Toward the complexity of the existence of wonderfully stable partitions and strictly core stable coalition structures in enemy-oriented hedonic games

Abstract

We study the computational complexity of the existence and the verification problem for wonderfully stable partitions (WSPE and WSPV) and of the existence problem for strictly core stable coalition structures (SCSCS) in enemy-oriented hedonic games. In this note, we show that WSPV is NP-complete and both WSPE and SCSCS are DP-hard, where DP is the second level of the boolean hierarchy, and we discuss an approach for classifying the latter two problems in terms of their complexity.

Hedonic Games with Ordinal Preferences and Thresholds

“…In these nontrivial cases, we ask how hard it is to decide whether for a given FEN-hedonic game a given coalition structure possibly or necessarily satisfies γ, and to decide whether there exists a coalition structure in a given FEN-hedonic game that possibly or necessarily satisfies γ. Similar questions are often analyzed in the context of hedonic games (Woeginger, 2013b;Aziz et al, 2013b;Rey et al, 2016). We now adapt the definition of the verification problem to the notions of possible and necessary verification, and we similarly adapt the definition of the existence problem to possible and necessary existence.…”

“…6. The friends-and-enemies encoding (Dimitrov et al, 2006;Sung & Dimitrov, 2007;Rey et al, 2016;Nguyen et al, 2016): Each agent partitions the set of other agents into two sets (her friends and her enemies); under the friend-oriented preference extension, coalition X is preferred to coalition Y if X contains more friends than Y , or as many friends as Y and fewer enemies than Y ; under the enemy-oriented preference extension, X is preferred to Y if X contains fewer enemies than Y , or as many enemies as Y and more friends than Y . 7.…”

Anchored Team Formation Games

“…ASHGs are of particular interest to our work, because they are generalized by ATFGs; there are many other hedonic games inspired by ASHGs [1,6,13,14,20]. There are many hardness results for ASHGs and their variants, which establish some baselines for our work [2,15,16,19,29].…”

Altruistic Hedonic Games

“…Also considering friends, enemies, and neutral agents, Kerkmann, Lang, Rey, Rothe, Schadrack, and Schend (2020) propose a bipolar extension of the responsive extension principle and use it to derive partial preferences over coalitions, characterize coalition structures that necessarily or possibly satisfy certain stability concepts, and study the related problems in terms of their complexity. Barrot, Ota, Sakurai, and Yokoo (2019) study the impact of additional unknown agents, and Rey, Rothe, Schadrack, and Schend (2016) study wonderful stability (a.k.a. perfectness) and strict core stability in enemy-oriented hedonic games.…”