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Lecture21

Lecture21 - C&O 355 Lecture 21 Notes Nicholas Harvey...

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C&O 355: Lecture 21 Notes Nicholas Harvey http://www.math.uwaterloo.ca/~harvey/ 1 Minimum s - t Cuts In Lecture 20, we looked at the minimum s - t cut problem. Let’s now look at a variant of this problem which introduces capacities on the arcs. Problem 1.1. Let G = ( V, A ) be a directed graph. Let s and t be particular vertices in V . For each arc a A , there is a capacity c a 0. A cut (or s - t cut) is a set F A such that there is no s - t path in G \ F . The goal is to find a cut F that minimizes its capacity, namely a F c a . Let P be the set of all simple directed paths from s to t . An integer program formulation of this problem is: min c T y (IP-MC) s.t. X a p y a 1 p ∈ P y ∈ { 0 , 1 } A linear program relaxation is: min c T y (LP-MC) s.t. X a p y a 1 p ∈ P y 0 Last time we proved the following theorem. Theorem 1.2. Assume c a = 1 for all a A . Then there is an optimal solution z such that, for some U V , we have z a = ( 1 (if a δ + ( U )) 0 (otherwise). That is, z is the characteristic vector of the set δ + ( U ). The previous theorem is actually true for any capacities c 0, and the same proof works. Our proof used the dual of LP-MC, complementary slackness, and Strong LP Duality. In this lecture we will prove a theorem very similar to Theorem 1.2, using a very different approach. We will again consider the dual of LP-MC, which is: 1

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max X p ∈P x p (LP-PF) s.t. X p : a p x p c a a A x 0 We will prove the following theorem. Theorem 1.3. For any non-negative, integral arc capacities c , there is an integral optimal solution to LP-PF. 2 Network Flows We will think of the solutions to LP-PF as specifying path-flows . For every directed s - t path, it specifies a “flow value” x p , such that every arc a has at most c a units of flow passing through it. These objects are studied in the area known as “network flows”, and the course C&O 351 focuses on this area. However, LP-MF is not the usual linear program to consider. It is more common to look at what I’ll call arc-flows , which are feasible solutions to the linear program LP-AF, defined next. For any u V , define δ + ( u ) = { ( u, v ) : ( u, v ) A } δ - ( u ) = { ( v, u ) : ( v, u ) A } So δ + ( u ) is the set of outgoing arcs from u , and δ - ( u ) is the set of incoming arcs to u . The arc-flow linear program is: max X a δ + ( s ) z a - X a δ - ( s ) z a (LP-AF) s.t.
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Lecture21 - C&O 355 Lecture 21 Notes Nicholas Harvey...

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