3-SAT with atmost 3 variables and variable occuring once per clause

Question

I've stumbled across this problem on CSES https://cses.fi/345/task/E/ and was wondering is it somehow reducible to 2-SAT with given constraints?

So, the problem states that you need to solve a 3-SAT given in a CNF form, albeit a special case, where:

1. Each clause has exactly 3 distinct variables
2. Each variable occurs at most 3 times in all clauses

I doubt that CSES would ask for a heuristic, solver like, approach but can't figure out how those new constraints help in creating a polynomial, possibly close to linear (due to size of input) solution.

Any help would be much appreciated :)

Answer 1

Consider a bipartite graph $G=(C+V, E)$, where $C$ is the set of clauses and $V$ is the set of variables in the SAT instance. There is an edge $(c,x)$ with $c \in C$ and $x \in V$ if $x$ appears (possibly negated) in $c$.

Let $S \subseteq C$ and define $N[S] = \{x \in V \mid \exists c \in C, (c,x) \in E \}$. I claim that $|N[S]| \ge |S|$. Indeed, if we had $|S| < |N[S]|$ then the average number of occurrences of a variable in a clause in $S$ would be $\frac{3 |S|}{|N[S]|} > 3$, showing that some variable must appear at least $4$ times among the clauses in $S$.

Then, by Hall's theorem, $G$ admits a $C$-perfect matching. Consider any such matching and let $x_c \in V$ be the variable matched to $c \in C$. A satisfying assignment sets $x_c$ to true if $x_c$ is positive in $c$, and sets $x_c$ otherwise. All unmatched variables can be set arbitrarily.

A matching in a bipartite graph can be found in time $O(m \sqrt{n})$, where $n$ (resp. $m$) is the number of vertices (resp. edges) of $G$. In your case $m = \Theta(n)$, so the above simplifies to $O(n^{3/2})$.