On the NP-Completeness of Computing the Commonality Among the Objects Upon Which a Collection of Agents Has Performed an Action

We prove the NP-completeness of the so-called Social Group Commonality (SGC) problem which queries the commonality among the objects ‘touched’ by collections of agents while executing an action. Although it naturally arises in several contexts, e.g., in profiling the behavior of a collection of system users, SGC (to the authors’ knowledge) has been ignored. Our proof of SGC NP-completeness consists of a Karp reduction from the well-known Longest Common Subsequence (LCS) problem to SGC. We also prove that a special case of SGC which we call 2-SGC, where the commonality among actions is limited to agent pairs, remains NP-complete. For proving NP-completeness of 2-SGC though, our reduction departs from the well-known Hitting Set problem. Finally, we hypothesize that the optimality version of SGC is NP-hard, hinting on how to deal with the proof obligation.