Efficient methods to find subsets for join conditions

Develop efficient algorithms to find subsets V_H ⊆ V and U ⊆ V that satisfy the join-condition prerequisites needed to apply the general-subgraph edge-join (Proposition “general-subgraph-edge-join”) and subset-join (Proposition “subset-join-proposition”) partial optimality results for cubic correlation clustering CP3 on a graph G=(V,E) with triple set T and costs c. Specifically, for Proposition “general-subgraph-edge-join,” identify V_H and cuts U ⊂ V_H that fulfill the inequality Σ_{pq∈δ(V_H)∩E^-} c_{pq} + Σ_{pqr∈T_{δ(V_H)}∩T^-} c_{pqr} ≥ Σ_{pq∈δ(U, V_H\U)} c_{pq} + Σ_{pqr∈T_{δ(U, V_H\U)}∩T_H} c_{pqr}, and for Proposition “subset-join-proposition” identify V_H allowing certification that all edges in E_H should be joined.

Background

The paper introduces certified join conditions that, when satisfied, allow merging nodes and thereby reducing cubic correlation clustering instances. Applying these conditions requires discovering specific subsets of nodes (V_H and U) that satisfy nontrivial inequalities linking edge and triangle costs.

While the authors provide special-case corollaries for very small V_H (|V_H| ∈ {2,3}) and practical heuristics, they explicitly note the absence of efficient general procedures to find these subsets for larger subgraphs, limiting the applicability of the stronger join results.