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Flows and Cuts
Considering the applications suggested at the beginning of the chapter, it is natural to ask for the maximum value of a flow in a given network.
Flows and Cuts
Considering the applications suggested at the beginning of the chapter, it is natural to ask for the maximum value of a flow in a given network. Put another way, we want to find the largest number \(v_0\) so that there exists a flow \(\phi\) of value \(v_0\) in the network. Of course, we not only want to find the maximum value \(v_0\), but we also want to find a flow \(\phi\) having this value. Although it may seem a bit surprising, we will develop an efficient algorithm which both finds a flow of maximum value and finds a certificate verifying the claim of optimality. This certificate makes use of the following important concept.
A partition \(V=L\cup U\) of the vertex set \(V\) of a network with \(S\in L\) and \(T\in U\) is called a cut.Our choice of \(L\) and \(U\) for the names of the two parts of the partition will make more sense later in the chapter. The capacity of a cut \(V=L\cup U\), denoted \(c(L,U)\), is defined by \[\begin{aligned}\end{aligned}\] Put another way, the capacity of the cut \(V=L\cup U\) is the total capacity of all edges from \(L\) to \(U\). Note that in computing the capacity of the cut \(V=L\cup U\), we only add the capacities of the edges from \(L\) to \(U\). We do not include the edges from \(U\) to \(L\) in this sum.
Example
Let's again take a look at the network in . Let's first consider the cut \(V=L_1\cup U_1\) with \[\begin{aligned}\end{aligned}\] Here we see that the capacity of the cut is \[\begin{aligned}\end{aligned}\] We must be a bit more careful, however, when we look at the cut \(V=L_2\cup U_2\) with \[\begin{aligned}\end{aligned}\] Here the capacity of the cut is \[\begin{aligned}\end{aligned}\] Notice that we do not include \(c(D,B)\) in the calculation as the directed edge \((D,B)\) is from \(U_2\) to \(L_2\).
The relationship between flows and cuts rests on the following fundamentally important theorem.
Condensed: the full section is in Keller & Trotter, Applied Combinatorics.
Sasa Hakuna kifaa cha kupigia hesabu kinachoweza kufanya hesabu, lakini kifaa hicho kinaweza kutumiwa katika hesabu.
Ishara zinazotumiwa hapa
Vaa alama yoyote ya ufafanuzi kamili, picha, na maana ya kila herufi.
Maswali ambayo watu huuliza
Permutation or combination?
Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).
What is a graph in this sense?
Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.
Sehemu za ukurasa huu zimebadilishwa kutoka kwa Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Tumekamatwa na kukosolewa tena hapa; makosa ni yetu.
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