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I'm trying to prove that $\mathrm{HALF}\text-\mathrm{FALSE}$ is NP-hard, where $\mathrm{HALF}\text-\mathrm{FALSE}$ is the following problem:

given a boolean formula $\phi(x_1,\dots,x_n)$, is there a satisfying assignament in which exactly $n/2$ variables have value false?

Now, i got the problem down if it was like this:

given a boolean formula $\phi(x_1,\dots,x_n)$, is there a satisfying assignament in which at least $n/2$ variables have value false?

the reduction for this modified problem from $\mathrm{SAT}$ to (let's call it $\mathrm{HALF}\text-\mathrm{FALSE}'$) i think is the following: $$f(\phi(x_1,\dots,x_n))=\phi(x_1,\dots,x_n)\bigwedge_{i=1}^{n}y_i'$$

it's correct because $\phi \in SAT \Leftrightarrow f(\phi) \in \mathrm{HALF}\text-\mathrm{FALSE}'$ and $f$ is computable in polynomial time.

Back to the original problem, my thoughts were something like this, if i can transform first the original formula in a formula that is satiasfied only by all true assignament iff the original formula is satisfied, then do the same thing as before. So something like this: $$f(\phi(x_1,\dots,x_n))=\psi(x_1,\dots,x_n)\bigwedge_{i=1}^{n}y_i'$$ where $\psi(x_1,\dots,x_n)$ has only all true assignament iff $\phi(x_1,\dots,x_n)$ is satisfied. But i don't know how to construct $\psi$.

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    $\begingroup$ There is an elementary reduction from SAT to HALF-FALSE. Hint: If $\psi$ is a tautology, then $\varphi \wedge \psi$ is satisfiable if and only if $\varphi$ is satisfiable. Define $\psi$ on fresh copies of all the variables in $\varphi$. $\endgroup$ Commented Jun 4, 2022 at 21:37
  • $\begingroup$ So you are saying a reduction like $f(\phi(x_1,\dots ,x_n))=\phi(x_1,\dots ,x_n) \wedge \bigvee_{i=1}^{n} (x_i \vee x_i')$ works? i don't see how though, if $\phi$ is satisfiable we don't know how many false variables the assignament has, and i don't understand the logic behind adding a tautology. $\endgroup$
    – giggiox
    Commented Jun 4, 2022 at 21:51
  • $\begingroup$ A tautology $\psi(x'_1, ..., x'_n)$ is satisfied by any assignment to its variables. If $\varphi(x_1, ..., x_n)$ is satisfiable, does $\varphi(x_1, ..., x_n) \wedge \psi(x'_1, ..., x'_n)$ have a satisfying assignment that sets exactly $n$ variables to true? If $\varphi$ is unsatisfiable, does $\varphi \wedge \psi$ have a satisfying assignment (setting exactly $n$ variables to true)? $\endgroup$ Commented Jun 4, 2022 at 21:58
  • $\begingroup$ Got it! this tricked me because i was thinking that we can't know how many variables are assigned false in the assignament that satisfy the formula, so i was thinking to first force the formula to have all assignament to one, then "mechanically" adding $n$ false assignaments. I think this new way of seeing the problem opened up me many possibilities even for other problems. $\endgroup$
    – giggiox
    Commented Jun 5, 2022 at 6:28

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Convert your formula into 3-CNF (not strictly necessary, I think). Duplicate every clause in the formula with new variables and every literal inverted. Then if there exists a satisfying assignment in the original formula one exists in the new one and exactly half of the variables are false. Clearly, if this HALF-FALSE problem is solvable in p-time then so is SAT.

Suppose your 3-CNF formula is $(x_1+x_2'+x_3')(x_1'+x_2+x_3)(x_1'+x_2'+x_3)$. Rewrite it so that it becomes: $$ (x_1+x_2'+x_3')(x_1'+x_2+x_3)(x_1'+x_2'+x_3)\land \\ (y_1'+y_2+y_3)(y_1+y_2'+y_3')(y_1+y_2+y_3') $$

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  • $\begingroup$ In the process of solving the problem i tried something similar but since i don't know how to prove that this is a correct reduction, i gave up. How can i prove this reduction is correct? I think the idea is that the satisfying assignament to the formula has $k$ false variables, the new formula will have $n-k$ false variables in the same assignament. But how can i prove it? $\endgroup$
    – giggiox
    Commented Jun 5, 2022 at 6:29
  • $\begingroup$ Write a truth table for the new formula. For every row where the output is 1 it must be so that the subformula over the $x$-variables outputs 1 and the subformula over the $y$-variables also outputs 1. Of those, the rows where all $y$-values are negations of the $x$-values must also output 1. If HALF-FALSE can find rows like that in p-time simply read the values of the $x$-variables in that row and you have a satisfying assignment for 3SAT, known to be NP-hard. $\endgroup$ Commented Jun 5, 2022 at 10:51

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