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EMIS 8374 Flows on Cycles Updated 18 March 2008. Example Max Flow Problem. (0,2). 2. 4. (0,4). (0,5). 1. 6. (0,6). (0,6). t. s. (0,6). 3. 5. (0,7). (0,5). A Feasible Flow with v = 3. (2,2). 2. 4. (0,4). (3,5). 1. 6. (5,6). (2,6). t. s. (0,6). 3. 5. (3,7). (5,5).
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Example Max Flow Problem (0,2) 2 4 (0,4) (0,5) 1 6 (0,6) (0,6) t s (0,6) 3 5 (0,7) (0,5)
A Feasible Flow with v = 3 (2,2) 2 4 (0,4) (3,5) 1 6 (5,6) (2,6) t s (0,6) 3 5 (3,7) (5,5)
Flow on a Cycle (2,2) 2 4 (0,4) (3,5) 1 6 (5,6) (2,6) t s (0,6) 3 5 (3,7) (5,5)
Reduce x23 from 5 to 3 out – in = -2 (2,2) 2 4 (0,4) (3,5) 1 6 (3,6) (2,6) t s (0,6) 3 5 (3,7) (5,5) out – in = 2
Reduce x35 from 5 to 3 out – in = -2 (2,2) 2 4 (0,4) (3,5) 1 6 (3,6) (2,6) t s (0,6) 3 5 (3,7) (3,5) out – in = 0 out – in = 2
Reduce x54 from 2 to 0 out – in = -2 out – in = 2 (2,2) 2 4 (0,4) (3,5) 1 6 (3,6) (0,6) t s (0,6) 3 5 (3,7) (3,5) out – in = 0 out – in = 0
Reduce x42 from 2 to 0 out – in = 0 out – in = 0 (0,2) 2 4 (0,4) (3,5) 1 6 (3,6) (0,6) t s (0,6) 3 5 (3,7) (3,5) out – in = 0 out – in = 0
Balanced Flow with v = 3 (0,2) 2 4 (0,4) (3,5) 1 6 (3,6) (0,6) t s (0,6) 3 5 (3,7) (3,5)
General Case • Let j be a node in a cycle and consider a feasible flow x. • The cycle contains exactly one arc (i, j) going into j and exactly one arc (j, k) going out of j. • Reducing xij and xjk by the same amount leaves node j balanced.
Example 2 1 3
Reducing the flow on the cycle by Claim: y is a feasible flow
Substitute y for x • xij = yij if (i, j) not in cycle • xij = yij +δif (i, j) in cycle
Substitute y for x • xij = yij if (i, j) not in cycle • xij = yij +δif (i, j) in cycle
Flow is Balanced Around the Cycle Since yij = xij for all arcs not in the cycle, y also satisfies the flow balance constraints for the nodes that are not in the cycle.
Conclusion • Reducing flow around a cycle yields another feasible flow • If the cycle doesn’t contain the source or the sink, then the value (v) of the flow isn’t changed • There always an acyclic maximum flow • An undirected edge {i, j} with net capacity uij can be represented by directed arcs (i, j) and (j, i) each with capacity uij