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PS03-100225 - MATH 682 Problem Set#3 This problem set is...

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MATH 682 Problem Set #3 This problem set is due at the beginning of class on February 25 . Below, “graph” means “simple finite graph” except where otherwise noted. 1. (15 points) Demonstrate the following facts about a directed graph D . (a) (5 points) Prove that sum v V ( D ) d - D ( v ) = v V ( D ) d + D ( v ). Recall that d - and d + represent the indegree and outdegree respectively. (b) (10 points) Prove that a directed Eulerian tour (i.e. a directed closed trail traversing every edge) exists if and only if d - ( v ) = d + ( v ) for all v V ( D ). 2. (10 points) Suppose each edge in a digraph D with a source s and sink t has not only a maximum capacity c ( e ) but also a minimum required flow b ( e ), so that a flow f must satisfy b ( e ) f ( e ) c ( e ). Assume we know of a legal but not necessarily maximal flow f 0 . Explain how the Ford-Fulkerson algorithm could be modified to produce a maximum flow under these constraints. 3. (10 points) Let us consider a problem of committee assignment akin to traditional matching problems. Suppose we have individuals x 1 , x 2 , . . . , x k and committees c 1 , c 2 , . . . , c , such that each individual
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