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A) show that SATISFIABILITY remains NP complete even if it is restri

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a) show that SATISFIABILITY remains NP-complete even if
it is restricted to instances in which each variable appears
at most three times. (Hint: Replace the occurrences of
variable, say, x, by new variables x1,...,xk. Then add a
formula in which each of these variables appears twice,
stating that \"all these variables are equivalent.\")
b) What happens if each variable appears at most twice?
Solution
a) For any sr3, either every Boolean expression with
nexactly 3 variables perclause and no more than s
occurrences per
NP-complete: given an arbitrary formula, it is NP-hard to
find a satisfying assignment for it. In fact, even satisfying
Boolean formulas where each variable xi appears at most
three times in the formula is NP-complete. (Here is a
reduction from the general case to this case: Suppose a
variable x appears k>3 times. Replace each of its
occurrences with fresh new variables x1,x2,,xk, and
constrain these k variables to all take on the same value,
by ANDing the formula with
(¬x1 ORx2) AND (¬x2 ORx3) ANDAND (¬xk1 ORxk) AND
(¬xkORx1).

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The total number of occurrences of each variable xj is now
three.) On the other hand, if each variable appears only
once in the formula, then the satisfiability algorithm is very
easy: since we have a monotone formula in
x1,,xn,¬x1,,¬xn, we set xi to 0 if ¬xi appears,
otherwise we set xi to 1
What happens if each variable appears at most twice
b) To clarify further, here is an example instance of the
problem:
((x1 ANDx3) OR (x2 ANDx4 ANDx5)) AND (¬x1 OR ¬x4)
AND (¬x2 OR (¬x3 AND ¬x5))
Note that when we restrict the class of formulas further to
conjunctive normal form (i.e. a depth-2 circuit, an AND of
ORs of literals) then this \"at most twice\" problem is
known to be solvable in polynomial time. In fact, applying
the \"resolution rule\" repeatedly will work. But it is not
clear (at least, not to me) how to extend resolution for the
class of general formulas to get a polytime algorithm. Note
when we reduce a formula to conjunctive normal form in
the usual way, this reduction introduces variables with
three occurrences. So it seems plausible that perhaps one
might be able to encode an NP
-complete problem in the additional structure provided by a
formula, even one with only two occurrences per variable.

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a) show that SATISFIABILITY remains NP -complete even if it is restricted to instances in which each variable appears at most three times. (Hint: Replace the occurrences of variable, say, x, by new variables x1,...,xk. Then add a formula in which each of these variables appears twice, stating that \"all these variables are equivalent.\") b) What happens if each variable appears at most twice? Solution a) For any sr3, either every Boolean expression with nexactly 3 variables perclause and no more th an s occurrences per NP-complete: given an arbitrary formula, it is NP -hard to find a satisfying assignment for it. In fact, even satisfying Boolean formulas where each variable xi appears at most three times in the formula is NP-complete. (Here is a reduction from the general case to this case: Suppose a variable x appears k>3 times. Replace each of its occurrences with fresh new variables ...
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