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721 An edge of a ow network is called critical if decreasing the capacity of this edge results in a decrease in the maximum ow Give an ef cient algorithm that nds a critical edge in a network 722 In a particular network G = (V, E) whose edges have integer capacities ce , we have already found the maximum ow f from node s to node t However, we now nd out that one of the capacity values we used was wrong: for edge (u, v) we used cuv whereas it should have been cuv 1 This is unfortunate because the ow f uses that particular edge at full capacity: fuv = cuv We could redo the ow computation from scratch, but there s a faster way Show how a new optimal ow can be computed in O(|V | + |E|) time
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723 A vertex cover of an undirected graph G = (V, E) is a subset of the vertices which touches every edge that is, a subset S V such that for each edge {u, v} E, one or both of u, v are in S Show that the problem of nding the minimum vertex cover in a bipartite graph reduces to maximum ow (Hint: Can you relate this problem to the minimum cut in an appropriate network )
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724 Direct bipartite matching We ve seen how to nd a maximum matching in a bipartite graph via reduction to the maximum ow problem We now develop a direct algorithm Let G = (V1 V2 , E) be a bipartite graph (so each edge has one endpoint in V1 and one endpoint in V2 ), and let M E be a matching in the graph (that is, a set of edges that don t touch) A vertex is said to be covered by M if it is the endpoint of one of the edges in M An alternating path is a path of odd length that starts and ends with a non-covered vertex, and whose edges alternate between M and E M (a) In the bipartite graph below, a matching M is shown in bold Find an alternating path
A B C D E F G H I
(b) Prove that a matching M is maximal if and only if there does not exist an alternating path with respect to it (c) Design an algorithm that nds an alternating path in O(|V | + |E|) time using a variant of breadth- rst search 725 The dual of maximum ow Consider the following network with edge capacities (d) Give a direct O(|V | |E|) algorithm for nding a maximal matching in a bipartite graph
(a) Write the problem of nding the maximum ow from S to T as a linear program (b) Write down the dual of this linear program There should be a dual variable for each edge of the network and for each vertex other than S, T Now we ll solve the same problem in full generality Recall the linear program for a general maximum ow problem (Section 72) (c) Write down the dual of this general ow LP, using a variable ye for each edge and xu for each vertex u = s, t (d) Show that any solution to the general dual LP must satisfy the following property: for any directed path from s to t in the network, the sum of the ye values along the path must be at least 1 (e) What are the intuitive meanings of the dual variables Show that any s t cut in the network can be translated into a dual feasible solution whose cost is exactly the capacity of that cut 726 In a satis able system of linear inequalities a11 x1 + + a1n xn am1 x1 + + amn xn b1 bm
we describe the jth inequality as forced-equal if it is satis ed with equality by every solution x = (x1 , , xn ) of the system Equivalently, i aji xi bj is not forced-equal if there exists an x that satis es the whole system and such that i aji xi < bj For example, in x1 + x 2 x1 x2 x1 x2 2
the rst two inequalities are forced-equal, while the third and fourth are not A solution x to the system is called characteristic if, for every inequality I that is not forced-equal, x satis es I without equality In the instance above, such a solution is (x1 , x2 ) = ( 1, 3), for which x1 < 1 and x2 < 0 while x1 + x2 = 2 and x1 x2 = 2 (a) Show that any satis able system has a characteristic solution (b) Given a satis able system of linear inequalities, show how to use linear programming to determine which inequalities are forced-equal, and to nd a characteristic solution 727 Show that the change-making problem (Exercise 617) can be formulated as an integer linear program Can we solve this program as an LP, in the certainty that the solution will turn out to be integral (as in the case of bipartite matching) Either prove it or give a counterexample
2 1
728 A linear program for shortest path Suppose we want to compute the shortest path from node s to node t in a directed graph with edge lengths le > 0 (a) Show that this is equivalent to nding an s t ow f that minimizes size(f ) = 1 There are no capacity constraints (b) Write the shortest path problem as a linear program (c) Show that the dual LP can be written as xu xv luv for all (u, v) E (d) An interpretation for the dual is given in the box on page 206 Why isn t our dual LP identical to the one on that page 729 Hollywood A lm producer is seeking actors and investors for his new movie There are n available actors; actor i charges si dollars For funding, there are m available investors Investor j will provide pj dollars, but only on the condition that certain actors Lj {1, 2, , n} are included in the cast (all of these actors Lj must be chosen in order to receive funding from investor j) The producer s pro t is the sum of the payments from investors minus the payments to actors The goal is to maximize this pro t (a) Express this problem as an integer linear program in which the variables take on values {0, 1} max xs xt
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