Syntactic Characterisations of Polynomial-Time Optimisation Classes (Syntactic Characterizations of Polynomial-Time Optimization Classes)

TL;DR

This paper explores the descriptive complexity of polynomial-time optimization problems, providing logical characterizations for classes within P and extending to non-polynomially bounded maximization problems.

Contribution

It offers new logical characterizations of polynomial-time optimization classes using second-order logic, including universal Horn formulas with successor relations.

Findings

Logical characterizations for polynomial-time optimization problems.

Extension of characterizations to non-polynomially bounded maximization problems.

Bridging the gap between decision problem complexity and optimization problem descriptions.

Abstract

In Descriptive Complexity, there is a vast amount of literature on decision problems, and their classes such as \textbf{P, NP, L and NL}. ~ However, research on the descriptive complexity of optimisation problems has been limited. Optimisation problems corresponding to the \textbf{NP} class have been characterised in terms of logic expressions by Papadimitriou and Yannakakis, Panconesi and Ranjan, Kolaitis and Thakur, Khanna et al, and by Zimand. Gr\"{a}del characterised the polynomial class \textbf{P} of decision problems. In this paper, we attempt to characterise the optimisation versions of \textbf{P} via expressions in second order logic, many of them using universal Horn formulae with successor relations. The polynomially bound versions of maximisation (maximization) and minimisation (minimization) problems are treated first, and then the maximisation problems in the "not necessarily polynomially bound" class.