We consider the question of the existence of variables with few occurrences
in boolean conjunctive normal forms (clause-sets). Let mvd(F) for a clause-set
F denote the minimal variable-degree, the minimum of the number of occurrences
of variables. Our main result is an upper bound mvd(F) <= nM(surp(F)) <=
surp(F) + 1 + log_2(surp(F)) for lean clause-sets F in dependency on the
surplus surp(F).
- Lean clause-sets, defined as having no non-trivial autarkies, generalise
minimally unsatisfiable clause-sets.
- For the surplus we have surp(F) <= delta(F) = c(F) - n(F), using the
deficiency delta(F) of clause-sets, the difference between the number of
clauses and the number of variables.
- nM(k) is the k-th "non-Mersenne" number, skipping in the sequence of
natural numbers all numbers of the form 2^n - 1.
We conjecture that this bound is nearly precise for minimally unsatisfiable
clause-sets.
As an application of the upper bound we obtain that (arbitrary!) clause-sets
F with mvd(F) > nM(surp(F)) must have a non-trivial autarky (so clauses can be
removed satisfiability-equivalently by an assignment satisfying some clauses
and not touching the other clauses). It is open whether such an autarky can be
found in polynomial time.
As a future application we discuss the classification of minimally
unsatisfiable clause-sets depending on the deficiency.Comment: 14 pages. Revision contains more explanations, and more information
regarding the sharpness of the boun
