Distributed Reasoning for Multiagent Simple Temporal Problems

Definition 4. We label lines 4-8 of the 4PPC algorithm as the reinstate procedure. This procedure reinstates a timepoint vk by, for every pair of previously-reinstated, neighboring timepoints vi and vj, tightening edge eki and edge ejk with respect to eij. Lemma 1. Let o and o′ be two distinct total orderings of the vertices, V for some graph G = 〈V,E〉. If o′ is consistent with the precedence relation ≺o , then G4 o = G4 o′ . Proof. Assume G4 o 6= G4 o′ . Since G4 o = 〈 V,E ∪ E4 o 〉 and G4 o′ = 〈 V,E ∪ E4 o′ 〉 , if G4 o 6= G4 o′ then E 4 o 6= E4 o′ . E4 o 6= E4 o′ implies that there exists at least one edge exy such that, either exy ∈ E 4 o and exy / ∈ E4 o′ , or exy / ∈ E 4 o and exy ∈ E4 o′ . WLOG, let exy ∈ E4 o be the first edge that is added to E4 o under elimination order o that is not added to E4 o′ under o ′. In order for edge exy to be added under elimination order o, there must be some vertex vz such that it is eliminated prior to vx and vy and shares an edge with both vx and vy (exz and eyz respectively) at the time of its elimination. By