# 12 - Algorithms Demands and Bounds Demands and Bounds...

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Algorithms – Demands and Bounds Demands and Bounds Design and Analysis of Algorithms Andrei Bulatov

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Algorithms – Demands and Bounds 12-2 Demands In our flow network model there is only 1 source and sink Can we do something for several sources and sinks? More general: Every vertex v has a demand If demand is positive, the vertex requires that the amount of flow tering the vertex is greater than the outgoing flow v d entering the vertex is greater If demand is negative, it requires that the amount of outgoing flow is greater. It provides supply. -3 -3 4 2 3 3 2 2 2
Algorithms – Demands and Bounds 12-3 Circulation with Demands Let S be the set of nodes with negative demand (sources) Let T be the set of nodes with positive demand (sinks) A circulation with demands is a function f that assigns a nonnegative real numbers to each arc: apacity condition) For each e we have } { v d (Capacity condition) E (Demand condition) For each v V, we have, The Circulation with Demands Problem Instance : A digraph G, capacities and demands Objective : Does there exist a circulation satisfying the Capacity and Demand conditions -3 -3 4 2 3 3 2 2 1 2 2 2 2 2 e c e f ) ( 0 v out in d v f v f = - ) ( ) (

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Algorithms – Demands and Bounds 12-4 Sum of Demands Lemma If there exists a feasible circulation with demands then roof } { v d 0 = V v v d Proof Suppose f is a feasible circulation Then In this expression the value f(e) for each e E counted exactly twice, once positively, and once negatively. Thus the sum equals 0 - = V v out in V v v v f v f d ) ( ) (
12-5 Flow from Circulation and Back The idea is to add new source and sink Let - = = S v v T v v d d D v u u d - d G G’ Lemma There is a feasible circulation with demands in G if and only if the maximum flow in G’ has value D. If all capacities and demands

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12 - Algorithms Demands and Bounds Demands and Bounds...

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