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Two hardness results for core stability in hedonic coalition formation games
Discrete Applied Mathematics 161 (2013) 1837–1842
Contents lists available at SciVerse ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Two hardness results for core stability in hedonic coalition formation games Vladimir G. Deineko a , Gerhard J. Woeginger b,∗ a
Warwick Business School, The University of Warwick, Coventry CV4 7AL, United Kingdom
b
Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
article
info
Article history: Received 6 October 2012 Received in revised form 26 February 2013 Accepted 4 March 2013 Available online 26 March 2013
abstract We establish NP-completeness of two problems on core stable coalitions in hedonic games. In the first problem every player has only two acceptable coalitions in his preference list, and in the second problem the preference structures arise from the distances in an underlying metric space. © 2013 Elsevier B.V. All rights reserved.
Keywords: Computational social choice Computational complexity Coalition formation Hedonic game
1. Introduction In many economic, political and social environments individuals carry out their activities in groups; typical examples are households, trade unions, political parties, families, sport clubs, carpools, and research networks. The underlying game theoretic framework is known as coalition formation, where individuals are called players and groups are called coalitions. In hedonic coalition formation each player’s happiness/satisfaction only depends on the other members of his coalition and not on how the remaining players outside his coalition are grouped together. This hedonic research line goes back to the seminal paper [12] of Drèze and Greenberg. A central issue in coalition formation is the stability of a system of coalitions: whenever there is a possibility of increasing one’s happiness/satisfaction by moving to another coalition or by merging or splitting or otherwise restructuring the coalition system, the players will react accordingly and the system will change. A stable system is not subject to such changes. The social choice literature knows a wide variety of stability concepts, such as for instance the core, the strict core, the Nash stable set, the individually stable set, and the contractually individually stable set; we refer the reader to Banerjee, Konishi and Sönmez [5] and Bogomolnaia and Jackson [7]. In this paper we will concentrate on so-called core stable systems of coalitions in hedonic games: a system is core stable, if no subset of players can strictly increase their happiness by splitting off from their current coalitions and forming a new coalition. Well-known special cases of the core stability problem are the stable matching problem [13], the stable roommate problem [15], and the three-sided stable matching problem [1]. There is a rich literature analyzing the computational complexity of various special cases, such as for instance discussed by Aziz, Brandt and Seedig [3], Ballester [4], Cechlárová and Hajduková [8,9], Dimitrov, Borm, Hendrickx and Sung [11], Irving and Manlove [16], Ng and Hirschberg [18], Ronn [19], Subramanian [20], and Woeginger [21,22].
∗
Corresponding author. Tel.: +31 40 247 2415; fax: +31 40 246 5995. E-mail addresses: [email protected] (V.G. Deineko), [email protected] (G.J. Woeginger).
0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.03.003
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V.G. Deineko, G.J. Woeginger / Discrete Applied Mathematics 161 (2013) 1837–1842
Contributions of this paper. After summarizing the central definitions and concepts in Section 2, we will derive two new hardness results on core stability in hedonic coalition formation games. Both proofs are concise, clean, and extremely simple. Our results close two gaps in our understanding of hedonic games. Our first result concerns hedonic games with short preference lists. A coalition is unacceptable for a player, if he prefers staying alone to participating in that coalition; otherwise the coalition is called acceptable. Irving, Manlove and O’Malley [17] consider the variant of Gale–Shapley matching where every man and every woman have only three acceptable partners; they establish NP-completeness of deciding whether in this scenario there exists a perfect stable matching (which pairs up all the players). Furthermore, [17] shows that the corresponding Gale–Shapley problem with only two acceptable partners is polynomially solvable. In Section 3 we discuss hedonic games in which every player has only two acceptable coalitions of arbitrary size, beyond the option of staying alone (whereas [17] consider the case of three acceptable coalitions of size two). We show that in this scenario deciding the existence of a core stable partition into coalitions is NP-complete. Our second result concerns the 3-dimensional stable roommate problem; in this problem the only feasible coalitions are those of size three. Arkin, Bae, Efrat, Okamoto, Mitchell and Polishchuk [2] discuss a special case with metric preference structures, where every player is some point in a metric space with distance function d(·, ·). Player p strictly prefers triple px1 x2 to triple py1 y2 if and only if d(p, x1 ) + d(p, x2 ) < d(p, y1 ) + d(p, y2 ). Arkin et al. [2] exhibit a highly structured instance that does not possess a core stable matching; the computational complexity of their special case however remained open. In Section 4 we establish NP-completeness of deciding the existence of a core stable partition for the metric 3-dimensional stable roommate problem. The complexity of this problem for concrete metric spaces (like the Euclidean plane) remains open. 2. Definitions and preliminaries Let N be a finite set of players. A coalition is a non-empty subset of N. Every player i ∈ N ranks all the coalitions containing i via his preference relation ≼i , which is reflexive, transitive and complete. The underlying strict order is denoted ≺i , where S ≺i T means that S ≼i T but not T ≼i S. If S ≺i T then player i (strictly) prefers participating in T to participating in S, and if S ≼i T then player i weakly prefers participating in T to participating in S. A partition Π is a collection of coalitions which partition N; hence every coalition in Π is non-empty, distinct coalitions are disjoint, and the union of all coalitions equals N. For a partition Π and a player i, we denote by Π (i) the unique coalition in Π containing i. The following definition introduces core stability, a key stability concept in hedonic games. Definition 2.1. A coalition S blocks a partition Π , if every player i ∈ S strictly prefers Π (i) ≺i S. A partition Π is core stable, if there is no blocking coalition. A coalition is a singleton coalition, if it consists of a single player. A coalition C is acceptable for player i, if it satisfies both C ≽i {i} and C ̸= {i}; in other words, player i does not prefer staying alone to being in an acceptable coalition C . Note that any core stable partition must entirely consist of singleton coalitions and of coalitions that are acceptable to all of their members. 3. Hardness for a variant with short preference lists Throughout this section we will investigate hedonic games with preference lists of length two that satisfy the following conditions. First, the ranking of every player i is of the form C1 (i) ≻i C2 (i) ≻i {i}. Secondly, all coalitions C1 (i) and C2 (i) are acceptable for all of their members. (Note that the first condition is the crucial one, whereas the second condition is purely technical and not restrictive at all: one could ignore coalitions C containing players for which C is unacceptable, as such coalitions will never be part of a core stable coalition.) We will model such a hedonic game in the following way as a directed multi-graph: the vertices of the graph are the acceptable coalitions of the players, and for every player i there is a corresponding arc going from vertex C2 (i) to vertex C1 (i). Note that every coalition (vertex) in this directed graph consists of exactly those players that correspond to its incident arcs. A subset W of the vertices is a kernel, if it satisfies the following two properties K1 and K2: K1. For every vertex w ∈ W , none of w ’s successors lies in W . K2. For every vertex w ̸∈ W , at least one of w ’s successors lies in W . The following theorem establishes a connection between kernels and core stable partitions, which might be of independent interest. Theorem 3.1. A hedonic game with preference lists of length two has a core stable partition, if and only if its underlying directed graph contains a kernel. Proof. First assume that the game has a core stable partition Π , and define the set W to contain all vertices corresponding to coalitions in Π that are not singleton. We claim that W is a kernel. For property K1, suppose for the sake of contradiction that W would contain a vertex together with one of its successors. Then the player corresponding to the connecting arc would be contained in two distinct coalitions in Π , which contradicts Π being a partition. For property K2, suppose that
V.G. Deineko, G.J. Woeginger / Discrete Applied Mathematics 161 (2013) 1837–1842
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there is a vertex w ̸∈ W so that none of w ’s successors lies in W . Then all the players that correspond to arcs going out of w are singleton in Π , and prefer coalition w to being alone. Furthermore all the players that correspond to arcs going into w have w as their top preference and would prefer it to their current coalition in Π . Hence w would be blocking, which contradicts our assumption that Π is core stable. Next assume that the directed graph contains a kernel W , and note that by property K1 the coalitions in W are pairwise disjoint. We define a partition Π of the players that uses of all the coalitions in W and that puts the remaining players into singleton coalitions. We claim that Π is core stable. By property K2, any coalition s ̸∈ W in the directed graph has some successor t with t ∈ W . The player that corresponds to the arc from s to t prefers his current coalition t in Π to his second choice s; this means that s is not blocking. Chvátal [10] proved that it is NP-complete to decide whether a given directed graph contains a kernel; see also problem [GT57] in Garey and Johnson [14]. Our next goal is to establish the NP-completeness of a highly restricted special case of the kernel problem, where every vertex in the directed graph is incident to exactly three (in-going and out-going) arcs. The reduction will be done from the following (NP-complete) version of the SATISFIABILITY problem; see Berman, Karpinski and Scott [6]. Problem: (3, B2)-SAT. Instance: A set X of Boolean variables; a Boolean formula φ over X in conjunctive normal form where (i) each clause consists of exactly three (distinct) literals, and (ii) every variable occurs exactly twice in negated and exactly twice in unnegated form. Question: Is formula φ satisfiable? For an instance of (3, B2)-SAT, we let set R contain all variable-clause pairs (x, c ) for which variable x occurs (negated or unnegated) in clause c. We construct the following directed graph.
• For every variable x ∈ X , we introduce a directed cycle on four vertices (which we call a variable-cycle). The four vertices correspond to the four variable-clause pairs for x in set R, so that negated and unnegated occurrences of x alternate along the cycle. • For every clause c ∈ φ , we introduce a directed cycle on three vertices (which we call a clause-cycle). The three vertices correspond to the three variable-clause pairs for c in set R. • For every variable-clause pair (x, c ) ∈ R, we create an arc that goes from the corresponding vertex in the clause-cycle to the corresponding vertex in the variable-cycle. Note that in the constructed directed graph, every vertex is indeed incident to exactly three (in-going and out-going) arcs. Lemma 3.2. The (3, B2)-SAT instance has a satisfying truth assignment, if and only if the constructed directed graph contains a kernel. Proof. (If). Consider a kernel W . It is easily seen from properties K1 and K2 that W contains exactly two, non-consecutive vertices from each variable-cycle. If W contains the two vertices that correspond to the unnegated occurrences of x, we set x to true; if W contains the vertices corresponding to negated occurrences, we set x to false. We claim that this truth assignment satisfies all clauses c ∈ φ . Indeed, consider the clause-cycle corresponding to c. Property K1 implies that at most one of the three vertices is in W , and we consider an arc (x, c ) → (y, c ) on the cycle whose endpoints are not in W . Property K2 implies that vertex (x, c ) ̸∈ W has a successor in W , and the only possible candidate for this is the successor on the variable-cycle for x. The truth value of variable x has been chosen such that clause c is true. (Only if). Let T be a satisfying truth assignment for φ . In a first phase we select from every variable-cycle two vertices for set W : if variable x is true under T then we pick the two vertices that correspond to unnegated occurrences, and if variable x is false under T then we pick the two vertices that correspond to negated occurrences. In the second phase, we consider a clause-cycle for some clause c ∈ φ . Every vertex in this cycle has a successor on some variable-cycle. As c contains one or two or three true literals, one or two or three of these successors have already been picked for W in the first phase. If all three successors have been picked, we do nothing. If one or two successors have been picked, we pick one additional vertex from the clause-cycle for W so that K1 and K2 are fulfilled; we leave the (straightforward) details to the reader. The resulting set W is a kernel. Lemma 3.2 yields the NP-completeness of deciding the existence of a kernel for the highly restricted special case where every vertex in the directed graph is incident to exactly three arcs. We start from such a restricted kernel instance, and translate it into a corresponding hedonic game by essentially reverting the multi-graph construction: every arc becomes a player, every vertex becomes a coalition of size three (that consists of the three incident arcs/players), and every arc/player prefers his head vertex/coalition to his tail vertex/coalition. Then Theorem 3.1 yields the following NP-completeness result: Theorem 3.3. It is NP-complete to decide whether a hedonic game with preference lists of length two has a core stable partition, even in the case where all acceptable coalitions have size three.
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The statement in Theorem 3.3 is essentially tight, as any decrease in the values of its crucial parameters leads to a tractable problem. Indeed, the corresponding problem for preference lists of length one (and acceptable coalitions of unrestricted size) is straightforward, as there always exists a core stable partition. One possibility is to take a maximal system of pairwise disjoint acceptable coalitions, and to add an appropriate system of singleton coalitions to this. And also the corresponding problem for preference lists of length two and acceptable coalitions of size two is polynomially solvable; see Irving, Manlove and O’Malley [17]. 4. Hardness for a metric variant In the 3-dimensional stable roommate problem, the allowed coalitions are exactly the coalitions of size three. The goal is to decide whether there is a core stable partition. In this section we analyze a metric 3-dimensional roommate variant, where every player is some point in a metric space equipped with a standard distance function d(·, ·) that is symmetric (so that d(x, y) = d(y, x) for all x, y) and that satisfies the triangle inequality (so that d(x, y) + d(y, z ) ≥ d(x, z ) for all x, y, z). Player p weakly prefers triple px1 x2 to triple py1 y2 if and only if d(p, x1 ) + d(p, x2 ) ≤ d(p, y1 ) + d(p, y2 ). If there is strict inequality, then the player strictly prefers triple px1 x2 . The crucial quantity d(p, x1 ) + d(p, x2 ) will be called the cost of triple px1 x2 for player p. In this section we establish the NP-completeness of deciding the existence of a core stable partition for such metric instances. The reduction is done from the following 2-dimensional stable roommate variant. Problem: Classical Roommate. Instance: A set M of 2m players; for every player i ∈ M a rank function ri : M → {1, . . . , 2m} such that player i strictly prefers player j′ to j′′ as roommate if and only if rj (j′ ) < rj (j′′ ). Question: Does there exist a core stable partition of M into pairs? Note that in Classical Roommate the only feasible coalitions are pairs. We also stress that the underlying rank functions rj are not necessarily injective and that ties rj (k′ ) = rj (k′′ ) are allowed. Irving [15] has shown that the problem variant without such ties is polynomially solvable, whereas Ronn [19] proved that the considered variant with ties is NP-complete. We start from an arbitrary instance of Classical Roommate, and we define a corresponding instance of the metric 3dimensional roommate problem. In order to distinguish clearly between the players in both games, we will sometimes speak of old players (in the 2-dimensional variant) and of new players (in the 3-dimensional variant). We introduce the following new players.
• For every old player i ∈ M, there is a corresponding new main-player P (i). • For every 2-element subset {i, j} ⊆ M, we create three corresponding new pair-players P {i, j}, P ′ {i, j}, and P ′′ {i, j}. • Furthermore there are m new dummy players D1 , . . . , Dm . Next we specify the distances in the underlying metric space. All distances will lie in the left-open–right-closed interval (2L, 4L] with L = 8m; note that distances from this range automatically satisfy the triangle inequality.
• • • •
The distance between P (i) and P (j) equals 3L. The distance between P (i) and P {i, j} equals 2L + ri (j). The distances from P {i, j} to P ′ {i, j} and from P {i, j} to P ′′ {i, j} are both 3L. All remaining distances are 4L.
The following two Lemmas 4.1 and 4.2 exhibit the close connections between the instance of Classical Roommate and the constructed 3-dimensional instance. Lemma 4.1. If the instance of Classical Roommate has a core stable partition into pairs, then there exists a core stable partition into triples for the constructed 3-dimensional roommate instance. Proof. We consider the stable matching µ for the old players in M, and construct from it the following partition of the new players: Whenever (i, j) ∈ µ, we use the triple consisting of P (i), P (j), P {i, j} and the triple that consists of P ′ {i, j}, P ′′ {i, j} together with some dummy player. Whenever (i, j) ̸∈ µ, we use the triple P {i, j}, P ′ {i, j}, P ′′ {i, j}. Note that in these triples every main-player P (i) incurs a cost strictly below 6L and every pair-player P {i, j} incurs a cost of at most 6L. We claim that these triples form a core stable partition Π for the 3-dimensional roommate instance. Suppose for the sake of contradiction that there exists a blocking triple. If some main-player P (a) participates in the blocking triple, then his cost must stay strictly below 6L; consequently P (a) can only block together with main-players or with pair-players of the type P {a, c }. We discuss three subcases. First, three main-players P (a), P (b), P (c ) cannot form a blocking triple, as the resulting cost for P (a) would be 6L. Secondly, three players P (a), P {a, b}, P {a, c } cannot form a blocking triple, as the resulting cost for the pair-player P {a, b} would be above 6L. Thirdly, three players P (a), P (b), P {a, b} cannot form a blocking triple: if (a, a′ ) and (b, b′ ) are the corresponding pairs in the stable matching µ, a blocking triple P (a), P (b), P {a, b} would imply that old player a prefers b to a′ and that old player b prefers a to b′ . Hence (a, b) would be a blocking pair for matching µ, so that µ itself is not stable. This contradiction closes the third subcase and shows that no main-player P (a) can participate in the blocking triple.
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Next, we observe that also none of the pair-players P {a, b} can participate in the blocking triple: the distance from P {a, b} to any remaining pair-player or dummy player equals 3L or 4L, so that there is no way of beating his cost 6L in partition Π . To summarize, all remaining candidates for a blocking triple are the dummy players and the pair-players P ′ {a, b} and P ′′ {a, b}. These players are pairwise at distance 4L, and can never form a blocking set. Lemma 4.2. If the constructed 3-dimensional roommate instance has a core stable partition into triples, then the instance of Classical Roommate has a core stable partition into pairs. Proof. We consider the stable partition Π that divides the new players into triples, and we derive some structural properties of Π . As a first property, we note that the cost of any pair-player P {i, j} in Π is at most 6L; otherwise Π would be blocked by the triple P {i, j}, P ′ {i, j}, P ′′ {i, j}. As the second property, we show the following: whenever partition Π puts two main-players P (a) and P (b) into a common triple, then the third partner in the triple must be the pair-player P {a, b}; such a triple is called canonical. Indeed, assume that some triple in the stable partition Π consists of two main-players P (a) and P (b) together with some third player X.
• If X is a main-player P (c ), a pair-player P ′ {i, j} or P ′′ {i, j}, or a dummy player, then the triple P (a), P (b), P {a, b} would be blocking. The cost of players P (a) and P (b) would improve from 6L or higher down to 5L + ra (b) and 5L + rb (a), respectively, and the cost of player P {a, b} would drop from 6L or higher to 4L + ra (b) + rb (a). • If X is a player P {i, j} with {i, j} ̸= {a, b}, then the cost of P {i, j} would be larger than 6L and thus contradict the first property. Hence the only remaining possibility is that X is the pair-player P {a, b}; this proves the second property. Next suppose that in partition Π , some main-player P (a) is not in a canonical triple. Then the partners of P (a) cannot be two pair-players P {a, i} and P {a, j}, as this would contradict the first property; we conclude that the cost of P (a) in Π lies above 6L. As the cardinality of M is even, there exists another main-player P (b) who also is not in a canonical triple and whose cost in Π also must lie above 6L. But then partition Π is blocked by the triple P (a), P (b), P {a, b}: the cost of P (a) and P (b) would drop from a value above 6L to a value below 5L, and the cost of P {a, b} would improve from above 5L to below 5L. All in all, this means that partition Π puts all the main-players into canonical triples. We define a matching µ for the old players in M that pairs up two old players i and j whenever the corresponding new main-players P (i) and P (j) in Π are together in a canonical triple. We claim that matching µ is stable. Suppose that there is a blocking pair (a, b) for µ, with corresponding pairs (a, a′ ) and (b, b′ ) in µ. This means ra (b) < ra (a′ ) and rb (a) < rb (b′ ), and a fortiori implies that the triple P (a), P (b), P {a, b} is blocking for the stable partition Π . This contradiction yields the stability of matching µ. Theorem 4.3. In the metric variant of the 3-dimensional stable roommate problem, it is strongly NP-complete to decide the existence of a core stable partition. Proof. The problem is clearly contained in NP: the YES-certificate is a core stable partition, and the verification is done by inspecting all O(n3 ) triples that are candidates for a blocking set. For NP-hardness, we note that the above reduction can be done in polynomial time and that all involved integers are polynomially bounded in |M |. Lemmas 4.1 and 4.2 establish correctness of the reduction: the constructed instance of the metric 3-dimensional roommate problem has a core stable partition into triples, if and only if the considered instance of Classical Roommate has a core stable partition into pairs. Arkin et al. [2] construct an instance of the metric 3-dimensional stable roommate problem that does not allow a core stable matching. The underlying metric space is the Euclidean plane. The computational complexity of this Euclidean special case is an open (and seemingly challenging) problem. Acknowledgments We thank the referees for their constructive comments which helped to improve the presentation of the paper. VD acknowledges support by Warwick University’s Centre for Discrete Mathematics and Its Applications (DIMAP) and by the EPSRC fund EP/F017871. GW acknowledges support by DIAMANT (a mathematics cluster of The Netherlands Organization for Scientific Research NWO), and by the Alexander von Humboldt Foundation, Bonn, Germany. References [1] A. Alkan, Non-existence of stable threesome matchings, Mathematical Social Sciences 16 (1988) 207–209. [2] E.M. Arkin, S.W. Bae, A. Efrat, K. Okamoto, J.S.B. Mitchell, V. Polishchuk, Geometric stable roommates, Information Processing Letters 109 (2009) 219–224. [3] H. Aziz, F. Brandt, H.G. Seedig, Computing desirable partitions in additively separable hedonic games, Artificial Intelligence 195 (2013) 316–334. [4] C. Ballester, NP-completeness in hedonic games, Games and Economic Behavior 49 (2004) 1–30. [5] S. Banerjee, H. Konishi, T. Sönmez, Core in a simple coalition formation game, Social Choice and Welfare 18 (2001) 135–153. [6] P. Berman, M. Karpinski, A.D. Scott, Hardness of short symmetric instances of MAX-3SAT, Electronic Colloquium on Computational Complexity (2003) 049.
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[7] A. Bogomolnaia, M.O. Jackson, The stability of hedonic coalition structures, Games and Economic Behavior 38 (2002) 201–230. [8] K. Cechlárová, J. Hajduková, Computational complexity of stable partitions with B-preferences, International Journal of Game Theory 31 (2002) 353–364. [9] K. Cechlárová, J. Hajduková, Stable partitions with W -preferences, Discrete Applied Mathematics 138 (2004) 333–347. [10] V. Chvátal, On the computational complexity of finding a kernel, Report CRM-300, Centre de Recherches Mathématiques, Université de Montréal, 1973. [11] D. Dimitrov, P. Borm, R. Hendrickx, S.-C. Sung, Simple priorities and core stability in hedonic games, Social Choice and Welfare 26 (2006) 421–433. [12] J. Drèze, J. Greenberg, Hedonic coalitions: optimality and stability, Econometrica 48 (1980) 987–1003. [13] D. Gale, L.S. Shapley, College admissions and the stability of marriage, American Mathematical Monthly 69 (1962) 9–15. [14] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979. [15] R.W. Irving, An efficient algorithm for the stable roommates problem, Journal of Algorithms 6 (1985) 577–595. [16] R.W. Irving, D.F. Manlove, The stable roommates problem with ties, Journal of Algorithms 43 (2002) 85–105. [17] R.W. Irving, D.F. Manlove, G. O’Malley, Stable marriage with ties and bounded length preference lists, Journal of Discrete Algorithms 43 (2009) 213–219. [18] C. Ng, D.S. Hirschberg, Three-dimensional stable matching problems, SIAM Journal on Discrete Mathematics 4 (1991) 245–252. [19] E. Ronn, NP-complete stable matching problems, Journal of Algorithms 11 (1990) 285–304. [20] A. Subramanian, A new approach to stable matching problems, SIAM Journal on Computing 23 (1994) 671–701. [21] G.J. Woeginger, A hardness result for core stability in additive hedonic games, Mathematical Social Sciences 65 (2013) 101–104. [22] G.J. Woeginger, Core stability in hedonic coalition formation, in: Proceedings of the 39th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM’2013, in: LNCS, vol. 7741, Springer-Verlag, 2013, pp. 33–50.
Contents lists available at SciVerse ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Two hardness results for core stability in hedonic coalition formation games Vladimir G. Deineko a , Gerhard J. Woeginger b,∗ a
Warwick Business School, The University of Warwick, Coventry CV4 7AL, United Kingdom
b
Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
article
info
Article history: Received 6 October 2012 Received in revised form 26 February 2013 Accepted 4 March 2013 Available online 26 March 2013
abstract We establish NP-completeness of two problems on core stable coalitions in hedonic games. In the first problem every player has only two acceptable coalitions in his preference list, and in the second problem the preference structures arise from the distances in an underlying metric space. © 2013 Elsevier B.V. All rights reserved.
Keywords: Computational social choice Computational complexity Coalition formation Hedonic game
1. Introduction In many economic, political and social environments individuals carry out their activities in groups; typical examples are households, trade unions, political parties, families, sport clubs, carpools, and research networks. The underlying game theoretic framework is known as coalition formation, where individuals are called players and groups are called coalitions. In hedonic coalition formation each player’s happiness/satisfaction only depends on the other members of his coalition and not on how the remaining players outside his coalition are grouped together. This hedonic research line goes back to the seminal paper [12] of Drèze and Greenberg. A central issue in coalition formation is the stability of a system of coalitions: whenever there is a possibility of increasing one’s happiness/satisfaction by moving to another coalition or by merging or splitting or otherwise restructuring the coalition system, the players will react accordingly and the system will change. A stable system is not subject to such changes. The social choice literature knows a wide variety of stability concepts, such as for instance the core, the strict core, the Nash stable set, the individually stable set, and the contractually individually stable set; we refer the reader to Banerjee, Konishi and Sönmez [5] and Bogomolnaia and Jackson [7]. In this paper we will concentrate on so-called core stable systems of coalitions in hedonic games: a system is core stable, if no subset of players can strictly increase their happiness by splitting off from their current coalitions and forming a new coalition. Well-known special cases of the core stability problem are the stable matching problem [13], the stable roommate problem [15], and the three-sided stable matching problem [1]. There is a rich literature analyzing the computational complexity of various special cases, such as for instance discussed by Aziz, Brandt and Seedig [3], Ballester [4], Cechlárová and Hajduková [8,9], Dimitrov, Borm, Hendrickx and Sung [11], Irving and Manlove [16], Ng and Hirschberg [18], Ronn [19], Subramanian [20], and Woeginger [21,22].
∗
Corresponding author. Tel.: +31 40 247 2415; fax: +31 40 246 5995. E-mail addresses: [email protected] (V.G. Deineko), [email protected] (G.J. Woeginger).
0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.03.003
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Contributions of this paper. After summarizing the central definitions and concepts in Section 2, we will derive two new hardness results on core stability in hedonic coalition formation games. Both proofs are concise, clean, and extremely simple. Our results close two gaps in our understanding of hedonic games. Our first result concerns hedonic games with short preference lists. A coalition is unacceptable for a player, if he prefers staying alone to participating in that coalition; otherwise the coalition is called acceptable. Irving, Manlove and O’Malley [17] consider the variant of Gale–Shapley matching where every man and every woman have only three acceptable partners; they establish NP-completeness of deciding whether in this scenario there exists a perfect stable matching (which pairs up all the players). Furthermore, [17] shows that the corresponding Gale–Shapley problem with only two acceptable partners is polynomially solvable. In Section 3 we discuss hedonic games in which every player has only two acceptable coalitions of arbitrary size, beyond the option of staying alone (whereas [17] consider the case of three acceptable coalitions of size two). We show that in this scenario deciding the existence of a core stable partition into coalitions is NP-complete. Our second result concerns the 3-dimensional stable roommate problem; in this problem the only feasible coalitions are those of size three. Arkin, Bae, Efrat, Okamoto, Mitchell and Polishchuk [2] discuss a special case with metric preference structures, where every player is some point in a metric space with distance function d(·, ·). Player p strictly prefers triple px1 x2 to triple py1 y2 if and only if d(p, x1 ) + d(p, x2 ) < d(p, y1 ) + d(p, y2 ). Arkin et al. [2] exhibit a highly structured instance that does not possess a core stable matching; the computational complexity of their special case however remained open. In Section 4 we establish NP-completeness of deciding the existence of a core stable partition for the metric 3-dimensional stable roommate problem. The complexity of this problem for concrete metric spaces (like the Euclidean plane) remains open. 2. Definitions and preliminaries Let N be a finite set of players. A coalition is a non-empty subset of N. Every player i ∈ N ranks all the coalitions containing i via his preference relation ≼i , which is reflexive, transitive and complete. The underlying strict order is denoted ≺i , where S ≺i T means that S ≼i T but not T ≼i S. If S ≺i T then player i (strictly) prefers participating in T to participating in S, and if S ≼i T then player i weakly prefers participating in T to participating in S. A partition Π is a collection of coalitions which partition N; hence every coalition in Π is non-empty, distinct coalitions are disjoint, and the union of all coalitions equals N. For a partition Π and a player i, we denote by Π (i) the unique coalition in Π containing i. The following definition introduces core stability, a key stability concept in hedonic games. Definition 2.1. A coalition S blocks a partition Π , if every player i ∈ S strictly prefers Π (i) ≺i S. A partition Π is core stable, if there is no blocking coalition. A coalition is a singleton coalition, if it consists of a single player. A coalition C is acceptable for player i, if it satisfies both C ≽i {i} and C ̸= {i}; in other words, player i does not prefer staying alone to being in an acceptable coalition C . Note that any core stable partition must entirely consist of singleton coalitions and of coalitions that are acceptable to all of their members. 3. Hardness for a variant with short preference lists Throughout this section we will investigate hedonic games with preference lists of length two that satisfy the following conditions. First, the ranking of every player i is of the form C1 (i) ≻i C2 (i) ≻i {i}. Secondly, all coalitions C1 (i) and C2 (i) are acceptable for all of their members. (Note that the first condition is the crucial one, whereas the second condition is purely technical and not restrictive at all: one could ignore coalitions C containing players for which C is unacceptable, as such coalitions will never be part of a core stable coalition.) We will model such a hedonic game in the following way as a directed multi-graph: the vertices of the graph are the acceptable coalitions of the players, and for every player i there is a corresponding arc going from vertex C2 (i) to vertex C1 (i). Note that every coalition (vertex) in this directed graph consists of exactly those players that correspond to its incident arcs. A subset W of the vertices is a kernel, if it satisfies the following two properties K1 and K2: K1. For every vertex w ∈ W , none of w ’s successors lies in W . K2. For every vertex w ̸∈ W , at least one of w ’s successors lies in W . The following theorem establishes a connection between kernels and core stable partitions, which might be of independent interest. Theorem 3.1. A hedonic game with preference lists of length two has a core stable partition, if and only if its underlying directed graph contains a kernel. Proof. First assume that the game has a core stable partition Π , and define the set W to contain all vertices corresponding to coalitions in Π that are not singleton. We claim that W is a kernel. For property K1, suppose for the sake of contradiction that W would contain a vertex together with one of its successors. Then the player corresponding to the connecting arc would be contained in two distinct coalitions in Π , which contradicts Π being a partition. For property K2, suppose that
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there is a vertex w ̸∈ W so that none of w ’s successors lies in W . Then all the players that correspond to arcs going out of w are singleton in Π , and prefer coalition w to being alone. Furthermore all the players that correspond to arcs going into w have w as their top preference and would prefer it to their current coalition in Π . Hence w would be blocking, which contradicts our assumption that Π is core stable. Next assume that the directed graph contains a kernel W , and note that by property K1 the coalitions in W are pairwise disjoint. We define a partition Π of the players that uses of all the coalitions in W and that puts the remaining players into singleton coalitions. We claim that Π is core stable. By property K2, any coalition s ̸∈ W in the directed graph has some successor t with t ∈ W . The player that corresponds to the arc from s to t prefers his current coalition t in Π to his second choice s; this means that s is not blocking. Chvátal [10] proved that it is NP-complete to decide whether a given directed graph contains a kernel; see also problem [GT57] in Garey and Johnson [14]. Our next goal is to establish the NP-completeness of a highly restricted special case of the kernel problem, where every vertex in the directed graph is incident to exactly three (in-going and out-going) arcs. The reduction will be done from the following (NP-complete) version of the SATISFIABILITY problem; see Berman, Karpinski and Scott [6]. Problem: (3, B2)-SAT. Instance: A set X of Boolean variables; a Boolean formula φ over X in conjunctive normal form where (i) each clause consists of exactly three (distinct) literals, and (ii) every variable occurs exactly twice in negated and exactly twice in unnegated form. Question: Is formula φ satisfiable? For an instance of (3, B2)-SAT, we let set R contain all variable-clause pairs (x, c ) for which variable x occurs (negated or unnegated) in clause c. We construct the following directed graph.
• For every variable x ∈ X , we introduce a directed cycle on four vertices (which we call a variable-cycle). The four vertices correspond to the four variable-clause pairs for x in set R, so that negated and unnegated occurrences of x alternate along the cycle. • For every clause c ∈ φ , we introduce a directed cycle on three vertices (which we call a clause-cycle). The three vertices correspond to the three variable-clause pairs for c in set R. • For every variable-clause pair (x, c ) ∈ R, we create an arc that goes from the corresponding vertex in the clause-cycle to the corresponding vertex in the variable-cycle. Note that in the constructed directed graph, every vertex is indeed incident to exactly three (in-going and out-going) arcs. Lemma 3.2. The (3, B2)-SAT instance has a satisfying truth assignment, if and only if the constructed directed graph contains a kernel. Proof. (If). Consider a kernel W . It is easily seen from properties K1 and K2 that W contains exactly two, non-consecutive vertices from each variable-cycle. If W contains the two vertices that correspond to the unnegated occurrences of x, we set x to true; if W contains the vertices corresponding to negated occurrences, we set x to false. We claim that this truth assignment satisfies all clauses c ∈ φ . Indeed, consider the clause-cycle corresponding to c. Property K1 implies that at most one of the three vertices is in W , and we consider an arc (x, c ) → (y, c ) on the cycle whose endpoints are not in W . Property K2 implies that vertex (x, c ) ̸∈ W has a successor in W , and the only possible candidate for this is the successor on the variable-cycle for x. The truth value of variable x has been chosen such that clause c is true. (Only if). Let T be a satisfying truth assignment for φ . In a first phase we select from every variable-cycle two vertices for set W : if variable x is true under T then we pick the two vertices that correspond to unnegated occurrences, and if variable x is false under T then we pick the two vertices that correspond to negated occurrences. In the second phase, we consider a clause-cycle for some clause c ∈ φ . Every vertex in this cycle has a successor on some variable-cycle. As c contains one or two or three true literals, one or two or three of these successors have already been picked for W in the first phase. If all three successors have been picked, we do nothing. If one or two successors have been picked, we pick one additional vertex from the clause-cycle for W so that K1 and K2 are fulfilled; we leave the (straightforward) details to the reader. The resulting set W is a kernel. Lemma 3.2 yields the NP-completeness of deciding the existence of a kernel for the highly restricted special case where every vertex in the directed graph is incident to exactly three arcs. We start from such a restricted kernel instance, and translate it into a corresponding hedonic game by essentially reverting the multi-graph construction: every arc becomes a player, every vertex becomes a coalition of size three (that consists of the three incident arcs/players), and every arc/player prefers his head vertex/coalition to his tail vertex/coalition. Then Theorem 3.1 yields the following NP-completeness result: Theorem 3.3. It is NP-complete to decide whether a hedonic game with preference lists of length two has a core stable partition, even in the case where all acceptable coalitions have size three.
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The statement in Theorem 3.3 is essentially tight, as any decrease in the values of its crucial parameters leads to a tractable problem. Indeed, the corresponding problem for preference lists of length one (and acceptable coalitions of unrestricted size) is straightforward, as there always exists a core stable partition. One possibility is to take a maximal system of pairwise disjoint acceptable coalitions, and to add an appropriate system of singleton coalitions to this. And also the corresponding problem for preference lists of length two and acceptable coalitions of size two is polynomially solvable; see Irving, Manlove and O’Malley [17]. 4. Hardness for a metric variant In the 3-dimensional stable roommate problem, the allowed coalitions are exactly the coalitions of size three. The goal is to decide whether there is a core stable partition. In this section we analyze a metric 3-dimensional roommate variant, where every player is some point in a metric space equipped with a standard distance function d(·, ·) that is symmetric (so that d(x, y) = d(y, x) for all x, y) and that satisfies the triangle inequality (so that d(x, y) + d(y, z ) ≥ d(x, z ) for all x, y, z). Player p weakly prefers triple px1 x2 to triple py1 y2 if and only if d(p, x1 ) + d(p, x2 ) ≤ d(p, y1 ) + d(p, y2 ). If there is strict inequality, then the player strictly prefers triple px1 x2 . The crucial quantity d(p, x1 ) + d(p, x2 ) will be called the cost of triple px1 x2 for player p. In this section we establish the NP-completeness of deciding the existence of a core stable partition for such metric instances. The reduction is done from the following 2-dimensional stable roommate variant. Problem: Classical Roommate. Instance: A set M of 2m players; for every player i ∈ M a rank function ri : M → {1, . . . , 2m} such that player i strictly prefers player j′ to j′′ as roommate if and only if rj (j′ ) < rj (j′′ ). Question: Does there exist a core stable partition of M into pairs? Note that in Classical Roommate the only feasible coalitions are pairs. We also stress that the underlying rank functions rj are not necessarily injective and that ties rj (k′ ) = rj (k′′ ) are allowed. Irving [15] has shown that the problem variant without such ties is polynomially solvable, whereas Ronn [19] proved that the considered variant with ties is NP-complete. We start from an arbitrary instance of Classical Roommate, and we define a corresponding instance of the metric 3dimensional roommate problem. In order to distinguish clearly between the players in both games, we will sometimes speak of old players (in the 2-dimensional variant) and of new players (in the 3-dimensional variant). We introduce the following new players.
• For every old player i ∈ M, there is a corresponding new main-player P (i). • For every 2-element subset {i, j} ⊆ M, we create three corresponding new pair-players P {i, j}, P ′ {i, j}, and P ′′ {i, j}. • Furthermore there are m new dummy players D1 , . . . , Dm . Next we specify the distances in the underlying metric space. All distances will lie in the left-open–right-closed interval (2L, 4L] with L = 8m; note that distances from this range automatically satisfy the triangle inequality.
• • • •
The distance between P (i) and P (j) equals 3L. The distance between P (i) and P {i, j} equals 2L + ri (j). The distances from P {i, j} to P ′ {i, j} and from P {i, j} to P ′′ {i, j} are both 3L. All remaining distances are 4L.
The following two Lemmas 4.1 and 4.2 exhibit the close connections between the instance of Classical Roommate and the constructed 3-dimensional instance. Lemma 4.1. If the instance of Classical Roommate has a core stable partition into pairs, then there exists a core stable partition into triples for the constructed 3-dimensional roommate instance. Proof. We consider the stable matching µ for the old players in M, and construct from it the following partition of the new players: Whenever (i, j) ∈ µ, we use the triple consisting of P (i), P (j), P {i, j} and the triple that consists of P ′ {i, j}, P ′′ {i, j} together with some dummy player. Whenever (i, j) ̸∈ µ, we use the triple P {i, j}, P ′ {i, j}, P ′′ {i, j}. Note that in these triples every main-player P (i) incurs a cost strictly below 6L and every pair-player P {i, j} incurs a cost of at most 6L. We claim that these triples form a core stable partition Π for the 3-dimensional roommate instance. Suppose for the sake of contradiction that there exists a blocking triple. If some main-player P (a) participates in the blocking triple, then his cost must stay strictly below 6L; consequently P (a) can only block together with main-players or with pair-players of the type P {a, c }. We discuss three subcases. First, three main-players P (a), P (b), P (c ) cannot form a blocking triple, as the resulting cost for P (a) would be 6L. Secondly, three players P (a), P {a, b}, P {a, c } cannot form a blocking triple, as the resulting cost for the pair-player P {a, b} would be above 6L. Thirdly, three players P (a), P (b), P {a, b} cannot form a blocking triple: if (a, a′ ) and (b, b′ ) are the corresponding pairs in the stable matching µ, a blocking triple P (a), P (b), P {a, b} would imply that old player a prefers b to a′ and that old player b prefers a to b′ . Hence (a, b) would be a blocking pair for matching µ, so that µ itself is not stable. This contradiction closes the third subcase and shows that no main-player P (a) can participate in the blocking triple.
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Next, we observe that also none of the pair-players P {a, b} can participate in the blocking triple: the distance from P {a, b} to any remaining pair-player or dummy player equals 3L or 4L, so that there is no way of beating his cost 6L in partition Π . To summarize, all remaining candidates for a blocking triple are the dummy players and the pair-players P ′ {a, b} and P ′′ {a, b}. These players are pairwise at distance 4L, and can never form a blocking set. Lemma 4.2. If the constructed 3-dimensional roommate instance has a core stable partition into triples, then the instance of Classical Roommate has a core stable partition into pairs. Proof. We consider the stable partition Π that divides the new players into triples, and we derive some structural properties of Π . As a first property, we note that the cost of any pair-player P {i, j} in Π is at most 6L; otherwise Π would be blocked by the triple P {i, j}, P ′ {i, j}, P ′′ {i, j}. As the second property, we show the following: whenever partition Π puts two main-players P (a) and P (b) into a common triple, then the third partner in the triple must be the pair-player P {a, b}; such a triple is called canonical. Indeed, assume that some triple in the stable partition Π consists of two main-players P (a) and P (b) together with some third player X.
• If X is a main-player P (c ), a pair-player P ′ {i, j} or P ′′ {i, j}, or a dummy player, then the triple P (a), P (b), P {a, b} would be blocking. The cost of players P (a) and P (b) would improve from 6L or higher down to 5L + ra (b) and 5L + rb (a), respectively, and the cost of player P {a, b} would drop from 6L or higher to 4L + ra (b) + rb (a). • If X is a player P {i, j} with {i, j} ̸= {a, b}, then the cost of P {i, j} would be larger than 6L and thus contradict the first property. Hence the only remaining possibility is that X is the pair-player P {a, b}; this proves the second property. Next suppose that in partition Π , some main-player P (a) is not in a canonical triple. Then the partners of P (a) cannot be two pair-players P {a, i} and P {a, j}, as this would contradict the first property; we conclude that the cost of P (a) in Π lies above 6L. As the cardinality of M is even, there exists another main-player P (b) who also is not in a canonical triple and whose cost in Π also must lie above 6L. But then partition Π is blocked by the triple P (a), P (b), P {a, b}: the cost of P (a) and P (b) would drop from a value above 6L to a value below 5L, and the cost of P {a, b} would improve from above 5L to below 5L. All in all, this means that partition Π puts all the main-players into canonical triples. We define a matching µ for the old players in M that pairs up two old players i and j whenever the corresponding new main-players P (i) and P (j) in Π are together in a canonical triple. We claim that matching µ is stable. Suppose that there is a blocking pair (a, b) for µ, with corresponding pairs (a, a′ ) and (b, b′ ) in µ. This means ra (b) < ra (a′ ) and rb (a) < rb (b′ ), and a fortiori implies that the triple P (a), P (b), P {a, b} is blocking for the stable partition Π . This contradiction yields the stability of matching µ. Theorem 4.3. In the metric variant of the 3-dimensional stable roommate problem, it is strongly NP-complete to decide the existence of a core stable partition. Proof. The problem is clearly contained in NP: the YES-certificate is a core stable partition, and the verification is done by inspecting all O(n3 ) triples that are candidates for a blocking set. For NP-hardness, we note that the above reduction can be done in polynomial time and that all involved integers are polynomially bounded in |M |. Lemmas 4.1 and 4.2 establish correctness of the reduction: the constructed instance of the metric 3-dimensional roommate problem has a core stable partition into triples, if and only if the considered instance of Classical Roommate has a core stable partition into pairs. Arkin et al. [2] construct an instance of the metric 3-dimensional stable roommate problem that does not allow a core stable matching. The underlying metric space is the Euclidean plane. The computational complexity of this Euclidean special case is an open (and seemingly challenging) problem. Acknowledgments We thank the referees for their constructive comments which helped to improve the presentation of the paper. VD acknowledges support by Warwick University’s Centre for Discrete Mathematics and Its Applications (DIMAP) and by the EPSRC fund EP/F017871. GW acknowledges support by DIAMANT (a mathematics cluster of The Netherlands Organization for Scientific Research NWO), and by the Alexander von Humboldt Foundation, Bonn, Germany. References [1] A. Alkan, Non-existence of stable threesome matchings, Mathematical Social Sciences 16 (1988) 207–209. [2] E.M. Arkin, S.W. Bae, A. Efrat, K. Okamoto, J.S.B. Mitchell, V. Polishchuk, Geometric stable roommates, Information Processing Letters 109 (2009) 219–224. [3] H. Aziz, F. Brandt, H.G. Seedig, Computing desirable partitions in additively separable hedonic games, Artificial Intelligence 195 (2013) 316–334. [4] C. Ballester, NP-completeness in hedonic games, Games and Economic Behavior 49 (2004) 1–30. [5] S. Banerjee, H. Konishi, T. Sönmez, Core in a simple coalition formation game, Social Choice and Welfare 18 (2001) 135–153. [6] P. Berman, M. Karpinski, A.D. Scott, Hardness of short symmetric instances of MAX-3SAT, Electronic Colloquium on Computational Complexity (2003) 049.
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[7] A. Bogomolnaia, M.O. Jackson, The stability of hedonic coalition structures, Games and Economic Behavior 38 (2002) 201–230. [8] K. Cechlárová, J. Hajduková, Computational complexity of stable partitions with B-preferences, International Journal of Game Theory 31 (2002) 353–364. [9] K. Cechlárová, J. Hajduková, Stable partitions with W -preferences, Discrete Applied Mathematics 138 (2004) 333–347. [10] V. Chvátal, On the computational complexity of finding a kernel, Report CRM-300, Centre de Recherches Mathématiques, Université de Montréal, 1973. [11] D. Dimitrov, P. Borm, R. Hendrickx, S.-C. Sung, Simple priorities and core stability in hedonic games, Social Choice and Welfare 26 (2006) 421–433. [12] J. Drèze, J. Greenberg, Hedonic coalitions: optimality and stability, Econometrica 48 (1980) 987–1003. [13] D. Gale, L.S. Shapley, College admissions and the stability of marriage, American Mathematical Monthly 69 (1962) 9–15. [14] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979. [15] R.W. Irving, An efficient algorithm for the stable roommates problem, Journal of Algorithms 6 (1985) 577–595. [16] R.W. Irving, D.F. Manlove, The stable roommates problem with ties, Journal of Algorithms 43 (2002) 85–105. [17] R.W. Irving, D.F. Manlove, G. O’Malley, Stable marriage with ties and bounded length preference lists, Journal of Discrete Algorithms 43 (2009) 213–219. [18] C. Ng, D.S. Hirschberg, Three-dimensional stable matching problems, SIAM Journal on Discrete Mathematics 4 (1991) 245–252. [19] E. Ronn, NP-complete stable matching problems, Journal of Algorithms 11 (1990) 285–304. [20] A. Subramanian, A new approach to stable matching problems, SIAM Journal on Computing 23 (1994) 671–701. [21] G.J. Woeginger, A hardness result for core stability in additive hedonic games, Mathematical Social Sciences 65 (2013) 101–104. [22] G.J. Woeginger, Core stability in hedonic coalition formation, in: Proceedings of the 39th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM’2013, in: LNCS, vol. 7741, Springer-Verlag, 2013, pp. 33–50.