Lower Bounds on Syntactic Logic Expressions for Optimization Problems and Duality using Lagrangian Dual to characterize optimality conditions

Prabhu Manyem

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arXiv · 0904.4331

Lower Bounds on Syntactic Logic Expressions for Optimization Problems and Duality using Lagrangian Dual to characterize optimality conditions

Prabhu Manyem

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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 ($Π_1$) Horn sentences in the presence of a successor relation. We show here that, on the other hand, if $P \neq NP$, optimisation problems defy such a characterisation, by demonstrating that even a $Π_0$ (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.

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Prabhu Manyem. 2011-07-24. Lower Bounds on Syntactic Logic Expressions for Optimization Problems and Duality using Lagrangian Dual to characterize optimality conditions. https://arxiv.org/abs/0904.4331

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