Project Selection Bin Li 10/29/2008

Contents Background Problem Formulation Algorithm Design

Background The telecommunication company is assessing the pros and cons of a project to offer some new type high-speed access service to residential customers. The revenue from the high-speed access service might not be enough to updating the routers; however, once the company has updated the routers, they’ll earn more money. These projects often interact each other. The question is: which projects should be pursued, and which should be passed up?

Problem Formulation(1) n projects 1,2,…,n Dependencies between projects Project i depends on project j implies project i cannot be done unless project j is done. Each project i has a cost/profit pi pi>0 implies project i generates pi profit pi<0 implies project i requires a cost pi

Problem Formulation (2) Constrains: For a set A of projects A is a valid solution if A is dependency closed, that is for every , all projects that project i depends on are also in A. Our goal Find valid A to maximize profit(A)

Algorithm Design Max-flow Min-cut theorem If f is a flow in a flow network G=(V,E) with source s and sink t, then the following conditions are equivalent: f is a maximum flow in G The residual network Gf contains no augmenting paths. |f|=c (S, T) for some cut (S, T) of G.

Algorithm Design(2) Build a flow network Projects represented as nodes in a graph If project i depends on project j, then (i, j) is an edge Add source s and sink t For each project i with pi>0, add edge (s, i) with capacity pi For each project i with pi<0, add edge (i, t) with capacity –pi For each dependency edge (i, j), we put capacity ∞ We can see that the capacity of the cut({s},PU{t}) is , so the maximum-flow value in this flow network is at most C.

Algorithm Design(3) Example There are four project{1,2,3,4},where p1>0,p2>0,p3<0,p4<0 project 1 is dependent on project 4 Project 2 is dependent on project 3

Algorithm Design(4) Lemma 1 Proof Suppose (A’, B’) is a an s-t cut of finite capacity (no ∞) edges. Then projects in A=A’-{s} are a valid solution. Proof If A=A’-{s} is not a valid solution then there is a project and a project such that i depends on j. Since (i, j) capacity is ∞, implies the cut(A’,B’) capacity is ∞, contradicting assumption.

Algorithm Design(5) Lemma 2 Proof The capacity of the cut (A’,B’), where A’=AU{s} and A is a valid solution, is Proof Edge in the flow network: The edges Leaving the source: The edges Entering the sink: Other edges: contribute Zero

Algorithm Design(6)

Algorithm Design(7) We have shown that if (A’,B’) is an s-t cut in G with finite capacity then A=A’-{s} is a valid set of projects Therefore a minimum s-t cut(A*,B*) gives a maximum set of projects A*-{s} since C is fixed.

Thank you for your attention! Questions?