Flows and Connectivity

Transcription

1 Chapter 4 Flows and Connectivit 4. Network Flows Capacit Networks A network N is a connected weighted loopless graph (G,w) with two specified vertices and, called the source and the sink, respectivel, denoted b N = (G,w). If w is a nonnegative capacit function c, then the network N = (G,c) is called a capacit network, and the value c is the capacit of the edge a. If c is an integer for an a E(G), then N is called an integral capacit network. We can, without loss of generalit, suppose that G is a simple digraph since we ma identif parallel edges with one edge whose weight is the sum of all weights on these edges if parallel edges eist. Figure 4. shows an integral capacit network Figure 4.: An integral capacit network N; an (, )-flow in N

2 Flows and Connectivit Applications of Capacit Networks Intuitivel, the capacit of an edge a = (u, z) ma be thought of as the maimum amount of some materials that can be transported along a from u to z per unit of time. For eample, the capacit of the edge a ma present the number of seats available on a direct flight from cit u to cit z in some airline sstem. One the other hand, this capacit might be the capacit of a pipeline from cit u to cit z in an oil network, or perhaps the maimum weight of items that can be transported b truck along a highwa from cit u to cit z. The problem in general, then, is to maimize the flow from the source to the sink without eceeding the capacit of the edges. We now introduce the notion of the flow. Flows and Maimum Flows Let N = (G,w) be a capacit network. A function f E (G) is called a flow in N from to, in short (, )-flow, if it satisfies the capacit constraint condition f c, a E(G), (4.) and the conservation condition f + (u) = f (u), u V (G) \ {, }. (4.) It is clear that there is at least one (, )-flow in ever capacit network, since the function f defined b f = for all a E(G), i.e., the zero flow, satisfies both (4.) and (4.) clearl. Figure 4. gives a nontrivial eample of a flow in the capacit network shown in Figure 4.. According to the condition (4.), it is easil verified that an (, )-flow f satisfies f + () f () = f () f + (). (4.) This common quantit is called the value of f, denoted b val f. For eample, the value of the flow f indicated in Figure 4. is 8, that is, val f = 8. An (, )-flow f in N is maimum if there is no (, )-flow f in N such that val f > val f.

3 4.. NETWORK FLOWS Cuts and Minimum Cuts An (, )-cut in N is a set of edges of the form (S, S), where S and S. The capacit of an (, )-cut B, denoted b cap B, is the sum of the capacities of edges in B, that is, capb = c(b) = a Bc. An (, )-cut B in N is called to be minimum, if there eists no (, )-cut B in N such that cap B < cap B. For eample, in the network of Figure 4., an (, )-cut B is indicated b the heav lines, and its capacit cap B = 8. Let f be an (, )-flow in N and B = (S, S) an (, )-cut. It is eas to prove that valf = capb f = { c, a (S, S);, a (S, S) Relationship between Maimum Flows and Minimum Cuts The maimum flow and the minimum cut are of obvious importance in the contet of transportation networks. We will, in Section 4.4, present an efficient algorithm for finding such flows and cuts in a given network. We now show an important and the best-known theorem on relationship between the maimum flow and the minimum cut, due to Ford and Fulkerson (956). Theorem 4. (ma-flow min-cut theorem) In an capacit network, the value of a maimum flow is equal to the capacit of a minimum cut. Proof: Let N = (G,c) be a capacit network, f be a maimum (, )-flow and B = (S, S) be a minimum (, )-cut in N. From the definition, for an u S, we have f + (u) f (u) = { val f, if u = ;, if u S \ {}. It is not difficult to show (the eercise 4..) that valf = f + () f () = u S(f + (u) f (u)) = f + (S) f (S) f + (S) c + (S) = cap B.

4 Flows and Connectivit (Note: f + (u) f + (S), u S Thus, it is sufficient to show that f (u) f (S).) u S valf capb. (4.4) To the end, we define a new digraph G obtained from G b adding a smmetric edge a for each edge a of G, where a is said an old edge and a a new edge. Define a function f E (G ) as follows: { c f, f = f, if a is old; if a is new. For eample, for the digraph G shown in Figure 4., the resulting digraph G according to this wa is shown in Figure 4., where edges indicated b curves are new, the digit nearb the edge a is f. 6 4 Figure 4.: G and f; H = G f Let H = G f, the support of f, for eample, in Figure 4., the digraph in is H corresponding to G and f in. We now claim that there eists no (, )-path in H. Suppose to the contrar that P is an (, )-path in H. Let σ = min{ f : a E(P)}. Then σ >. Define a function f E (G) as follows: For each a E(G), f + σ, if a E(P); f = f σ, if a E(P); f, otherwise. It is not difficult to show that f is an (, )-flow and val f = val f + σ > val f (the eercise 4..), which contradicts the maimalit of f. Therefore, there eists no (, )-path in H. Let

5 4.. NETWORK FLOWS S = {u V (H) : there eists an (, u)-path in H}. Then S and / S since there eists no (, )-path in H. This implies that B = (S, S ) is an (, )-cut in G. Thus, in G we have (the eercise 4..) {, for a (S f, S ); = f, for a (S, S ), (4.5) that is, in G we have This implies that f = { c, for a (S, S );, for a (S, S ). valf = f + (S ) f (S ) = f + (S ) = capb capb. The inequalit (4.4) is proved, and so the theorem follows. Corollar 4. In an integral capacit network, there must be an integral maimum flow, and its value is equal to the capacit of a minimum cut. The proof is left to the reader as an eercise (the eercise 4..4). The analtical method of network flow is an important one in graph theor. The ma-flow min-cut theorem, that is, Theorem 4. is not onl the base of network analsis b flows, but also of central importance in graph theor. We will later find that man well-known theorems on graph theor, in particular, Menger s theorem stated in the net section, can be deduced from it. In application part of this chapter, we will make use of network flow techniques to describe efficient algorithms for solving three classes of real-world problems.

6 4 Flows and Connectivit 4. Menger s Theorem In this section, we will present the best-known Menger s theorem in graph theor. To state this theorem we need some notation. Let and be two vertices of a graph G. We said (, )-paths P, P,,P n in G to be internall disjoint if V (P i ) V (P j ) = {, }, and edge-disjoint if E(P i ) E(P j ) = for an i and j with i j n. Edge Version of Menger s Theorem η G (, ): the maimum numbers of edge-disjoint (, )-paths in G λ G (, ): the minimum number of edges in an (, )-cut in G, which is call the local edge-connectivit of G. The following inequalit holds clearl η G (, ) λ G (, ) (4.6) as to destro all edge-disjoint (, )-paths in G we need to delete at least one edge from each of η G (, ) (, )-paths. We will show that the equalit in (4.6) alwas holds b making use of Corollar 4., due to Ford and Fulkerson (956), Elians, Feinstein and Shannon (956). Theorem 4. Let and be two distinct vertices in a graph G. Then η G (, ) = λ G (, ). Proof: B the inequalit (4.6), it is sufficient to prove the following inequalit: η G (, ) λ G (, ). (4.7) To the end, we consider a capacit network N = (G,c) with c for each a E(G). B Corollar 4., there eist a maimum (, )-integral flow f and a minimum (, )-cut B = (S, S) such that val f = cap B (see Figure 4., where B is depicted b the heav edges). Thus, it is clear that λ G (, ) B = capb = valf. In order to prove (4.7), therefore, we need to onl prove valf η G (, ).

7 4.. MENGER S THEOREM 5 Let H = G f, the support of f (see Figure 4. ). Since c for an a E(G),f for an a E(H). This implies that { d + H () d H () = valf = d H () d+ H (), d + H (u) = d H (u), u V (H) \ {, }. Therefore there are val f edge-disjoint (, )-paths in H (the eercise.8.). This means val f η G (, ) and, thus, the theorem follows. Figure 4.: A maimum (, )-flow f and a minimum (, )-cut B; H = G f We note that η G (, ) = η G (, ) ma be not, in general, alwas true for a digraph (see the eercise 4..). However, Lováz (97) obtained the following result, which gives another characterization of an eulerian digraph. if Corollar 4. A connected digraph G is eulerian if and onl η G (, ) = η G (, ),, V (G). Proof: The necessit is eas b Theorem 4., but the proof of the sufficienc is more difficult, and is omitted. Verte Version of Menger s Theorem A nonempt set S V (G) \ {, } is said to be an (, )-separating set in G if there eists no (, )-path in G S. S S Figure 4.4: An (, )-separating set in a digraph or an undirected graph Figure 4.4 illustrates an (, )-separating set in a digraph and an (, )-separating in an undirected graph.

8 6 Flows and Connectivit κ G (, ): the minimum cardinalit of an (, )-separating set in G, which is called the local (verte-)connectivit of G. ζ G (, ): the maimum numbers of internall disjoint (, )-paths in G. The following inequalit holds clearl ζ G (, ) κ G (, ) (4.8) as to destro all internall disjoint (, )-paths in G we need to delete at least one verte from each of ζ G (, ) (, )-paths. Menger (97) found that the equalit in (4.8) alwas holds. We will deduce it from Theorem 4.. w u w u r r u s s Figure 4.5: The split of a verte u of G The proof of this result needs a new operation on graphs, split of a verte. Let u V (G). The split of u is such an operation. First replace u b two new vertices u and u, and join them b an edge (u, u ), then replace each edge of G with head u b a new edge with head u, and each edge of G with tail u b a new edge with tail u. This operation is illustrated in Figure 4.5. u v u u v v w z w w z z Figure 4.6: G; H obtained from G b splitting all vertices ecept and Theorem 4. (Menger s theorem) Let and be two distinct vertices of G without edges from to. Then ζ G (, ) = κ G (, ). Proof: B the inequalit (4.8), we onl need to prove the inequalit ζ G (, ) κ G (, ). (4.9)

9 4.. MENGER S THEOREM 7 To the end, let H be the graph obtained from G b splitting all u V (G)\{, } (see Figure 4.6). B Theorem 4., we have η H (, ) = λ H (, ). B construction of H, it is not difficult to observe that η H (, ) edge-disjoint (, )-paths in H correspond η H (, ) internall disjoint (, )-paths in G obtained b contracting all edges of tpe (u, u ) (see Figure 4.6). It follows that ζ G (, ) η H (, ) = λ H (, ). Thus, we onl need to prove the inequalit λ H (, ) κ G (, ). Let B be an (, )-cut in H with B = λ H (, ). Then there eists S V (H) such that B = (S, S), and S and S. Let S be the set of the tails of edges in B. Then S B, and there is no (, )-path in H S. Let S be the set of the vertices in G obtained b contracting all edges of tpe (u, u ), where u, u S. Then S S, and there is no (, )-path in G S. It follows that κ G (, ) S S B = λ H (, ) as desired and so the theorem follows. Equivalence of Three Theorems Corollar 4. Theorem 4. Theorem 4.. Direct proofs of Theorem 4. and Theorem 4. Theorem 4. Theorem 4. Corollar 4.. Eercises: 4..; 4..5; 4..; 4..4; 4..5 Thank You!