We are feeling experimental and want to create a new dish. There are various ingredients we can choose from and we’d like to use as many of them as possible, but some ingredients don’t go well with others. If there are n possible ingredients (numbered 1 to n), we write down an n × n matrix giving the discord between any pair of ingredients. This discord is a real number between 0.0 and 1.0, where 0.0 means “they go together perfectly” and 1.0 means “they really don’t go together.” Here’s an example matrix when there are five possible ingredients.
| 1 | 2 | 3 | 4 | 5 |
1 | 0.0 | 0.4 | 0.2 | 0.9 | 1.0 |
2 | 0.4 | 0.0 | 0.1 | 1.0 | 0.2 |
3 | 0.2 | 0.1 | 0.0 | 0.8 | 0.5 |
4 | 0.9 | 1.0 | 0.8 | 0.0 | 0.2 |
5 | 1.0 | 0.2 | 0.5 | 0.2 | 0.0 |
In this case, ingredients 2 and 3 go together pretty well whereas 1 and 5 clash badly. Notice that this matrix is necessarily symmetric; and that the diagonal entries are always 0.0. Any set of ingredients incurs a penalty which is the sum of all discord values between pairs of ingredients. For instance, the set of ingredients {1, 3, 5} incurs a penalty of 0.2 + 1.0 + 0.5 = 1.7. We want this penalty to be small.
EXPERIMENTAL CUISINE
Input: n, the number of ingredients to choose from; D, the n× n “discord” matrix; some number p ≥ 0
OUTPUT: The maximum number of ingredients we can choose with penalty ≤ p.
Show that if EXPERIMENTAL CUISINE is solvable in polynomial time, then so is 3SAT.
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