at node i negative b i represents external demand from the network must satisfy 1 T b = 0 (total supply = total demand) i x j A ij =- 1 x k A ik = 1 b i balance equations: Ax = b reduced balance equations: Ax = ( b 1 , . . . , b m- 1 ) Network optimization 85 Minimum cost network flow problem...
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networks - EE236A (Fall 2007-08) Lecture 8 Network...

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