hw6.pdf - CSOR W4231.002 – Spring 2019 Homework 6 Out Monday Due 11:59pm Monday May 6 2019 For all algorithms you design you should always describe

hw6.pdf - CSOR W4231.002 – Spring 2019 Homework 6 Out...

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CSOR W4231.002 – Spring, 2019 Homework 6 Out: Monday, Apr 22, 2019 Due: 11:59pm, Monday, May 6, 2019 For all algorithms you design, you should always describe them clearly in English, give pseudocode, prove correctness and give the best upper bound that you can for the running time. If you give a reduction, you should (a) state the input instances to both problems, (b) clearly describe the reduction transformation and argue that it requires polynomial time, and (c) prove equivalence of the original and the derived instances. Collaboration is limited to discussion of ideas only. You should write up the solutions entirely on your own. You should list your collaborators on your write-up. If you do not type your solutions, be sure that your hand-writing is legible, your scan is high-quality and your name is clearly written on your homework. 1. (24 points) Recall problem 5 from hw5 on flow networks with supplies . Alternatively, we can introduce flow networks with demands : instead of node supplies , each node v V has an integer demand d v ; if d v < 0, then v is a source , while if d v > 0, it is a sink . A circulation with demands is a function f : E R + that satisfies capacity constraints (for each e E , 0 f ( e ) c e ), and demand constraints (for each v V , f in ( v ) - f out ( v ) = d v ). You can answer whether a circulation with demands exists using the same methodology you used in hw5. However, in many applications, on top of the node demands, the flow must also make use of certain edges. To capture such constraints, consider the following variant of a flow network with demands.
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