3SAT to 1-in-3SAT reduction with additonal constraints

Question

The simplest Reduction for 3-SAT to 1-in-3-SAT reduction is as follows:

For each 3SAT clause: $x+y+z=1$ Introduce 4 new variables $\{a, b, c, d\}$ and replace original clause with below 3 clauses:

1. $R(x^-, a, b)=1$
2. $R(y, b, c) =1$
3. $R(z^-, c, d)=1$

Query: Given a 3SAT instance how can we reduce it into (as simple as possible) 1-in-3SAT instance with the following additional constraints:

1. All new variables occur in at least 2 clauses.
2. There are no dummy/redundant clauses in the 1-in-3-SAT reduction. A dummy/redundant clause is whose addition or removal in the problem does not change the set of solutions of the problem. Thus, all clauses are essential.

I have been stuggling with this for some time without much success.

Answer 1

Edit: This answers a related problem, in which "as simple as possible" means that the number of clauses is minimal.

Let $A$ be the given 3SAT instance. I will assume that you are looking for a corresponding 1-in-3SAT instance $B$ with as few clauses as possible, rather than some local minimum. It also seems that you restrict yourself to instances $B$ whose variables include those of $A$, plus potentially new variables, and that any satisfying assignment for $B$ is also a satisfying assignment for $A$, and that any satisfying assignment for $A$ can be extended to a satisfying assignment for $B$ using some assignment for the new variables.

Let $c$ be the number of clauses in $B$. If $A$ is not satisfiable, then $B$ will be an unsatisfiable 1-in-3SAT instance of constant length ($c=1$ or $c=2$ depending on whether a single clause can contain a literal and its negation).

We can in polynomial time tell whether $A$ is satisfiable based on $B$ as follows. If $c>2$, then $A$ is satisfiable. If $B$ has length $c$, we can find all its satisfying assignments in $O(c2^{3c})$ time, so if $c\leq 2$, we can find all satisfying assignments in constant time and hence determine whether $A$ is satisfiable. Therefore, a reduction to a 1-in-3SAT instance with a minimum number of clauses is NP-hard.