You are consulting for an environmental statistics firm. They collect statistics and publish the collected data in a book. The statistics are about populations of different regions in the world and are recorded in multiples of one million. Examples of such statistics would look like the following table.
Country | A | B | C | Total |
grown-up men | 11.998 | 9.083 | 2.919 | 24.000 |
grown-up women | 12.983 | 10.872 | 3.145 | 27.000 |
children | 1.019 | 2.045 | 0.936 | 4.000 |
Total | 26.000 | 22.000 | 7.000 | 55.000 |
We will assume here for simplicity that our data is such that all row and column sums are integers. The Census Rounding Problem is to round all data to integers without changing any row or column sum. Each fractional number can be rounded either up or down. For example, a good rounding for our table data would be as follows.
Country | A | B | C | Total |
grown-up men | 11.000 | 10.000 | 3.000 | 24.000 |
grown-up women | 13.000 | 10.000 | 4.000 | 27.000 |
children | 2.000 | 2.000 | 0.000 | 4.000 |
Total | 26.000 | 22.000 | 7.000 | 55.000 |
(a) Consider first the special case when all data are between 0 and 1. So you have a matrix of fractional numbers between 0 and 1, and your problem is to round each fraction that is between 0 and 1 to either 0 or 1 without changing the row or column sums. Use a flow computation to check if the desired rounding is possible.
(b) Consider the Census Rounding Problem as defined above, where row and column sums are integers, and you want to round each fractional number a to either \a\ or \a\. Use a flow computation to check if the desired rounding is possible.
(c) Prove that the rounding we are looking for in (a) and (b) always exists.
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