[Paper Review] NP-completeness of Partial Chirotope Extendibility

Patrick Baier|ArXiv.org|Apr 21, 2005

graph theory and CDMA systems2 references3 citations

TL;DR

This paper establishes the NP-completeness of extending a partially defined chirotope to a full chirotope by leveraging Donald Knuth's proof of NP-completeness for extending partial CC-systems. It demonstrates that the problem reduces to extending a directed graph to a vortex-free tournament, which is known to be NP-complete via 3SAT reduction, thereby proving chirotope extendibility is NP-complete as well.

ABSTRACT

In the monograph "Axioms and Hulls" (1992) 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.

Motivation & Objective

• To clarify the connection between Knuth's CC-system extendibility and chirotope extendibility.
• To establish that extending a partial chirotope to a complete chirotope is NP-complete.
• To show that the NP-completeness of CC-system extendibility directly implies the same complexity for chirotopes.
• To formalize the equivalence between uniform chirotopes and pre-CC-systems under the given axioms.
• To provide a clear reduction path from 3SAT to chirotope extendibility via vortex-free tournaments.

Proposed method

• Reduction of 3SAT to the problem of extending a directed graph to a vortex-free tournament.
• Use of Knuth’s axioms for CC-systems, particularly Axioms 1–3 and 5/5’ (interiority and transitivity), to define pre-CC-systems.
• Construction of a tournament from a partial triple system using the rule: p→q iff tpq for a fixed point t.
• Proof that a pre-CC-system exists if and only if the associated tournament is vortex-free.
• Demonstration that the Grassmann-Plücker relations are equivalent to the absence of forbidden subgraphs (vortices) in the tournament.
• Application of Knuth’s NP-completeness result for vortex-free tournament extension to conclude NP-completeness for chirotope extension.

Experimental results

Research questions

• RQ1 Is the problem of extending a partially defined chirotope to a complete chirotope computationally hard?
• RQ2 Does Knuth’s NP-completeness result for CC-system extendibility imply the same complexity for chirotopes?
• RQ3 Can the extendibility of a partial chirotope be reduced to the problem of extending a directed graph to a vortex-free tournament?
• RQ4 Are the Grassmann-Plücker relations logically equivalent to the vortex-freeness condition in tournaments derived from chirotopes?
• RQ5 What is the computational complexity of deciding whether a partial function on triples can be extended to a chirotope?

Key findings

• The extendibility of a partial chirotope to a full chirotope is NP-complete.
• The NP-completeness follows directly from Knuth’s proof of NP-completeness for extending partial CC-systems.
• A partial CC-system can be extended to a complete CC-system if and only if the associated tournament is vortex-free.
• The equivalence between uniform chirotopes and pre-CC-systems is established via the Grassmann-Plücker relations and the absence of forbidden subgraphs.
• The reduction from 3SAT to vortex-free tournament extension implies that chirotope extendibility is NP-complete.
• The problem remains NP-complete even when restricting to partial functions defined on triples with a common point.