NP-completeness of Partial Chirotope Extendibility

In the monograph “Axioms and Hulls” (1992) [2] Donald Knuth studies some axiomatizations of geometric situations. The structures described by one of the axiom systems are called CC-systems. Knuth proves that it is NP-complete to decide, whether a partially defined CC-system can be extended to a complete CC-system. The aim of this note is to show that Knuth’s proof of this result also implies that it is NP-complete to decide the extendability of partially defined chirotopes. Together with Stefan Felsner I’ve been interested in the question, whether partially given signotopes are extendable (cf. [1] for signotopes). Through a talk of Bernd Gärtner in Berlin we learned about a proof for the NPcompleteness of chirotope extendibility by Falk Tschirschnitz [3, 4]. Remarks of Ileana Streinu and Stefan Felsner brought my attention to an NPcompleteness proof of Knuth [2]. In fact, the work of Knuth implies that it is NP-complete to decide the extendability of partially defined chirotopes. The aim of this note is to make the connections clear. Knuth defines CC-systems as a boolean function of all ordered triples of a given finite groundset (whose elements are usually referred to as points) following five Axioms, where an expression of the form pqr means that the value associated with (p, q, r) is true (p. 3-4): Axiom 1: pqr =⇒ qrp (cyclic symmetry) Axiom 2: pqr =⇒ ¬ prq (antisymmetry) Axiom 3: pqr ∨ prq (nondegeneracy) Axiom 4: tqr ∧ ptr ∧ pqt =⇒ pqr (interiority) Axiom 5: tsp ∧ tsq ∧ tsr ∧ tpq ∧ tqr =⇒ tpr (transitivity) Axiom 5’: tps ∧ tqs ∧ trs ∧ tpq ∧ tqr =⇒ tpr (dual transitivity) (Each of these axioms is to be read with an implied quantification “for all Page numbers always refer to Knuth’s monograph [2].