SAT - Problems and reductions with respect to the number of variables

Conference · Sat Dec 31 04:00:00 EST 1994

OSTI ID:35867

Buening, H K; Grandjean, E

In many algorithms or heuristics for the SAT problem the number of distinct variables occurring in the formula seems to play a central role, e.g. in Davis-Putnam algorithms. For deterministic algorithms, Schiermeyer has proved that 3-SAT is solvable in deterministic time O({vert_bar} {Phi} {vert_bar} c{sup n}), where n is the number of variables of {Phi} and c = 1.579. In contrast, no such algorithm is known (for any c < 2 ) for SAT. A pol.-time reduction f from SAT to 3-SAT, where f ({Phi}) contains n variables, if n is the number of variables of {Phi}, would lead immediately to an upper bound O({vert_bar} {Phi} {vert_bar} c{sup n}) for SAT for some c < 2. But if P{ne}NP, such a reduction cannot exist. The complexity of the Unique Satisfiability problem is an open and hard problem. It is known that Unique SAT is coNP-complete and in D{sup P}, but it is conjectured to be neither coNP-complete nor D{sup P}-complete. By means of polynomial reductions we show that for each c > 1 Unique SAT is decidable in deterministic time O({vert_bar} {Phi} {vert_bar} c{sup n}) iff SAT is decidable in deterministic time O({vert_bar} {Phi} {vert_bar} c{sup n}), where n is the number of variables. With slight modifications this holds for k-Unique SAT and k-SAT, too. Finally we present a proof that for each problem in NTIME(n) there is a polynomial reduction to SAT such that the number of variables in f({Phi}) is only O(n) improving Schnorr-Cook`s reduction with O(nlogn) variables.

• OSTI ID: 35867
• Report Number(s): CONF-9408161--
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGORITHMS
EFFICIENCY
MATRICES
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
OPTIMIZATION