Clique and Vertex Cover are solvable in polynomial time if the input structure is ordered and contains a successor predicate

TL;DR

This paper initially claimed polynomial-time solutions for Clique and Vertex Cover problems under certain ordered structures with successor predicates, but was later withdrawn due to errors in the logic and expressibility assumptions.

Contribution

The paper proposed a novel approach to solve Clique and Vertex Cover in polynomial time under specific logical and structural conditions, which was later retracted.

Findings

Initial claims of polynomial-time algorithms were invalidated.

Logical expressibility constraints prevent the proposed solutions.

The manuscript was withdrawn due to errors in the core assumptions.

Abstract

In this manuscript, assuming that Graedel's 1991 results are correct (which implies that bounds on the solution values for optimization problems can be expressed in existential second order logic where the first order part is universal Horn), I will show that Clique and Vertex Cover can be solved in polynomial time if the input structure is ordered and contains a successor predicate. In the last section, we will argue about the validity of Graedel's 1991 results. Update: Manuscript withdrawn, because results are incorrect. If phi = phi_1 AND phi_2, and phi is a Horn formula, it does NOT mean that both phi_1 and phi_2 are Horn formulae. Furthermore, the cardinality constraint CANNOT be expressed as a universal Horn sentence in ESO (NOT even when the structure is ordered).