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

TL;DR

This paper explores the limitations of simple syntactic logical expressions in characterizing NP-hard optimization problems and introduces a duality-based method to solve such problems efficiently.

Contribution

It demonstrates the expressive limitations of existential second order Horn formulas for NP-hard optimization problems and connects duality with descriptive complexity for efficient solutions.

Findings

Simple syntactic expressions cannot guarantee polynomial-time solvability for NP-hard optimization problems.

All polynomially solvable decision problems can be expressed as ESO universal Horn sentences.

A duality-based approach allows solving optimization problems with a single decision machine call.

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 ($\Pi_1$) 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 $\Pi_0$ (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.