Stable marriage problems with quantitative preferences
Abstract
The stable marriage problem is a wellknown problem of matching men to women so that no man and woman, who are not married to each other, both prefer each other. Such a problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools or more generally to any twosided market. In the classical stable marriage problem, both men and women express a strict preference order over the members of the other sex, in a qualitative way. Here we consider stable marriage problems with quantitative preferences: each man (resp., woman) provides a score for each woman (resp., man). Such problems are more expressive than the classical stable marriage problems. Moreover, in some reallife situations it is more natural to express scores (to model, for example, profits or costs) rather than a qualitative preference ordering. In this context, we define new notions of stability and optimality, and we provide algorithms to find marriages which are stable and/or optimal according to these notions. While expressivity greatly increases by adopting quantitative preferences, we show that in most cases the desired solutions can be found by adapting existing algorithms for the classical stable marriage problem.
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1 Introduction
The stable marriage problem (SM) [5] is a wellknown problem of matching the elements of two sets. It is called the stable marriage problem since the standard formulation is in terms of men and women, and the matching is interpreted in terms of a set of marriages. Given men and women, where each person expresses a strict ordering over the members of the opposite sex, the problem is to match the men to the women so that there are no two people of opposite sex who would both rather be matched with each other than their current partners. If there are no such people, all the marriages are said to be stable. In [4] Gale and Shapley proved that it is always possible to find a matching that makes all marriages stable, and provided a polynomial time algorithm which can be used to find one of two extreme stable marriages, the socalled maleoptimal or femaleoptimal solutions. The GaleShapley algorithm has been used in many reallife scenarios, such as in matching hospitals to resident doctors [12], medical students to hospitals, sailors to ships [8], primary school students to secondary schools [13], as well as in market trading [14].
In the classical stable marriage problem, both men and women express a strict preference order over the members of the other sex in a qualitative way. Here we consider stable marriage problems with quantitative preferences. In such problems each man (resp., woman) provides a score for each woman (resp., man). Stable marriage problems with quantitative preferences are interesting since they are more expressive than the classical stable marriage problems, since in classical stable marriage problem a man (resp., a woman) cannot express how much he (resp., she) prefers a certain woman (resp., man). Moreover, they are useful in some reallife situations where it is more natural to express scores, that can model notions such as profit or cost, rather than a qualitative preference ordering. In this context, we define new notions of stability and optimality, we compare such notions with the classical ones, and we show algorithms to find marriages which are stable and/or optimal according to these notions. While expressivity increases by adopting quantitative preferences, we show that in most cases the desired solutions can be found by adapting existing algorithms for the classical stable marriage problem.
Stable marriage problems with quantitative preferences have been studied also in [6, 7]. However, they solve these problems by looking at the stable marriages that maximize the sum of the weights of the married pairs, where the weights depend on the specific criteria used to find an optimal solution, that can be minimum regret criterion [6], the egalitarian criterion [7] or the Lex criteria [7]. Therefore, they consider as stable the same marriages that are stable when we don’t consider the weights. We instead use the weights to define new notions of stability that may lead to stable marriages that are different from the classical case. They may rely on the difference of weights that a person gives to two different people of the other sex, or by the strength of the link of the pairs (man,woman), i.e., how much a person of the pair wants to be married with the other person of the pair. The classical definition of stability for stable marriage problems with quantitative preferences has been considered also in [2] that has used a semiringbased soft constraint approach [3] to model and solve these problems.
The paper is organized as follows. In Section 2 we give the basic notions of classical stable marriage problems, stable marriage problems with partially ordered preferences and stable marriage problems with quantitative preferences (SMQs). In Section 3 we introduce a new notion of stability, called stability for SMQs, which depends on the difference of scores that every person gives to two different people of the other sex, and we compare it with the classical notion of stability. Moreover, we give a new notion of optimality, called lexoptimality, to discriminate among the new stable marriages, which depends on a voting rule. We show that there is a unique optimal stable marriage and we give an algorithm to find it. In Section 4 we introduce other notions of stability for SMQs that are based on the strength of the link of the pairs (man,woman), we compare them with the classical stability notion, and we show how to find marriages that are stable according to these notions with the highest global link. In Section 5 we summarize the results contained in this paper, and we give some hints for future work.
2 Background
We now give some basic notions on classical stable marriage problems, stable marriage problems with partial orders, and stable marriage problems with quantitative preferences.
2.1 Stable marriage problems
A stable marriage problem (SM) [5] of size is the problem of finding a stable marriage between men and women. Such men and women each have a preference ordering over the members of the other sex. A marriage is a onetoone correspondence between men and women. Given a marriage , a man , and a woman , the pair is a blocking pair for if prefers to his partner in and prefers to her partner in . A marriage is said to be stable if it does not contain blocking pairs.
The sequence of all preference orderings of men and women is usually called a profile. In the case of classical stable marriage problem (SM), a profile is a sequence of strict total orders.
Given a SM , there may be many stable marriages for . However, it is interesting to know that there is always at least one stable marriage.
Given an SM , a feasible partner for a man (resp., a woman ) is a woman (resp., a man ) such that there is a stable marriage for where and are married.
The set of all stable marriages for an SM forms a lattice, where a stable marriage dominates another stable marriage if men are happier (that is, are married to more or equally preferred women) in w.r.t. . The top of this lattice is the stable marriage where men are most satisfied, and it is usually called the maleoptimal stable marriage. Conversely, the bottom is the stable marriage where men’s preferences are least satisfied (and women are happiest, so it is usually called the femaleoptimal stable marriage). Thus, a stable marriage is maleoptimal iff every man is paired with his highest ranked feasible partner.
The GaleShapley (GS) algorithm [4] is a wellknown algorithm to solve the SM problem. At the start of the algorithm, each person is free and becomes engaged during the execution of the algorithm. Once a woman is engaged, she never becomes free again (although to whom she is engaged may change), but men can alternate between being free and being engaged. The following step is iterated until all men are engaged: choose a free man , and let propose to the most preferred woman on his preference list, such that has not already rejected . If is free, then and become engaged. If is engaged to man ’, then she rejects the man ( or ’) that she least prefers, and becomes, or remains, engaged to the other man. The rejected man becomes, or remains, free. When all men are engaged, the engaged pairs form the male optimal stable matching. It is female optimal, of course, if the roles of male and female participants in the algorithm were interchanged.
This algorithm needs a number of steps that, in the worst case, is quadratic in (that is, the number of men), and it guarantees that, if the number of men and women coincide, and all participants express a strict order over all the members of the other group, everyone gets married, and the returned matching is stable.
Example 1
Assume . Let and be respectively the set of women and men. The following sequence of strict total orders defines a profile:

(i.e., man prefers woman to woman ),

,

,

For this profile, the maleoptimal solution is . For this specific profile the femaleoptimal stable marriage coincides with the maleoptimal one.
2.2 Stable marriage problems with partially ordered preferences
In SMs, each preference ordering is a strict total order over the members of the other sex. More general notions of SMs allow preference orderings to be partial [9]. This allows for the modelling of both indifference (via ties) and incomparability (via absence of ordering) between members of the other sex. In this context, a stable marriage problem is defined by a sequence of partial orders, over the men and over the women. We will denote with SMP a stable marriage problem with such partially ordered preferences.
Given an SMP, we will sometimes use the notion of a linearization of such a problem, which is obtained by linearizing the preference orderings of the profile in a way that is compatible with the given partial orders.
A marriage for an SMP is said to be weaklystable if it does not contain blocking pairs. Given a man and a woman , the pair is a blocking pair if and are not married to each other in and each one strictly prefers the other to his/her current partner.
A weakly stable marriage dominates a weakly stable marriage iff for every man , and there is a man s.t. . Notice that there may be more than one undominated weakly stable marriage for an SMP.
2.3 Stable marriage problems with quantitative preferences
In classical stable marriage problems, men and women express only qualitative preferences over the members of the other sex. For every pair of women (resp., men), every man (resp., woman) states only that he (resp., she) prefers a woman (resp., a man) more than another one. However, he (resp., she) cannot express how much he (resp., she) prefers such a woman (resp., a man). This is nonetheless possible in stable marriage problems with quantitative preferences.
A stable marriage problem with quantitative preferences (SMQ) [7] is a classical SM where every man/woman gives also a numerical preference value for every member of the other sex, that represents how much he/she prefers such a person. Such preference values are natural numbers and higher preference values denote a more preferred item. Given a man and a woman , the preference value for man (resp., woman ) of woman (resp., man ) will be denoted by (resp., ).
Example 2
Let and be respectively the set of women and men. An instance of an SMQ is the following:

(i.e., man prefers woman to woman , and he prefers with value and with value ),

,

,

The numbers written into the round brackets identify the preference values.
In [7] they consider stable marriage problems with quantitative preferences by looking at the stable marriage that maximizes the sum of the preference values. Therefore, they use the classical definition of stability and they use preference values only when they have to look for the optimal solution. We want, instead, to use preference values also to define new notions of stability and optimality.
3 stability
A simple generalization of the classical notion of stability requires that there are not two people that prefer with at least degree (where is a natural number) to be married to each other rather than to their current partners.
Definition 1 (stability)
Let us consider a natural number with . Given a marriage , a man , and a woman , the pair is an blocking pair for if the following conditions hold:

prefers to his partner in , say , by at least (i.e., ),

prefers to her partner in , say , by at least (i.e., ).
A marriage is stable if it does not contain blocking pairs. A man (resp., woman ) is feasible for woman (resp., man ) if is married with in some stable marriage.
3.1 Relations with classical stability notions
Given an SMQ , let us denote with , the classical SM problem obtained from by considering only the preference orderings induced by the preference values of .
Example 3
If is equal to , then the stable marriages of coincide with the stable marriages of . However, in general, stability allows us to have more marriages that are stable according to this definition, since we have a more relaxed notion of blocking pair. In fact, a pair is an blocking if both and prefer each other to their current partner by at least and thus pairs where and prefer each other to their current partner of less than are not considered blocking pairs.
The fact that stability leads to a larger number of stable marriages w.r.t. the classical case is important to allow new stable marriages where some men, for example the most popular ones, may be married with partners better than all the feasible ones according to the classical notion of stability.
Given an SMQ , let us denote with the set of the stable marriages of and with the set of the stable marriages of . We have the following results.
Proposition 1
Given an SMQ , and a natural number with ,

if , ;

if , .
Given an SMP , the set of stable marriages of contains the set of stable marriages of , since the blocking pairs of are a subset of the blocking pairs of .
Let us denote with the stable marriage with incomparable pairs obtained from an SMQ by setting as incomparable every pair of people that don’t differ for at least , and with the set of the weakly stable marriages of . It is possible to show that the set of the weakly stable marriages of coincides with the set of the stable marriages of .
Theorem 1
Given an SMQ , .
Proof:
We first show that .
Assume that a marriage , we now show that .
If , then there is a pair (man,woman), say , in
such that prefers to his partner in , say ,
and prefers to her partner in , say .
By definition of ,
this means that prefers to by at least degree
and prefers to by at least degree in , and so .
Similarly, we can show that .
In fact, if , then there is a pair (man,woman), say , in
such that prefers to by at least degree
and prefers to by at least degree .
By definition of , this means that
prefers to and prefers to in
and so , i.e.,
is not a weakly stable marriage for .
This means that, given an SMQ , every algorithm that is able to find a weakly stable marriage for provides an stable marriage for .
Example 4
Assume that is . Let us consider the following instance of an SMQ, say .


,

,

The SMP is the following:

(where means incomparable),

,

,

The set of the stable marriages of , that coincides with the set of the weakly stable marriages of , by Theorem 1, contains the following marriages: and .
On the other hand, not all stable marriage problems with partially ordered preferences can be expressed as stable marriage problems with quantitative preferences such that the stable marriages in the two problems coincide. More precisely, given any SMP problem , we would like to be able to generate a corresponding SMQ problem and a value such that, in , the weights of elements ordered in differ more than , while those of elements that are incomparable in differ less than . Consider for example the case of a partial order over six elements, defined as follows: and . Then there is no way to choose a value and a linearization of the partial order such that the weights of and differ for at least , for any i,j between 1 and 5, while at the same time the weight of and each of the ’s differ for less than .
3.2 Dominance and lexmaleoptimality
We recall that in SMPs a weaklystable marriage dominates another weaklystable marriage if men are happier (or equally happy) and there is at least a man that is strictly happier. The same holds for stable marriages. As in SMPs there may be more than one undominated weaklystable marriage, in SMQs there may be more than one undominated stable marriage.
Definition 2 (dominance)
Given two stable marriages, say and , dominates if every man is married in to more or equally preferred woman than in and there is at least one man in married to a more preferred woman than in .
Example 5
Let us consider the SMQ shown in Example 4. We recall that is and that the stable marriages of this problem are and . does not dominate since, for , and does not dominate since, for , .
We now discriminate among the stable marriages of an SMQ, by considering the preference values given by women and men to order pairs that differ for less than .
We will consider a marriage optimal when the most popular men are as happy as possible and they are married with the most popular feasible women.
To compute a strict ordering on the men where the most popular men (resp., the most popular women) are ranked first, we follow a reasoning similar to the one considered in [11, 10], that is, we apply a voting rule [1] to the preferences given by the women (resp., by the men) . More precisely, such a voting rule takes in input the preference values given by the women over the men (resp., given by the men over the women) and returns a strict total order over the men (resp., women).
Definition 3 (lexmaleoptimal)
Consider an SMQ , a natural number , and a voting rule . Let us denote with (resp., ) the strict total order over the men (resp., over the women) computed by applying to the preference values that the women give to the men (resp., the men give to the women). An stable marriage is lexmaleoptimal w.r.t. and , if, for every other stable marriage , the following conditions hold:

there is a man such that ,

for every man , .
Proposition 2
Given an SMQ , a strict total ordering (resp., ) over the men (resp., women),

there is a unique lexmaleoptimal stable marriage w.r.t. and , say .

may be different from the maleoptimal stable marriage of ;

if has a unique undominated weakly stable marriage, say , then coincides with , otherwise is one of the undominated weakly stable marriages of .
Example 6
Let us consider the SMQ, , shown in Example 4. We have shown previously that this problem has two weakly stable marriages that are undominated. We now want to discriminate among them by considering the lexmaleoptimality notion. Let us consider as voting rule the rule that takes in input the preference values given by the women over the men (resp., by the men over the women) and returns a strict preference ordering over the men (resp., women). This preference ordering is induced by the overall score that each man (resp., woman) receives: men (women) that receive higher overall scores are more preferred. The overall score of a man (resp., woman ), say (resp., ), is computed by summing all the preference values that the women give to him (the men give to her). If two candidates receive the same overall score, we use a tiebreaking rule to order them. If we apply this voting rule to the preference values given by the women in , then we obtain , , and thus the ordering is such that . If we apply the same voting rule to the preference values given by the men in , , , and thus the ordering is such that . The lexmaleoptimal stable marriage w.r.t. and is the marriage .
3.3 Finding the lexmaleoptimal stable marriage
It is possible to find optimal stable marriages by adapting the GSalgorithm for classical stable marriage problems [4].
Given an SMQ and a natural number , by Theorem 1, to find an stable marriage it is sufficient to find a weakly stable marriage of . This can be done by applying the GS algorithm to any linearization of .
Given an SMQ , a natural number , and two orderings and over men and women computed by applying a voting rule to as described in Definition 3, it is possible to find the stable marriage that is lexmaleoptimal w.r.t and by applying the GS algorithm to the linearization of where we order incomparable pairs, i.e., the pairs that differ for less than in , in accordance with the orderings and .
Proposition 3
Given an SMQ , a natural number , (resp., ) an ordering over the men (resp., women), algorithm LexmalestableGS returns the lexmaleoptimal stable marriage of w.r.t. and .
4 Stability notions relying on links
Until now we have generalized the classical notion of stability by considering separately the preferences of the men and the preferences of the women. We now intend to define new notions of stability that take into account simultaneously the preferences of the men and the women. Such a new notion will depend on the strength of the link of the married people, i.e., how much a man and a woman want to be married with each other. This is useful to obtain a new notion of stable marriage, that looks at the happiness of the pairs (man,woman) rather than at the happiness of the members of a single sex.
A way to define the strength of the link of two people is the following.
Definition 4 (link additivestrength)
Given a man and a woman , the link additivestrength of the pair , denoted by , is the value obtained by summing the preference value that gives to and the preference value that gives to , i.e., . Given a marriage , the additivelink of , denoted by , is the sum of the links of all its pairs, i.e., .
Notice that we can use other operators beside the sum to define the link strength, such as, for example, the maximum or the product.
We now give a notion of stability that exploit the definition of the link additivestrength given above.
Definition 5 (linkadditivestability)
Given a marriage , a man , and a woman , the pair is a linkadditiveblocking pair for if the following conditions hold:

,

,
where is the partner of in and is the partner of in . A marriage is linkadditivestable if it does not contain linkadditiveblocking pairs.
Example 7
Let and be, respectively, the set of women and men. Consider the following instance of an SMQ, :

,

,

,

In this example there is a unique linkadditivestable marriage, that is , which has additivelink . Notice that such a marriage has an additivelink higher than the maleoptimal stable marriage of that is which has additivelink .
The strength of the link of a pair (man,woman), and thus the notion of link stability, can be also defined by considering the maximum operator instead of the sum operator.
Definition 6 (link maximalstrength)
Given a man and a woman , the link maximalstrength of the pair , denoted by , is the value obtained by taking the maximum between the preference value that gives to and the preference value that gives to , i.e., . Given a marriage , the maximallink of , denoted by , is the maximum of the links of all its pairs, i.e., .
Definition 7 (linkmaximalstability)
Given a marriage , a man , and a woman , the pair is a linkmaximalblocking pair for if the following conditions hold:

,

,
where is the partner of in and is the partner of in . A marriage is linkmaximalstable if it does not contain linkmaximalblocking pairs.
4.1 Relations with other stability notions
Given an SMQ , let us denote with (resp., ) the stable marriage problem with ties obtained from by changing every preference value that a person gives to a person with the value (resp., ), by changing the preference rankings accordingly, and by considering only these new preference rankings.
Let us denote with (resp., ) the set of the linkadditivestable marriages (resp., linkmaximalstable marriages) of and with (resp., ) the set of the weakly stable marriages of (resp., ). It is possible to show that these two sets coincide.
Theorem 2
Given an SMQ , and .
Proof: Let us consider a marriage . We first show that if then . If , there is a pair that is a linkadditiveblocking pair, i.e., and , where (resp., ) is the partner of (resp., ) in . Since , prefers to in the problem , and, since , prefers to in the problem . Hence is a blocking pair for the problem . Therefore, .
We now show that if then . If , there is a pair that is a blocking pair for , i.e., prefers to in the problem , and prefers to in the problem . By definition of the problem , and . Therefore, is a linkadditiveblocking pair for the problem . Hence, .
It is possible to show similarly that .
When no preference ordering changes in (resp., ) w.r.t. , then the linkadditivestable (resp., linkmaximalstable) marriages of coincide with the stable marriages of .
Proposition 4
Given an SMQ , if () , then (resp., ).
If there are no ties in (resp., ), then there is a unique linkadditivestable marriage (resp., linkmaximalstable marriage) with the highest link.
Proposition 5
Given an SMQ , if (resp., ) has no ties, then there is a unique linkadditivestable (resp., linkmaximalstable) marriage with the highest link.
If we consider the definition of linkmaximalstability, it is possible to define a class of SMQs where there is a unique linkmaximalstable marriage with the highest link.
Proposition 6
In an SMQ where the preference values are all different, there is a unique linkmaximalstable marriage with the highest link.
4.2 Finding linkadditivestable and linkmaximalstable marriages with the highest link
We now show that for some classes of preferences it is possible to find optimal linkadditivestable marriages and linkmaximalstable marriages of an SMQ by adapting algorithm GS, which is usually used to find the maleoptimal stable marriage in classical stable marriage problems.
By Proposition 2, we know that the set of the linkadditivestable (resp., linkmaximalstable) marriages of an SMQ coincides with the set of the weakly stable marriages of the SMP (resp., ). Therefore, to find a linkadditivestable (resp., linkmaximalstable) marriage, we can simply apply algorithm GS to a linearization of (resp., ).
Proposition 7
Given an SMQ , the marriage returned by algorithm linkadditivestableGS (linkmaximalstableGS) over , say , is linkadditivestable (resp., linkmaximalstable). Moreover, if there are not ties in (resp., ), is linkadditivestable (resp., linkmaximalstable) and it has the highest link.
When there are no ties in (resp., ), the marriage returned by algorithm linkadditivestableGS (resp., linkmaximalstableGS) is maleoptimal w.r.t. the profile with links. Such a marriage may be different from the classical maleoptimal stable marriage of , since it considers the happiness of the men reordered according to their links with the women, rather than according their single preferences.
This holds, for example, when we assume to have an SMQ with preference values that are all different and we consider the notion of link2stability.
Proposition 8
Given an SMQ where the preference values are all different, the marriage returned by algorithm linkmaximalstableGS algorithm over is linkmaximalstable and it has the highest link.
5 Conclusions and future work
In this paper we have considered stable marriage problems with quantitative preferences, where both men and women can express a score over the members of the other sex. In particular, we have introduced new stability and optimality notions for such problems and we have compared them with the classical ones for stable marriage problems with totally or partially ordered preferences. Also, we have provided algorithms to find marriages that are optimal and stable according to these new notions by adapting the GaleShapley algorithm.
We have also considered an optimality notion (that is, lexmaleoptimality) that exploits a voting rule to linearize the partial orders. We intend to study if this use of voting rules within stable marriage problems may have other benefits. In particular, we want to investigate if the procedure defined to find such an optimality notion inherits the properties of the voting rule with respect to manipulation: we intend to check whether, if the voting rule is NPhard to manipulate, then also the procedure on SMQ that exploits such a rule is NPhard to manipulate. This would allow us to transfer several existing results on manipulation complexity, which have been obtained for voting rules, to the context of procedures to solve stable marriage problems with quantitative preferences.
Acnowledgements
This work has been partially supported by the MIUR PRIN 20089M932N project “Innovative and multidisciplinary approaches for constraint and preference reasoning”.
References
 [1] K.J. Arrow, A.K. Sen, and K. Suzumura. Handbook of Social Choice and Welfare. North Holland, Elsevier, 2002.
 [2] S. Bistarelli, S. N. Foley, B. O’Sullivan, and F. Santini. From marriages to coalitions: A soft csp approach. In CSCLP, pages 1–15, 2008.
 [3] S. Bistarelli, U. Montanari, and F. Rossi. Semiringbased constraint solving and optimization. Journal of the ACM, 44(2):201–236, 1997.
 [4] D. Gale and L. S. Shapley. College admissions and the stability of marriage. Amer. Math. Monthly, 69:9–14, 1962.
 [5] D. Gusfield and R. W. Irving. The Stable Marriage Problem: Structure and Algorithms. MIT Press, Boston, Mass., 1989.
 [6] Dan Gusfield. Three fast algorithms for four problems in stable marriage. SIAM J. Comput., 16(1):111–128, 1987.
 [7] Robert W. Irving, Paul Leather, and Dan Gusfield. An efficient algorithm for the “optimal” stable marriage. J. ACM, 34(3):532–543, 1987.
 [8] J. Liebowitz and J. Simien. Computational efficiencies for multiagents: a look at a multiagent system for sailor assignment. Electonic government: an International Journal, 2(4):384–402, 2005.
 [9] D. Manlove. The structure of stable marriage with indifference. Discrete Applied Mathematics, 122(13):167–181, 2002.
 [10] M. S. Pini, F. Rossi, K. B. Venable, and T. Walsh. Manipulation complexity and gender neutrality in stable marriage procedures. Jornal of Autonomous Agents and MultiAgent Systems. To appear.
 [11] M. S. Pini, F. Rossi, K. B. Venable, and T. Walsh. Manipulation and gender neutrality in stable marriage procedures. In Proc. AAMAS’09, volume 1, pages 665–672, 2009.
 [12] A. E. Roth. The evolution of the labor market for medical interns and residents: a case study in game theory. Journal of Political Economy, 92:991–1016, 1984.
 [13] ChungPiaw Teo, Jay Sethuraman, and WeePeng Tan. Galeshapley stable marriage problem revisited: Strategic issues and applications. Manage. Sci., 47(9):1252–1267, 2001.
 [14] L. Tesfatsion. Galeshapley matching in an evolutionary trade network game. Economic Report, 43, 1998.
Maria Silvia Pini 
Department of Pure and Applied Mathematics 
University of Padova, Italy 
Email: mpini@math.unipd.it 
Francesca Rossi 
Department of Pure and Applied Mathematics 
University of Padova, Italy 
Email: frossi@math.unipd.it 
K. Brent Venable 
Department of Pure and Applied Mathematics 
University of Padova, Italy 
Email: kvenable@math.unipd.it 
Toby Walsh 
NICTA and UNSW, Sydney, Australia 
Email: Toby.Walsh@nicta.com.au 