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Can we transform a CNF to ILP without introducing new variables?

My question can be seen as a follow up to Express boolean logic operations in zero-one integer linear programming (ILP) as the solution proposed in that post introduces additional variables.

Also, my current understanding is that each 0 value in the truth table is represented by one linear inequality. This means that if my boolean function is of 16 input variables and balanced, there will be 2^15 linear inequalities so is there a way to reduce the number of inequalities?

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Moati
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1 Answers1

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No. Usually (for most functions / CNF formulas) you'll need additional variables. Linear inequalities form a convex space. If the set of solutions such that $f(x)=1$ is not convex, you'll need additional variables.

The exact number of additional variables and inequalities required depends on the function / on the CNF formula.

D.W.
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