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how to generate a barcode using asp.net c# f (i) is a valid ow from si to ti The size of each ow f in Software
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(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 forcedequal if it is satis ed with equality by every solution x = (x1 , , xn ) of the system Equivalently, i aji xi bj is not forcedequal 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 forcedequal, while the third and fourth are not A solution x to the system is called characteristic if, for every inequality I that is not forcedequal, 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 forcedequal, and to nd a characteristic solution 727 Show that the changemaking 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

