There are similarities between the TCPC and the classical matching theory problem known as the Stable Roommate Problem (SRP). SRP is about finding a stable match-ing for an even-sized set. A matchmatch-ing is a partition of the set into disjoint pairs of roommates. We say a matching is stable if there are not two individuals who are not roommates and both prefer each other to their roommate under the current matching.
TCPC and SRP try to match elements within a single set in groups of a given size (size two for SRP).
Irving [109] described an algorithm to solve SRP with a time complexity ofO(n2).
Nevertheless, this algorithm solves a decision problem. It determines whether a stable matching exists, and if so, it returns that matching. We are not solving a decision prob-lem but an optimization one. Furthermore, the TCPC is about matching eprob-lements in groups of any given size or a combination of sizes. Standard SRP uses rooms of size two but this problem becomes complete for rooms of size three [110]. The NP-completeness proof uses the partition into triangles problem as in the NP-NP-completeness proof of TCPC.
Note that the TCPC is about finding an optimal assignment given a certain criterion, but matching theory algorithms deal with the notion of stability. An optimization ver-sion of SRP has not always an optimal solution among the stable solutions, so that they can miss optimality. We show below an example.
Given a totally ordered preference list of possible mates for each student, the desir-ability ofAto be with B in a team k is calculated as the size fork minus the position in that list, starting by an index equal to one. Thus, the last position in the list has a desirability of zero. We show this information in the graph of Figure 5.1 for four students.
Table 5.5 shows desirability degrees for every possible partition of size two of the graph. Partition suitability is calculated as the sum of arities for each node into the subgraphs for each partition. We observe, for a standard SRP and a usual team suitability degree calculation, by summing satisfactions, that neither stability implies optimality nor optimality implies stability.
Section 5.7 : Concluding remarks 99
Figure 5.1: Degree of individual convenience of students in a team.
Partition Desirability Calculation Optimal Stable DA , BC (1 + 0) + (1 + 1) =3 7 7 DB , AC (2 + 0) + (1 + 2) =5 3 7
DC , AB (0 + 0) + (2 + 2) =4 7 3
Table 5.5: Optimality and stability of size-two partitions of graph in Figure 5.1.
There are criteria for the evaluation of team suitability that are different from the ones described in the previous section for STCM. Another psychological theory, posed by Belbin [52], insists on the importance of roles in team composition pro-cesses [44]. Belbin exposes nine important roles that an individual can play in a team:
plant, resource investigator, coordinator, shaper, monitor evaluator, implementer, team-worker, specialist and completer-finisher. According to Belbin’s theory, people play such roles with three different performances: preferred team roles, manageable roles and least preferred roles. An effective team needs to be role balanced, having at least one individual playing any role with a given minimum performance, which can be pre-ferred or manageable. Based exclusively on this theory, Alberola et al. [12] also pose this problem as a multi-agent coalition structure generation problem, developing a tool for practical use in an educational environment [11]. This tool uses Bayesian learning to estimate the predominant roles for each student from the peer-evaluation history made by their former teammates.
5.7 Concluding remarks
We have presented two different ways of solving the TCPC as a Weighted Partial MaxSAT problem, proved that the TCPC is NP-hard, and conducted experiments to evaluate the proposed approach using an exact MaxSAT solver. The experimental results show that the minimizing encoding outperforms the maximizing encoding, and MaxSAT solving is a good solving method for the TCPC.
The proposed MaxSAT approach has the following advantages: it is generic, and so does not need a dedicated algorithm; it is declarative, and so is easier to understand the way the problem is specified; it is flexible, and so different classroom configurations and other team formation problems can be solved similarly; and it is efficient, and so provides an optimal solution in a reasonable amount of time. Moreover, the idea of
creating a minimizing MaxSAT encoding from a maximizing MaxSAT encoding is new, and can be applied to many similar optimization problems.
Chapter 6
Conclusions and Future Work
This chapter summarizes the main contributions of the thesis and outlines a few research directions that we plan to pursue in the future.
6.1 Conclusions
The scientific contributions of this PhD thesis advance the state of the art and open new research directions in MaxSAT and MinSAT solving. The thesis proposes novel logic-based methods for solving challenging optimization problems. Moreover, it reinforces the idea that MaxSAT and MinSAT are competitive generic problem solving approaches in a wide range of application domains.
The main original contributions of this PhD thesis can be summarized as follows:
• The definition of a complete tableau calculus for Weighted Partial MaxSAT.
• The definition of a complete tableau calculus for Weighted Partial MinSAT.
• The definition of a complete calculus that is valid for both Weighted Partial MaxSAT and Weighted Partial MinSAT.
• The definition of the direct, improved and Tseitin-based transformations from non-clausal MaxSAT to clausal MaxSAT, and its conversion into cost-preserving transformations from non-clausal MinSAT to clausal MinSAT.
• The definition of a complete tableau calculus for solving non-clausal MaxSAT, which can also be used to solve non-clausal MinSAT.
• The resolution of the team composition problem in a classroom as a MaxSAT problem.
We believe that the inference systems defined in the thesis have contributed to the creation of a robust logical framework for MaxSAT and MinSAT. Before starting this thesis, to the best of our knowledge, there were no tableau-based approaches for
MaxSAT and MinSAT and there was no approach for solving non-clausal MaxSAT and non-clausal MinSAT.
We also believe that the proposed tableau calculi for MaxSAT and MinSAT could be used in logic courses to illustrate the differences between decision and optimization problems from a logic perspective. Semantic tableaux are very intuitive and could be helpful to explain and understand basic concepts of model and proof theory in Boolean optimization.
It seems clear that new logical approaches to MaxSAT and MinSAT will help to improve MaxSAT and MinSAT solvers. The challenge is to find out if these logical approaches can be used to devise novel solving methods for SAT and develop more powerful SAT solvers.
Finally, we would like to highlight that the application of MaxSAT solving to the area of team formation is promising. The usual constraints in team formation admit efficient MaxSAT encodings, and the challenge is to select the right encoding for each constraint. Moreover, the insights derived from the maximizing and minimizing encod-ings can be used to encode other optimization problems.