Definable Inapproximability: New Challenges for Duplicator

TL;DR

This paper demonstrates that certain NP-hard optimization problems cannot be approximated by algorithms expressible in fixed-point logic with counting, establishing logical inapproximability bounds without relying on complexity assumptions.

Contribution

It introduces a logical framework to prove inapproximability results for NP-hard problems, linking definability in fixed-point logic with counting to approximation hardness.

Findings

No FPC-definable algorithms can approximate certain NP-hard problems within fixed bounds.

Lower bounds on the number of variables needed in first-order logic with counting.

Establishment of logical inapproximability bounds independent of P vs NP assumptions.

Abstract

We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P = NP, no polynomial-time algorithm can give an approximate solution guaranteed to be within a fixed constant factor of the optimum. We show, in several such instances and without any complexity theoretic assumption, that no algorithm that is expressible in fixed-point logic with counting (FPC) can compute an approximate solution. Since important algorithmic techniques for approximation algorithms (such as linear or semidefinite programming) are expressible in FPC, this yields lower bounds on what can be achieved by such methods. The results are established by showing lower bounds on the number of variables required in first-order logic with counting to separate instances with a high optimum from those with a low optimum for fixed-size instances.