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

Es NP-completo calcular la comunidad entre los objetos sobre los que una colección de agentes ha realizado una acción

Roberto Alonso, Raúl Monroy

Department of Computer Science, Tecnologico de Monterrey, Campus Estado de Mexico, Carr. lago de Guadalupe Km 3.5, Atizapan, Estado de Mexico, Mexico. roberto.alonso@itesm.mx, raúlm@itesm.mx

Abstract

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.

Keywords: Social Group Commonality, complexity theory, social networks, graphs.

Resumen

En este trabajo demostramos que el problema que llamamos Comunalidad de grupos sociales (SGC por sus siglas en inglés) es NP-completo. Este problema consulta la comunalidad entre los objetos tocados por una colección de agentes que ejecutan acciones. Aunque se presenta naturalmente en varios contextos e.g., perfilar el comportamiento de un conjunto de usuarios de un sistema, SGC ha sido, acorde al conocimiento de los autores, ignorado. Nuestra demostración consiste en una reducción de Karp a partir del problema conocido como Longest Common Subsequence (LCS). Probamos también que un caso especial, al que llamamos 2-SGC, donde la comunalidad entre las acciones esta limitada a pares de agentes, sigue siendo NP-completo. Para probar 2-SGC, nuestra reducción parte del problema conocido como Hitting Set. Antes de concluir con el articulo, especulamos que la versión de optimización de SGC es NP-duro, dando indicaciones de como realizar la demostración necesaria.

Palabras clave: Comunalidad de grupos sociales, teoría de la complejidad, redes sociales, grafos.

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Acknowledgement

This paper has largely benefited from numerous discussions with Luis Angel Trejo-Rodriguez. We thank to the members of the NetSec group at Tecnologico de Monterrey, Estado de Mexico, for their constructive comments on an earlier version of this paper. The first author was supported by CONACYT student scholarship 45904, while the second author was in part supported by CONACyT grant 105698.

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