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Linear Optimization: Network Problems

                             Brady Hunsaker


                          November 14...
Integrality Property of Min Cost Network Flows


   A min cost network flow instance with integer supplies, demands, and
  ...
Assignment Problems
   Consider a transportation problem in which each supply is +1 and
   each demand is −1.
   The solut...
Max Flow-Min Cut
   In this problem we have two special nodes and upper bounds on
   edges, but no edge costs. The objecti...
Matching

   In this problem we have edge weights and wish to “match” or “pair
   up” edges.
   A solution consists of a s...
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Linear Optimization: Network Problems

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Linear Optimization: Network Problems

  1. 1. Linear Optimization: Network Problems Brady Hunsaker November 14, 2006 Brady Hunsaker () Linear Optimization: Network Problems November 14, 2006 1 / 10 Network Problems There are several kinds of network problems that are related to network flows. Many may be considered special cases of network flows. Many of these problems have specialized algorithms faster than network simplex. Many exhibit the integrality property of network flow problems. (On next slide.) Brady Hunsaker () Linear Optimization: Network Problems November 14, 2006 2 / 10
  2. 2. Integrality Property of Min Cost Network Flows A min cost network flow instance with integer supplies, demands, and upper bounds (if any) has an optimal solution with integer flow values, and the network simplex algorithm will find such a solution. That is, LPs with this special form will always have integer optimal solutions. This is not generally true for LPs! This result can also be proved based on the structure of the constraint matrix for a network flow problem. They fall in a class of matrices called totally unimodular matrices, which possess this integrality property. Brady Hunsaker () Linear Optimization: Network Problems November 14, 2006 3 / 10 Transportation Problems Consider a network flow instance in which all nodes have either supply or demand and edges only go between supply nodes and demand nodes. We have already seen several examples of such transportation problems. This is a special class of network flow problems. There exist special algorithms for this class. Brady Hunsaker () Linear Optimization: Network Problems November 14, 2006 4 / 10
  3. 3. Assignment Problems Consider a transportation problem in which each supply is +1 and each demand is −1. The solution to such a problem can be considered a pairing of supply nodes and demand nodes. In graph theory, this problem is usualy referred to as bipartite weighted matching. There are very fast algorithms for this problem. What are examples where such an instance may arise? Brady Hunsaker () Linear Optimization: Network Problems November 14, 2006 5 / 10 Shortest Path In these problems we have edge costs and two special nodes. The problem is to determine the path between the two nodes for which the sum of edge costs along the path is minimum. The graph may be directed or undirected. What are examples of this problem? What does a solution to this problem look like? How can this be modeled as a network flow problem? The integrality theorem allows us to use the network simplex algorithm, but there are much faster algorithms for this problem. Brady Hunsaker () Linear Optimization: Network Problems November 14, 2006 6 / 10
  4. 4. Max Flow-Min Cut In this problem we have two special nodes and upper bounds on edges, but no edge costs. The objective is to send as much flow as possible from the source node to the sink node. (Cost is irrelevant.) This problem may be modeled as an LP, though not as a min cost network flow problem. It is an excellent example of LP duality. What is the primal and dual? How can we interpret the dual problem? Brady Hunsaker () Linear Optimization: Network Problems November 14, 2006 7 / 10 Max Flow-Min Cut Brady Hunsaker () Linear Optimization: Network Problems November 14, 2006 8 / 10
  5. 5. Matching In this problem we have edge weights and wish to “match” or “pair up” edges. A solution consists of a set of edges such that no two edges are incident with the same node. The objective could be to find a maximum weight or minimum weight matching. Or it could be to find a matching with the maximum number of edges (called cardinality matching). This general problem may not be modeled as a min cost network flow problem. There are special relatively fast algorithms for this problem. There is a special case in which the nodes may be divided into two sets with all edges going between the two sets. Such a graph is called a bipartite graph and leads to bipartite matching. Brady Hunsaker () Linear Optimization: Network Problems November 14, 2006 9 / 10 Min Cost Network Flow with Upper Bounds Like the simplex algorithm, the network simplex algorithm may also be used when edges have upper bounds. How must the network simplex algorithm be modified to handle this change? Brady Hunsaker () Linear Optimization: Network Problems November 14, 2006 10 / 10

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