110 5 2 4 3 2 4 5 -15 1 2 4 7 10 3 6 6 3 5 6 -5 11 3 6 4 5 2 7 1 3 6 4 5 2 7 1 3 6 4 5 2 7 0 0 0 1 3 6 4 5 2 7 1 3 -6 -4 1 2 3 0 0 0 What is the flow in arc (4,3)? 1 3 6 4 5 2 7 1 3 -6 -4 1 2 3 0 0 0 What is the flow in arc (5,3)? 2 21 3 6 4 5 2 7 1 3 -6 -4 1 2 3 0 0 0 What is the flow in arc (3,2)? 2 3 1 3 6 4 5 2 7 1 3 -6 -4 1 2 3 0 0 0 What is the flow in arc (2,6)? 2 3 6 1 3 6 4 5 2 7 1 3 -6 -4 1 2 3 0 0 0 What is the flow in arc (7,1)? 2 3 6 4 1 3 6 4 5 2 7 1 3 -6 -4 1 2 3 0 0 0 What is the flow in arc (1,2)? 2 3 6 4 3 1 3 6 4 5 2 7 1 3 -6 -4 1 2 3 0 0 0 Note: there are two different ways of calculating the flow on (1,2), and both ways give a flow of 4. Is this a coincidence? 2 3 6 4 34 33 6 5 2 1 7 3 4 5 4 4 2 4 2 33 1 1 3 6 5 2 1 7 3 5 4 4 2 4 3 6 5 2 1 7 2 6 5 3 1 3 3 6 5 2 1 7 1 7 4 2 0 2 3 6 5 2 1 7 1 7 4 2 2 41 3 6 4 5 2 7 5 -6 3 -4 -2 1 Here is a spanning tree with arc costs. How can one choose node potentials so that reduced costs of tree arcs are 0? 1 3 6 4 5 2 7 5 -6 3 -4 -2 1 There is a redundant constraint in the minimum cost flow problem. One can set p1 arbitrarily. We will let p1 = 0. What is the node potential for 2? 0 1 3 6 4 5 2 7 5 -6 3 -4 -2 1 What is thenode potential for 7? 0 -5 1 3 6 4 5 2 7 5 -6 3 -4 -2 1 What is the potential for node 3? 0 -5 -6 1 3 6 4 5 2 7 5 -6 3 -4 -2 1 What is the potential for node 6? 0 -5 -6 -2 1 3 6 4 5 2 7 5 -6 3 -4 -2 1 What is the potential for node 4? 0 -5 -6 -2 -1 51 3 6 4 5 2 7 5 -6 3 -4 -2 1 What is the potential for node 5? 0 -5 -6 -2 -1 -4 These are the node potentials associated with this tree. They do not depend on arc flows, nor on costs of non-tree arcs. 1 3 6 4 5 2 7 5 -6 3 -4 -2 1 0 -5 -6 -2 -1 -4 -1 1 3 6 4 5 2 7 0 -5 -6 -2 -1 -4 -1 7 2 -3 Node potentials Original costs 1 3 6 4 5 2 7 2 5 -3 Flow on arcs Reduced costs 2 3 6 4 34 1 3 6 4 5 2 7 0 Flow on arcs 2 3 6 4 34 1 3 6 4 5 2 7 32 0 3 1 34 61 3 6 4 5 2 7 71 3 6 4 5 2 7 1 3 6 4 5 2 7 1 3 6 4 5 2 7 89MIT OpenCourseWarehttp://ocw.mit.edu For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 6.251J / 15.081J Introduction to Mathematical ProgrammingFall