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%3Stanford University Computer Science Reports%1
List Number 10↔December 1980
Listed here are abstracts of the most recent reports published by the
Department of Computer Science at Stanford University.
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%2Efficient Algorithms for certain Satisfiability and Linear Programming
Problems%1 by Bengt Aspvall (Thesis, ? pages, September 1980)
We present efficient algorithms for certain cases of Linear Equations solving,
Linear Programming, and SATisfiability testing. LE, LP, and SAT have all been
studied in both the theoretical and the applied areas of Computer Science.
Although many algorithms have been developed for these problems, there are
large gaps between today's best algorithms and the best theoretical lower
bounds. We do not eliminate these gaps, but instead present efficient
algorithms for LE, LP, and SAT (and for testing the truth of Quantified
Boolean Formulas) for the cases in which there are at most two variables or
literals per equation, inequality, or clause. These subcases seem to be
the "hardest" ones that are not as hard as the general problems.
The algorithms for LE(2), 2-SAT, and 2-QBF are linear-time algorithms, and they
are optimal except for constant factors. Our algorithm for the subcase of LP has
a nonlinear worst-case bound, but the bound is better than for any general LP
algorithm such as the polynomial-time algorithm by Khachiyan, and the algorithm
seems to be of practical as well as theoretical importance. In the algorithms,
we associate graphs with given problem instances. By performing searches on
the graphs (in the LP case combined with a binary search technique), we construct
a solution or a satisfying truth assignment if one exists. We present also a
linear-time algorithm, which uses the 2-SAT algorithm, for mapping certain Boolean
expressions in Conjunction Normal Form to equivalent CNF expressions having at
most one negated variable per clause (without increasing the length of the
expression); such CNF expressions can be tested for satisfiability in linear time.