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Questions from May 11, 2022
(Breadth-First Search) Let i and j be two nodes of a directed graph (N , A). (a) Consider the following algorithm, known as breadth-first search, for finding and mark each node n ∈ Tk+1 with the...
May 12, 2022
(Feasible Distribution Theorem) Show that the minimum cost flow problem introduced in Section 1.2.1, has a Show also that feasibility of the problem can be determined by solving a max- flow problem...
May 12, 2022
Describe an algorithm of the Ford-Fulkerson type for checking the feasibility and finding a feasible solution of a minimum cost flow problem (cf., Section 1.2.1). If the supplies si and the arc flow...
May 12, 2022
Consider a graph with arc flow range [0, cij ] for each arc (i, j), and let x be a capacity-feasible flow vector. (a) Consider any subset S of nodes all of which have nonpositive divergence and at...
May 12, 2022
This counterexample (from Chvatal [1983]) illustrates how the version of the FordFulkerson method where augmenting paths need not have as few arcs as possible may not terminate for a problem with...
May 12, 2022
(Graph Connectivity – Menger’s Theorem) Let s and t be two nodes in a directed graph. Use the max-flow/min-cut theorem to show that: (a) The maximum number of forward paths from s to t that do not...
May 12, 2022
Prove the following generalizations of Prop. 3.1 (a) Let x be a capacity-feasible flow vector, and let N + and N − be two disjoint subsets of nodes. Then exactly one of the following two alternatives...
May 12, 2022
Consider the Ford-Fulkerson algorithm as described in Section 3.2 (augmenting paths have as few arcs as possible). This exercise shows that the algorithm terminates and solves the max-flow problem in...
May 12, 2022
Layered Network Algorithm) Consider the algorithm described near the end of Section 3.2, which uses phases and augmentations through a layered network. (a) Provide an algorithm for constructing the...
May 12, 2022
(O(N2/3A) Complexity for Unit Capacity Graphs) Consider the max-flow problem in the special case where the arc flow range is [0,1] for all arcs. (a) Show that each path from the source to the sink...
May 12, 2022
(Relation of Path Construction and Assignment) The purpose of this exercise (from Bertsekas [1995c]) is to show the connection of the path construction algorithm of Section 3.3.1 with the assignment...
May 12, 2022
Consider the problem of matching n persons with n objects on a one-to-one basis (cf. Exercises 1.21 and 3.9). For each person i there is a given set of objects A(i) that can be matched with i. A...
May 12, 2022
A town has m dating agencies that match men and women. Agency i has a list of men and a list of women, and may match a maximum of ci man/woman pairs from its lists. A person may be in the list of...
May 12, 2022
Consider an n × n chessboard and let A and B be two given squares. (a) Consider the problem of finding the maximal number of knight paths that start at A, end at B, and do not overlap, in the sense...
May 12, 2022
Consider the “opposite” to the max-flow problem, which is to minimize the divergence out of s over all capacity-feasible flow vectors having zero divergence for all nodes other than s and t. (a)...
May 12, 2022
(Duality for Nonnegativity Constraints) Consider the version of the minimum cost flow problem where there are nonnegativity constraints which supports from below the set S, and is normal to the...
May 12, 2022
(Duality and the Max-Flow/Min-Cut Theorem) Consider a feasible max-flow problem and let Q = [S, N −S] be a minimum capacity cut separating s and t. Consider also the minimum cost flow problem...
May 12, 2022
Consider the minimum cost flow problem with nonnegativity constraints given in Fig. 5.20 (supplies and demands are shown next to the nodes, arc costs are immaterial). Find all basic flow vectors and...
May 12, 2022
From a Feasible to a Basic Feasible Flow Vector) Consider a feasible minimum cost flow problem such that the corresponding graph is connected. Suppose we are given a feasible flow vector x. Construct...
May 12, 2022
(Alternative Construction of a Basic Feasible Flow Vector) Consider the following algorithm that tries to construct a flow vector that has a given divergence vector s, and is zero on arcs which are...
May 12, 2022
Consider a transportation problem involving the set of sources S and the set of sinks T (cf. Example 1.4 in Ch. 1). Suppose that there is no strict subset S of S and strict subset T of T such that ...
May 12, 2022
Consider an assignment problem with sources 1, 2, 3, 4 and sinks 5, 6, 7, 8. There is an arc between each source and each sink. The arc costs are as follows: Suppose that the simplex method is...
May 12, 2022
(Rank of the Conservation of Flow Equations) Let us say that the conservation of flow equations (a) Show that if the graph is connected, the conservation of flow equations have rank N − 1, where N...
May 12, 2022
For Milestone One, you will complete a draft of the campaign overview of your high-level campaign goals. You will define the campaign goals and include details as to the intention of the campaign and...
Solved
May 12, 2022
Hansson Private Label, Inc.: Evaluating an Investment in ExpansionDoNotCopy orPostThis document is authorized for use only by David Schmitz until September 2011. Copying or...
Solved
May 12, 2022
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