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

Prabhu Manyem

Arxiv ID: 0909.5521•Last updated: 6/28/2022

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 Update: Manuscript withdrawn, because results are incorrect.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).

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