Lower Bounds on Syntactic Logic Expressions for Optimization Problems and Duality using Lagrangian Dual to characterize optimality conditions
Abstract
We show that simple syntactic expressions such as existential second order (ESO) universal Horn formulae can express NP-hard optimisation problems. There is a significant difference between the expressibilities of decision problems and optimisation problems. This is similar to the difference in computation times for the two classes of problems; for example, a 2SAT Horn formula can be satisfied in polynomial time, whereas the optimisation version in NP-hard. It is known that all polynomially solvable decision problems can be expressed as ESO universal () Horn sentences in the presence of a successor relation. We show here that, on the other hand, if , optimisation problems defy such a characterisation, by demonstrating that even a (quantifier free) Horn formula is unable to guarantee polynomial time solvability. Finally, by connecting concepts in optimisation duality with those in descriptive complexity, we will show a method by which optimisation problems can be solved by a single call to a "decision" Turing machine, as opposed to multiple calls using a classical binary search setting.
Cite
@article{arxiv.0904.4331,
title = {Lower Bounds on Syntactic Logic Expressions for Optimization Problems and Duality using Lagrangian Dual to characterize optimality conditions},
author = {Prabhu Manyem},
journal= {arXiv preprint arXiv:0904.4331},
year = {2011}
}
Comments
An expansion of the previous version to include: a single call to a decision Turing machine to solve optimization problems obeying strong duality in polynomial time