Temel Kavramlar
FMplex is a novel variable elimination method that combines the strengths of the Fourier-Motzkin and simplex algorithms, reducing the worst-case complexity from doubly to singly exponential while maintaining the ability to perform quantifier elimination.
Özet
The paper presents FMplex, a new variable elimination method for linear real arithmetic (LRA) constraints. FMplex is derived from the Fourier-Motzkin (FM) variable elimination procedure, but it uses a divide-and-conquer approach to reduce the worst-case complexity from doubly to singly exponential.
The key ideas are:
- FMplex performs a case split on the lower or upper bounds of a variable, rather than considering all bound combinations at once as in FM. This avoids certain redundancies that FM might generate.
- FMplex has a strong correspondence to the simplex algorithm, providing interesting theoretical insights into the relation between the two established methods.
- The authors adapt FMplex for satisfiability modulo theories (SMT) solving, including methods to prune the search tree based on structural observations.
- The authors provide a formal theorem connecting FMplex and the simplex algorithm, as well as a comprehensive experimental evaluation.
The paper extends the authors' previous work by providing additional explanations, more detailed examples, and full proofs.