How would I write the flow of the following graph as a linear combination of flows along s,t-paths and t,s-paths and cycles? The values of the edges in the graph represent the flow along that edge.
I'm not sure how to actually express flow in terms of a linear combination of a path/cycle. Would it be something like...
$sat=2.5+.5$
$sact=2.5+3+3$
$sbat=1+1+.5$
... for the paths, and...
$sats=2.5+0.5+2$
$sbacs=1+1+3+1$
... for the cycles?
Thanks,
Hristo
Each of the basic flows you're using should conserve flow at nodes other than s and t. For convenience, write e.g [acba] as a flow of 1 unit on the cycle $a \to b \to c \to a$, and [sat] as 1 unit on $s \to a \to t$. So you could end up with something like $\frac{3}{2} [sat] + 2 [acba] + [abta]$ (just for illustration; that's not the answer here)