The Consequences of Eliminating NP Solutions

TL;DR

This paper explores the theoretical possibilities and implications of reducing or eliminating solutions for NP functions, discussing complexity consequences and the role of advice strings in ambiguity reduction.

Contribution

It provides a comprehensive analysis of solution elimination in NP functions, highlighting the importance of advice strings and robustness in complexity-theoretic contexts.

Findings

Small advice strings are crucial for solution reduction.

Increasing advice size enhances robustness in ambiguity reduction.

Complexity-theoretic consequences depend on advice and problem framing.

Abstract

Given a function based on the computation of an NP machine, can one in general eliminate some solutions? That is, can one in general decrease the ambiguity? This simple question remains, even after extensive study by many researchers over many years, mostly unanswered. However, complexity-theoretic consequences and enabling conditions are known. In this tutorial-style article we look at some of those, focusing on the most natural framings: reducing the number of solutions of NP functions, refining the solutions of NP functions, and subtracting from or otherwise shrinking #P functions. We will see how small advice strings are important here, but we also will see how increasing advice size to achieve robustness is central to the proof of a key ambiguity-reduction result for NP functions.