I was given a problem to find the least number of popular votes to win an election through an electoral college. They gave me $N$ number of counties with $K$ required electoral votes to win. Each county had a population and a number of electoral votes given to the winner if he won the county. If you won about half of the county's popular votes you would get all electoral votes. How would you go about this sort of problem? Would you compare vote ratios or something else?
2026-03-25 22:05:08.1774476308
How would you find the least number of popular votes to win an election given it is split in electoral districts?
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You can solve the problem via integer linear programming as follows. For each county $i\in\{1,\dots,N\}$, let $p_i$ be the population and let $e_i$ be the number of electoral votes. Let binary variable $x_i$ indicate whether county $i$ is won. The problem is to minimize $$\sum_{i=1}^N \left(\left\lfloor\frac{p_i}{2}\right\rfloor +1\right) x_i$$ subject to $$\sum_{i=1}^N e_i x_i \ge K.$$