Combinatorial Characterisations of Graphical Computational Search Problems
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A Graphical Search problem, denoted Π(X,γ), where X is the vertex set or edge set of a graph G, consists of finding a solution Y, where Y ⊆ X and Y satisfies the predicate γ. Let Π̂ be the decision problem associated with Π(X,γ). A sub-solution of Π(X,γ) is a subset Y' that is a solution of the problem Π(X',γ), where X' ⊂ X. To Π(X,γ) we associate the set system (X, I), where I denotes the set of all the solutions and sub-solutions of Π(X,γ). The predicate γ is an accessible predicate if, given Y = ∅, Y is a solution of Π(X,γ) implies that there is an element y ∈ Y such that Y ∖ y is a sub-solution of Π(X,γ). If γ is an accessible property, we then show in Theorem 1 that a decision problem Π̂ is in P if and only if, for all input X, (X, I) satisfies Axioms M2', where M2', called the Augmentability Axiom, is an extension of both the Exchange Axiom and the Accessibility Axiom of a greedoid. We also show that a problem Π̂ is in P-complete if and only if, for all input X, (X, I) satisfies Axioms M2' and M1, where M1 is the Heredity Axiom of a matroid. A problem Π̂ is in NP if and only if , for all input X, (X, I) satisfies Axioms M2", where M2" is an extension of the Augmentability Axiom. Finally, the problem Π̂ is in NP-complete if and only if, for all input X, (X, I) satisfies Axiom M1. Using the fact that Hamiltonicity is an accessible property that satisfies M2", but does not satisfies Axiom M2', in Corollary 1 we get that P = NP.
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