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# s02lec13 - 15.053 The Minimum Cost Flow Problem Tuesday...

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1 15.053 Tuesday, April 2 z The Shortest Path Problem z Dijkstra’s Algorithm for Solving the Shortest Path Problem Handouts: Lecture Notes 2 The Minimum Cost Flow Problem 1 2 3 4 -\$3, 6 \$8,5 \$7,2 \$3, 4 \$2, 7 3 4 -5 -2 Directed Graph G = (N, A). Node set N, arc set A; Capacities u ij on arc (i,j) lower bound 0 on arc (i,j) Cost c ij on arc (i,j) Supply/demand b i for node i. (Positive indicates supply) A network with costs, capacities, supplies, demands Minimize the cost of sending flow s.t. Flow out of i - Flow into i = b i 0 x ij u ij 3 Formulation In general the LP formulation is given as Minimize subject to 11 1 0 ,, , nn ij ij ij ij ki i jk ij ij cx xx b i n xu == −= = ≤≤ ∑∑ 4 The Shortest Path Problem 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 What is the shortest path from a source node (often denoted as s) to a sink node, (often denoted as t)? What is the shortest path from node 1 to node 6? Assumptions for this lecture: 1. There is a path from the source to all other nodes. 2. All arc lengths are non-negative 5 Formulation as a linear program In general the LP formulation for the shortest path from a source, s, to a sink, t, is given as Minimize subject to 1 0 0 , , , , ij ij ij ki is i t i n otherwise xi j = = = ≥∀ 6 Another Formulation 1 1 0 ,{ } , ij ij ij ki ni s iN s j −∈ The LP formulation for the shortest path from a source, s, to all other nodes is given as Minimize subject to

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7 Some Questions Concerning the Shortest Path Problem z Where does it arise in practice? Direct applications Indirect (and often subtle) applications z How does one solve the shortest path problem? Dijkstra’s algorithm z How does one measure the performance of an algorithm? CPU time measurements Performance Guarantees z How does one establish that a solution is really the shortest path? Connection to LP duality 8 Possible sports scores z Flumbaya is an unusual water sport in which there are two types of scores possible. One can score a gymbol , which is worth 7 points, or one can score a quasher , which is worth 5 points. An announcer on TV states that a recent game was won by a score of 19 to 18. Is this possible? 9 More on Flumbaya 1 2 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 4 0 There is no path from node 0 to node 18. A score of 18 is impossible.
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s02lec13 - 15.053 The Minimum Cost Flow Problem Tuesday...

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