Connectivity and Menger s theorems

Transcription

1 Connectiity and Menger s theorems We hae seen a measre of connectiity that is based on inlnerability to deletions (be it tcs or edges). There is another reasonable measre of connectiity based on the mltiplicity of alternatie paths. We will see (in Menger s thms) that these two notions are, in fact, the same. G 1 G 2 G 3 G 4 G 1 : 1 niqe path between and G 2 : 4 paths between and G 3 : 10 paths between and G 4 : many paths between and Definition Two paths from to are internally disjoint if they hae no common internal ertex. The paths are edge disjoint if they hae no common edge. G 1 G 2 G 3 G 4 κ(g 1 )=1 κ (G 1 )=1 κ(g 2 )=1 κ (G 2 )=2 κ(g 3 )=3 κ (G 3 )=3 κ(g 4 )=4 κ (G 4 )=4 Between and, we hae... G 1 : 1 internally disjoint path; 1 edge disjoint path Math Prof. Kindred - Lectre 10 Page 1

2 G 2 : 1 internally disjoint path; 2 edge disjoint paths G 3 : 3 internally disjoint paths; 3 edge disjoint paths G 4 : 4 internally disjoint path; 4 edge disjoint paths Theorem (Menger - ertex, ndirected, global ersion). A graph G is k-connected if and only if eery pair of ertices are joined by k pairwise internally disjoint paths. Theorem (Menger - edge, ndirected, global ersion). A graph G is k-edge-connected if and only if eery pair of ertices are joined by k pairwise edge disjoint paths. There are seeral ersions of Menger s theorem, all of which can be deried from the max-flow, min-ct theorem. Directed graphs and network flows Definitions In a directed graph, or digraph, each edge has a direction: (, ) is different than (, ). So a digraph is a graph whose edges are ordered pairs of ertices. Directed edges are often called arcs. For edge (, ), is called the tail and is called the head. We define the in-degree and ot-degree of a ertex as in-degree of = d () = # of edges with head = {(, ) : V (G)} ot-degree of = d + () = # of edges with tail = {(, ) : V (G)} Math Prof. Kindred - Lectre 10 Page 2

3 Definitions A network is a digraph G together with a capacity fnction c : E(G) R 0 and two distingished tcs s, t V (G) known as the sorce ertex and the sink ertex. Definition A feasible flow in a network is a fnction f : E(G) R 0 sch that (a) 0 f(e) c(e) for all directed edges e. [capacity constraints] (b) For all tcs V (G) {s, t}, f () = f(, ) }{{} flow in [conseration constraints] The flow ale is f + (s) f (s). = f(, w) = f + (). w }{{} flow ot sorce ertex flow on an edge 3/3 a 1/10 capacity of an edge b sink ertex 4/4 s 2/2 3/10 t 5/5 4/4 x 7/7 tail of (x, y) head of (x, y) y flow ale = f + (s) - f - (s) = 8-0 = 8 Figre 1: A network is shown, along with a feasible flow on the network. Max flow problem The max flow problem is to find a feasible flow f with the maximm flow ale. Math Prof. Kindred - Lectre 10 Page 3

4 Definition A network ct, or an s t ct, is an edge ct [S, S] where s S, t S. The capacity of a network ct [S, S] is the sm of the capacities of the edges in the ct, cap[s, S] = c(e). Min ct problem The min ct problem is to find an s t ct with the minimm possible capacity. Lemma. For any feasible flow f and s t ct [S, S] in a network G, we hae flow ale = f(e) f(e). Proof. flow ale = f + (s) f (s) = f + (s) f (s) + = S f + () S Let e = (, ) be any edge in G. Then S {s} f (). if, S, then f(e) is not conted at all; (f + () f ()) }{{} =0 if S, S, then f(e) is conted once in the first sm bt not in the second sm; if S, S, then f(e) is conted once in the second sm bt not in the first sm; and if, S, then f(e) is conted once in the first sm and once in the second sm, so these two terms cancel. Math Prof. Kindred - Lectre 10 Page 4 ( )

5 Ths, we can express ( ) as flow ale = S f + () S f () = f(e) f(e). Max flow and min ct problems are dal optimization problems. (Recall that min ertex coer and max matching problems were dals to one another). Theorem (Weak dality). If f is a feasible flow and [S, S] is an s t- ct, then flow ale of f cap[s, S]. Proof. flow ale of f = f(e) f(e) c(e) = cap[s, S]. f(e) since f nonnegatie by preios lemma by capacity constraints of f Remark In particlar, max flow ale min ct capacity. Certificate of optimality: If f is a flow and [S, S] is an s t ct sch that flow ale of f = cap[s, S], then both the flow and the ct mst be optimal. Theorem (max flow, min ct strong dality). Let G be a network. The maximm ale of a flow eqals the minimm capacity of a ct. Math Prof. Kindred - Lectre 10 Page 5

6 We proe this strong dality reslt ia the Ford-Flkerson algorithm. Basics of Ford-Flkerson algorithm (1) Start with a zero flow eerywhere. (2) Form residal network. residal capacities: r(e) = c(e) f(e) + f(e 1 ) measres how mch we can increase flow along an arc or irtally do so (by decreasing flow along reerse arc) (3) Agment flow along s t path P in residal network by min e E(P ) r(e). (4) Go to step 2 and repeat. Contine repeating ntil there is no s t path in residal network. Ford-Flkerson algorithm retrns a feasible flow f and an s t ct [S, S] sch that the flow ale of f is eqal to the capacity of the s t ct, thereby proing the optimality of both. Lemma. If a flow f has a residal network with an agmenting path P, then f is not maximm. Frthermore, there is another flow f whose ale is r(p ) greater than the ale of f. Otline of proof. Form a new flow by agmenting along path P. Check that the capacity constraint holds, namely the flow on eery arc is between 0 and capacity. Check that flow conseration constraint holds for new flow. Math Prof. Kindred - Lectre 10 Page 6