Efficient decision procedure for the subset-join inequality

Develop an efficient algorithm to decide, for arbitrary subsets V_H ⊆ V and costs c in a CP3 instance, whether the subset-join certification inequality holds: max_{x ∈ X_G : x_{ij}=0} { Σ_{pqr∈T_H} c_{pqr} (1 − x_{pq} x_{pr} x_{qr}) + Σ_{pq∈E_H} c_{pq} (1 − x_{pq}) } ≤ min_{x ∈ X_G : x_{ij}=0} { Σ_{pqr∈T_{δ(V_H)}} c_{pqr} x_{pq} x_{pr} x_{qr} + Σ_{pq∈δ(V_H)} c_{pq} x_{pq} }, thereby enabling application of the subset-join partial optimality condition.

Background

Proposition “subset-join-proposition” provides a powerful certification to join all pairs inside a subset V_H when an inequality comparing worst-case interior separation against boundary coupling costs is satisfied.

The authors note that checking this inequality efficiently for general V_H and arbitrary costs is currently lacking, and they instead present a special-case corollary (under nonpositive interior costs) plus a heuristic search. An efficient general decision procedure would substantially broaden the condition’s practical utility.