Algorithm Design

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Chapter 7 Network Flow
Figure 7.13 (a) Aninstance of the Circulation Problem together with a solution: Numbers
inside the nodes are demands; numbers labeling the edges are capadties and flow
values, with the flow values inside boxes. (b) The result of reducing this instance to an
equivalent instance of the Maximum-Flow Problem.
In this setting, we say that a circulation with demands {d,} is a function f
that assigns a nonnegative real number to each edge and satisfies the following
two conditions.
O) (Capacity conditions) For each e E E, we have 0 < f(e) < Ce.
(ii) (Demand conditions) For each u ~ V, we have u, fin(u) - f°ut(u) = du.
Now, instead of considering a maximization problem, we are concerned with
a feasibility problem: We want to know whether there exists a circulation that
meets conditions (i) and (ii).
For example, consider the instance in Figure 7.13(a). Two of the nodes
are sources, with demands -3 and -3; and two of the nodes are sinks, with
demands 2 and 4. The flow values in the figure constitute a feasible circulation,
indicating how all demands can be satisfied while respecting the capacities.
If we consider an arbitrary instance of the Circulation Problem, here is a
simple condition that must hold in order for a feasible circulation to exist: The
total supply must equal the total demand.
(7.49) If there exists a feasible circulation with demands {du}, then ~u du = O.
Proof. Suppose there exists a feasible circulation f in this setting. Then
~-~.v du = ~u f in(u) - f °ut(u)" Now, in this latter expression, the value f(e) for
each edge e = (u, u) is counted exactly twice: once in f°Ut(u) and once in fin(u).
These two terms cance! out; and since this holds for all values f(e), the overall
sum is 0. []
supplies source~
ith flow.
7.7 Extensions to the Maximum-Flow Problem
o t* siphons flow~
ut of sinks. J
Figure 7.14 Reducing the Circulation Problem to the Maximum-Flow Problem.
Thanks to (7.49), we know that
E
v:dv>O v:du 0--we add
an edge (u, t*) with capacity du. For each node u ~ S--that is, each node with
d u < 0--we add an edge (s*, u) with capacity -du. We carry the remaining
structure of G over to G’ unchanged.
In this graph G’, we will be seeking a maximum s*-t* flow. Intuitively,
we can think of this reduction as introducing a node s* that "supplies" all the
sources with their extra flow, and a node t* that "siphons" the extra flow out
of the sinks. For example, part (b) of Figure 7.13 shows the result of applying
this reduction to the instance in part (a).
Note that there cannot be an s*-t* flow in G’ of value greater than D, since
the cut (A, B) with A = {s*} only has capacity D. Now, if there is a feasible
circulation f with demands {du} in G, then by sending a flow value of -du on
each edge (s*, u), and a flow value of du on each edge (u, t*), we obtain an s*t*
flow in G’ of value D, and so this is a maximum flow. Conversely, suppose
there is a (maximum) s*-t* flow in G’ of value D. It must be that every edge
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