Optimal Partitions in Additively Separable Hedonic Games^{1}^{1}1A preliminary version of this work was invited for presentation in the session ‘Cooperative Games and Combinatorial Optimization’ at the 24th European Conference on Operational Research (EURO 2010) in Lisbon. This material is based on work supported by the Deutsche Forschungsgemeinschaft under grants BR2312/61 (within the European Science Foundation’s EUROCORES program LogICCC) and BR 2312/71.
Abstract
We conduct a computational analysis of fair and optimal partitions in additively separable hedonic games. We show that, for strict preferences, a Pareto optimal partition can be found in polynomial time while verifying whether a given partition is Pareto optimal is coNPcomplete, even when preferences are symmetric and strict. Moreover, computing a partition with maximum egalitarian or utilitarian social welfare or one which is both Pareto optimal and individually rational is NPhard. We also prove that checking whether there exists a partition which is both Pareto optimal and envyfree is complete. Even though an envyfree partition and a Nash stable partition are both guaranteed to exist for symmetric preferences, checking whether there exists a partition which is both envyfree and Nash stable is NPcomplete.
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1 Introduction
Ever since the publication of von Neumann and Morgenstern’s Theory of Games and Economic Behavior in 1944, coalitions have played a central role within game theory. The crucial questions in coalitional game theory are which coalitions can be expected to form and how the members of coalitions should divide the proceeds of their cooperation. Traditionally the focus has been on the latter issue, which led to the formulation and analysis of concepts such as Gillie’s core, the Shapley value, or the bargaining set. Which coalitions are likely to form is commonly assumed to be settled exogenously, either by explicitly specifying the coalition structure, a partition of the players in disjoint coalitions, or, implicitly, by assuming that larger coalitions can invariably guarantee better outcomes to its members than smaller ones and that, as a consequence, the grand coalition of all players will eventually form.
The two questions, however, are clearly interdependent: the individual players’ payoffs depend on the coalitions that form just as much as the formation of coalitions depends on how the payoffs are distributed.
Coalition formation games, as introduced by Drèze and Greenberg (1980), provide a simple but versatile formal model that allows one to focus on coalition formation as such. In many situations it is natural to assume that a player’s appreciation of a coalition structure only depends on the coalition he is a member of and not on how the remaining players are grouped. Initiated by Banerjee et al. (2001) and Bogomolnaia and Jackson (2002), much of the work on coalition formation now concentrates on these socalled hedonic games.
The main focus in hedonic games has been on notions of stability for coalition structures such as Nash stability, individual stability, contractual individual stability, or core stability and characterizing conditions under which they are guaranteed to be nonempty (see, e.g., Bogomolnaia and Jackson, 2002). The most prominent examples of hedonic games are twosided matching games in which only coalitions of size two are admissible (Roth and Sotomayor, 1990).
General coalition formation games have also received attention from the artificial intelligence community, where the focus has generally been on computing partitions that give rise to the greatest social welfare (see, e.g., Sandholm et al., 1999). The computational complexity of hedonic games has been investigated with a focus on the complexity of computing stable partitions for different models of hedonic games (Ballester, 2004; Dimitrov et al., 2006; Cechlárová, 2008). We refer to Hajduková (2006) for a critical overview.
Among hedonic games, additively separable hedonic games (ASHGs) are a particularly natural and succinct representation in which each player has a value for every other player and the value of a coalition to a particular player is computed by simply adding his values of the players in his coalition.
Additive separability satisfies a number of desirable axiomatic properties (Barberà et al., 2004). ASHGs are the nontransferable utility generalization of graph games studied by Deng and Papadimitriou (1994). Sung and Dimitrov (2010) showed that for ASHGs, checking whether a core stable, strictcore stable, Nash stable, or individually stable partition exists is NPhard. Dimitrov et al. (2006) obtained positive algorithmic results for subclasses of additively separable hedonic games in which each player divides other players into friends and enemies. Branzei and Larson (2009) examined the tradeoff between stability and social welfare in ASHGs.
Contribution
In this paper, we analyze concepts from fair division in the context of coalition formation games. We present the first systematic examination of the complexity of computing and verifying optimal partitions of hedonic games, specifically ASHGs. We examine various standard criteria from the social sciences: Pareto optimality, utilitarian social welfare, egalitarian social welfare, and envyfreeness (see, e.g., Moulin, 1988).
In Section 3, we show that computing a partition with maximum egalitarian social welfare is NPhard. Similarly, computing a partition with maximum utilitarian social welfare is NPhard in the strong sense even when preferences are symmetric and strict.
In Section 4, the complexity of Pareto optimality is studied. We prove that checking whether a given partition is Pareto optimal is coNPcomplete in the strong sense, even when preferences are strict and symmetric. By contrast, we present a polynomialtime algorithm for computing a Pareto optimal partition when preferences are strict.^{2}^{2}2Thus, we identify a natural problem in coalitional game theory where verifying a possible solution is presumably harder than actually finding one. Interestingly, computing an individually rational and Pareto optimal partition is NPhard in general.
In Section 5, we consider complexity questions regarding envyfree partitions. Checking whether there exists a partition which is both Pareto optimal and envyfree is shown to be complete. We present an example which exemplifies the tradeoff between satisfying stability (such as Nash stability) and envyfreeness and use the example to prove that checking whether there exists a partition which is both envyfree and Nash stable is NPcomplete even when preferences are symmetric.
Our computational hardness results imply computational hardness of equivalent problems for hedonic coalition nets (Elkind and Wooldridge, 2009).
2 Preliminaries
In this section, we provide the terminology and notation required for our results.
2.1 Hedonic games
A hedonic coalition formation game is a pair where is a set of players and is a preference profile which specifies for each player the preference relation , a reflexive, complete and transitive binary relation on set .
denotes that strictly prefers over and that is indifferent between coalitions and . A partition is a partition of players into disjoint coalitions. By , we denote the coalition in which includes player .
A game is separable if for any player and any coalition and for any player not in we have the following: if and only if ; if and only if ; and if and only if .
In an additively separable hedonic game , each player has value for player being in the same coalition as and if is in coalition , then gets utility . For coalitions , if and only if .
A preference profile is symmetric if for any two players and is strict if for all such that . We consider ASHGs (additively separable hedonic games) in this paper. Unless mentioned otherwise, all our results are for ASHGs.
2.2 Fair and optimal partitions
In this section, we formulate concepts from the social sciences, especially the literature on fair division, for the context of hedonic games. A partition satisfies individual rationality if each player does as well as by being alone, i.e., for all , . For a utilitybased hedonic game and partition , we will denote the utility of player by . The different notions of fair or optimal partitions are defined as follows.^{3}^{3}3All welfare notions considered in this paper (utilitarian, elitist, and egalitarian) are based on the interpersonal comparison of utilities. Whether this assumption can reasonably be made is debatable.

The utilitarian social welfare of a partition is defined as the sum of individual utilities of the players: . A maximum utilitarian partition maximizes the utilitarian social welfare.

The elitist social welfare is given by the utility of the player that is best off: . A maximum elitist partition maximizes the utilitarian social welfare.

The egalitarian social welfare is given by the utility of the agent that is worst off: . A maximum egalitarian partition maximizes the egalitarian social welfare.

A partition of is Pareto optimal if there exists no partition of which Pareto dominates , that is for all , and there exists at least one player such that , .

Envyfreeness is a notion of fairness. In an envyfree partition, no player has an incentive to replace another player.
For the sake of brevity, we will call all the notions described above “optimality criteria” although envyfreeness is rather concerned with fairness than optimality. We consider the following computational problems with respect to the optimality criteria defined above.
Optimality: Given and a partition of , is optimal?
Existence: Does an optimal partition for a given exist?
Search: If an optimal partition for a given exists, find one.
Existence is trivially true for all criteria of optimality concepts. By the definitions, it follows that there exist partitions which satisfy maximum utilitarian social welfare, elitist social welfare, and egalitarian social welfare respectively.
3 Complexity of maximizing social welfare
In this section, we examine the complexity of maximizing social welfare in ASHGs. We first observe that computing a maximum utilitarian partition for strict and symmetric preferences is NPhard because it is equivalent to the NPhard problem of maximizing agreements in the context of correlation clustering (Bansal et al., 2004).
Theorem 1.
Computing a maximum utilitarian partition is NPhard in the strong sense even with symmetric and strict preferences.
Computing a maximum elitist partition is much easier. For any player , let be the set of players which strictly likes and . Both and can be computed in linear time. Let be the player such that for all . Then is a partition which maximizes the elitist social welfare. As a corollary, we can verify whether a partition has maximum elitist social welfare by computing a partition with maximum elitist social welfare and comparing with . Just like maximizing the utilitarian social welfare, maximizing the egalitarian social welfare is hard.
Theorem 2.
Computing a maximum egalitarian partition is NPhard in the strong sense.
Proof.
We provide a polynomialtime reduction from the NPhard problem MaxMinMachineCompletionTime (Woeginger, 1997) in which an instance consists of a set of identical machines , a set of independent jobs where job has processing time . The problem is to allot jobs to the machines such that the minimum processing time (without machine idle times) of all machines is maximized. Let be an instance of MaxMinMachineCompletionTime and let . From we construct an instance of EgalSearch. The ASHG for instance consists of and the preferences of the players are as follows: for all and all let and . Also, for let and for let . Each player corresponds to machine and each player corresponds to job .
Let be the partition which maximizes . We show that players are in separate coalitions and each player is in for some . We can do so by proving two claims. The first claim is that for such that , we have that . The second claim is that each player is in a coalition with a player . The proofs of the claims are omitted due to space limitations.
A job allocation corresponds to a partition where is in if job is assigned to for all and . Note that the utility of a player corresponds to the total completion time of all jobs assigned to according to . Let be a maximum egalitarian partition. Assume that there is another partition and induces a strictly greater minimum completion time. We know that for all and for all . But then from the assumption we have which is a contradiction.∎
4 Complexity of Pareto optimality
We now consider the complexity of computing a Pareto optimal partition. The complexity of Pareto optimality has already been considered in several settings such as house allocation (Abraham et al., 2005). Bouveret and Lang (2008) examined the complexity of Pareto optimal allocations in resource allocation problems. We show that checking whether a partition is Pareto optimal is hard even under severely restricted settings.
Theorem 3.
The problem of checking whether a partition is Pareto optimal is coNPcomplete in the strong sense, even when preferences are symmetric and strict.
Proof.
The reduction is from the NPcomplete problem E3C (EXACT3COVER) to deciding whether a given partition is Pareto dominated by another partition or not. Recall that in E3C, an instance is a pair , where is a set and is a collection of subsets of such that for some positive integer and for each . The question is whether there is a subcollection which is a partition of .
It is known that E3C remains NPcomplete even if each occurs in at most three members of (Garey and Johnson, 1979). Let be an instance of E3C. can be reduced to an instance , where is an ASHG defined in the following way. Let . The players preferences are symmetric and strict and are defined as follows: for all ; for all ; if and if ; for any ; and for any and for which is not already defined.
The partition in the instance is We see that the utilities of the players are as follows: for all ; for all ; and for all .
Assume that there exists such that is a partition of . Then we prove that is not Pareto optimal and there exists another partition of which Pareto dominates . We form another partition
In that case, for all ; for all ; for all ; and for all . Whereas the utilities of no player in decreases, the utility of some players in is more than in . Since Pareto dominates , is not Pareto optimal.
We now show that if there exists no such that is a partition of , then is Pareto optimal. We note that is a sufficiently large negative valuation to ensure that if , then cannot be in the same coalition in a Pareto optimal partition. For the sake of contradiction, assume that is not Pareto optimal and there exists a partition which Pareto dominates . We will see that if there exists a player such that , then there exists at least one such that . The only players whose utility can increase (without causing some other player to be less happy) are , or . We consider these player classes separately. If the utility of player increases, it can only increase from to so that is in the same coalition as and . However, this means that gets a decreased utility. The utility of can increase or stay the same only if it forms a coalition with some s. However in that case, to satisfy all s, there needs to exist an such that is a partition of .
Assume the utility of a player for increases. This is only possible if is in the same coalition as . Clearly, the coalition formed is because coalition brings a utility of 2 to . In that case needs to form a coalition where . If forms a coalition , then all players for need to form coalitions of the form such that . Otherwise, their utility of decreases. This is only possible if there exists a set of such that is a partition of .
Assume that there exists a partition that Pareto dominates and the utility of a player for some . This is only possible if each forms the coalition of the form where . This can only happen if there exists a set of such that is a partition of .∎
The fact that checking whether a partition is Pareto optimal is coNPcomplete has no obvious implications on the complexity of computing a Pareto optimal partition. In fact there is a simple polynomialtime algorithm to compute a partition which is Pareto optimal for strict preferences.
Theorem 4.
For strict preferences, a Pareto optimal partition can be computed in polynomial time.
Proof.
The statement follows from an application of serial dictatorship. Serial dictatorship (Abdulkadiroğlu and Sönmez, 1998) is a wellknown mechanism in resource allocation in which an arbitrary player is chosen as the ‘dictator’ who is then given his most favored allocation and the process is repeated until all players or resources have been dealt with. In the context of coalition formation, serial dictatorship is welldefined if preferences of players over coalitions are strict. Serial dictatorship is also welldefined for ASHGs with strict preferences as the dictator forms a coalition with all the players he strictly likes who have been not considered as dictators or are not already in some dictator’s coalition. The resulting partition is such that for any other partition , at least one dictator will strictly prefer to . Therefore is Pareto optimal. ∎
A standard criticism of Pareto optimality is that it can lead to inherently unfair allocations. To address this criticism, the algorithm can be modified to obtain less lopsided partitions. Whenever an arbitrary player is selected to become the dictator among the remaining players, choose a player that does not get extremely high elitist social welfare among the remaining players. Nevertheless, even this modified algorithm may output an partition that fails to be individually rational.
We know that the set of partitions which are both Pareto optimal and individually rational is nonempty. Repeated Pareto improvements on individually rational partition consisting of singletons leads to a Pareto optimal and individually rational partition. We show that computing a Pareto optimal and individually rational partition for ASHGs is weakly NPhard.
Theorem 5.
Computing a Pareto optimal and individually rational partition is weakly NPhard.
Proof.
Consider the decision problem SubsetSumZero in which an instance consists of a set of integer weights and the question is whether there exists a nonempty such that ? Since SubsetSum for positive integers is NPcomplete, it follows that SubsetSumZero is also NPcomplete.^{4}^{4}4We note that in any instance of SubsetSum all zeros in the set can be omitted to obtain an equivalent problem. Reduce SubsetSum to SubsetSumZero by adding to . Therefore, MaximalSubsetSumZero, the problem of finding a maximal cardinality subset such that is NPhard.
We prove the theorem by a reduction from MaximalSubsetSumZero. Reduce an instance of of MaximalSubsetSumZero to an instance where is an ASHG defined in the following way: where ; ; for all ; for all ; and for any for which is not already defined.
First, we show that in an individually rational partition , no player except gets positive utility, i.e., for all . Assume that w.l.o.g gets positive utility in . This implies there exist a subset such that . Then there exists such that which means that . Due to individual rationality, . But if , then and is not individually rational.
Assume that there exists a such that . Then without loss of generality and due to individual rationality . Again due to individual rationality, needs to be with another such that . And again due to individual rationality, needs to be with . This means, that for each , .
We show that in every Pareto optimal and individually rational partition , we have . For any other partition , in which this does not hold, .
Consider an and let be any partition of . The claim is that is a Pareto optimal and individually rational partition if and only if is of the form where is the maximal subset such that . Assume that is not a maximal subset such that . If , there exists a such that . If is not maximal then there is a larger set and a corresponding partition with and for all . For any other such that , we know that which implies that there is a which gets negative utility. ∎
5 Complexity of envyfreeness
Envyfreeness is a desirable property in resource allocation, especially in cake cutting settings. Lipton et al. (2004) proposed envyminimization in different ways and examined the complexity of minimizing envy in resource allocation settings. Bogomolnaia and Jackson (2002) mentioned envyfreeness in hedonic games but focused on stability. We already know that envyfreeness can be easily achieved by the partition of singletons.^{5}^{5}5The partition of singletons also satisfies individual rationality. Therefore, in conjunction with envyfreeness, we seek to satisfy other properties such as stability or Pareto optimality. A partition is Nash stable if there is no incentive for a player to be deviate to another (possibly empty) coalition. For symmetric ASHGs, it is known that Nash stable partitions always exist and they correspond to partitions for which the utilitarian social welfare is a local optimum (see, e.g., Bogomolnaia and Jackson, 2002). We now show that for symmetric ASHGs, there may not exist any partition which is both envyfree and Nash stable.
Example 1.
Consider an ASHG where and is defined as follows: , and . Then there exists no partition which is both envyfree and Nash stable.
We use the game in Example 1 as a gadget to prove the following.^{6}^{6}6Example 1 and the proof of Theorem 6 also apply to the combination of envyfreeness and individual stability where individual stability is a variant of Nash stability (Bogomolnaia and Jackson, 2002).
Theorem 6.
For symmetric preferences, checking whether there exists a partition which is both envyfree and Nash stable is NPcomplete in the strong sense.
Proof.
The problem is clearly in NP since envyfreeness and Nash stability can be verified in polynomial time. We reduce the problem from E3C. Let be an instance of E3C where is a set and is a collection of subsets of such that for some positive integer and for each . We will use the fact that E3C remains NPcomplete even if each occurs in at most three members of . can be reduced to an instance where is an ASHG defined in the following way. Let . We set all preferences as symmetric. The players preferences are as follows: for all , , and ; for all , and ; and for all for which valuations have not been defined,
We note that is a sufficiently large negative valuation to ensure that if , then and will get negative utility if they are in the same coalition. We show that there exists an envyfree and Nash stable partition for if and only if is a ‘yes’ instance of E3C.
Assume that there exists such that is a partition of . Then there exists a partition . It is easy to see that partition is Nash stable and envyfree. Players and both had an incentive to be with each other when they are singletons. However, each now gets utility 3 by being in a coalition with , and where . Therefore has no incentive to be with and has no incentive to join because . Similarly, no player is envious of another player.
Assume that there exists no partition of such that is a partition of . Then, there exists at least one such that is not in the coalition of the form where . Then the only individually rational coalitions which can form and get utility at least are the following , . In the first case, wants to deviate to . In the second case, is envious and wants to replace . Therefore, there exists no partition which is both Nash stable and envyfree. ∎
While the existence of a Pareto optimal partition and an envyfree partition is guaranteed, we show that checking whether there exists a partition which is both envyfree and Pareto optimal is hard.
Theorem 7.
Checking whether there exists a partition which is both Pareto optimal and envyfree is complete.
Proof.
The problem has a ‘yes’ instance if there exists an envyfree partition that Pareto dominates every other partition. Therefore the problem is in the complexity class .
We prove hardness by a reduction from a problem concerning resource allocation (with additive utilities) (de Keijzer et al., 2009). A resource allocation problem is a tuple where is a set of agents, is a set of indivisible objects and is a weight function. An is an allocation if for all such that , we have . The resultant utility of each agent is then . It was shown by de Keijzer et al. (2009) that the problem EEFADD of checking the existence of an envyfree and Pareto optimal allocation is complete.
Now, consider an instance of EEFADD and reduce it to an instance of an ASHG where and is specified by the following values: and for all , ; for all ; and for all where . It can then be shown that there exists a Pareto optimal and envyfree partition in if and only if is a ‘yes’ instance of EEFADD. The proof is omitted due to space limitations.∎
The results of this section show that, even though envyfreeness can be trivially satisfied on its own, it becomes much more delicate when considered in conjunction with other desirable properties.
6 Conclusions
We studied the complexity of partitions that satisfy standard criteria of fairness and optimality in additively separable hedonic games. We showed that computing a partition with maximum egalitarian or utilitarian social welfare is NPhard in the strong sense and computing an individually rational and Pareto optimal partition is weakly NPhard. A Pareto optimal partition can be computed in polynomial time when preferences are strict. Interestingly, checking whether a given partition is Pareto optimal is coNPcomplete even in the restricted setting of strict and symmetric preferences.
We also showed that checking the existence of partition which satisfies not only envyfreeness but an additional property like Nash stability or Pareto optimality is computationally hard. The complexity of computing a Pareto optimal partition for ASHGs with general preferences is still open. Other directions for future research include approximation algorithms to compute maximum utilitarian or egalitarian social welfare for different representations of hedonic games.
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Haris Aziz, Felix Brandt, and Hans Georg Seedig 
Department of Informatics 
Technische Universität München 
85748 Garching bei München, Germany 
{aziz,brandtf,seedigh}@in.tum.de 