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Improved WPM encoding for coalition structure generation under MC-nets

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Abstract

The Coalition Structure Generation (CSG) problem plays an important role in the domain of coalition games. Its goal is to create coalitions of agents so that the global welfare is maximized. To date, Weighted Partial MaxSAT (WPM) encoding has shown high efficiency in solving the CSG problem, which encodes a set of constraints into Boolean propositional logic and employs an off-the-shelf WPM solver to find out the optimal solution. However, in existing WPM encodings, a number of redundant encodings are asserted. This results in additional calculations and correspondingly incurs performance penalty. Against this background, this paper presents an Improved Rule Relation-based WPM (I-RWPM) encoding for the CSG problem, which is expressed by a set of weighted rules in a concise representation scheme called Marginal Contribution net (MC-net). In order to effectively reduce the constraints imposed on encodings, we first identify a subset of rules in an MC-net, referred as a set of freelance rules. We prove that solving the problem made up of all freelance rules can be achieved with a straightforward means without any extra encodings. Thus the set of rules requiring to be encoded is downsized. Next, we improve the encoding of transitive relations among rules. To be specific, compared with the existing rule relation-based encoding that generates transitive relations universally among all rules, I-RWPM only considers the transitivity among rules with particular relationship. In this way, the number of constraints to be encoded can be further decreased. Experiments suggest that I-RWPM significantly outperforms other WPM encodings for solving the same set of problem instances.

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Notes

  1. S ({i, j}) = S (i, j) if i < j

    = S (j, i) if j < i

  2. If ri and rj are compatible on different coalitions or independent, and there exists no rule compatible on the same coalition with ri or rj.

  3. We have examined several versions of Maxino and QMaxSAT and find that the performance of various versions of the same solver is quite similar. Therefore, in Table 3, we prefer to exhibit the performance of only one version while omitting other versions since they show similar results.

  4. To be exact, the average computation time is 5.765 seconds, as shown in Table 3.

  5. In Fig. 6a, the value of d, i.e. the probability that two rules are independent, is quite large. We explain the reason why d is extremely large from a theoretic view in Appendix B.

  6. We preferred Pwbo2.0 to SAT4j since Pwbo2.0 was evaluated as the most efficient solver for AWPM in the work [18].

  7. To be more precise, depending on the number of agents and the value of α, the average number \(\overline m\) will either be \(\lfloor {\frac {{1 - {\alpha ^{m}}}}{{1 - \alpha }}} \rfloor \) or \(\lceil {\frac {{1 - {\alpha ^{m}}}}{{1 - \alpha }}} \rceil \). However, to ease the discussion without loss of generality, we will assume that \(\overline m\) is simply \({\frac {{1 - {\alpha ^{m}}}}{{1 - \alpha }}}\).

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Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (Grant Number 61806171), Ministry of Education in China Project of Humanities and Social Sciences (Grant Number 17YJCZH260), JSPS KAKENHI (Grant Numbers JP16K00304, JP17K00307, JP17H00761 and JP18H03299).

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Authors and Affiliations

  1. Chengdu University of Technology, Chengdu, China

    Xiaojuan Liao

  2. Kyushu University, Fukuoka, Japan

    Miyuki Koshimura, Kazuki Nomoto & Makoto Yokoo

  3. Saga University, Saga, Japan

    Suguru Ueda

  4. National Institute of Advanced Industrial Science and Technology, Tokyo, Japan

    Yuko Sakurai

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  1. Xiaojuan Liao
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  2. Miyuki Koshimura
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  4. Suguru Ueda
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Correspondence to Xiaojuan Liao.

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Appendices

Appendix A: Performance Comparison of WPM Solves for RWPM and AWPM

Table 6 Average runtime (seconds) required by various solvers for RWPM
Full size table
Table 7 Average runtime (seconds) required by various solvers for AWPM
Full size table

Footnote 6This part shows the performance comparison of WPM solvers for RWPM and AWPM, exhibited in Table 6 and 7, respectively. From Table 6, it is clear that QMaxSAT stands out from the rest, in terms of both execution time and the number of successfully solved instances. Thus we select QMaxSAT as the solver for RWPM.

By contrast, statistics in Table 7 reveals that the comparison result heavily depends on the scale of problem. Specifically, when #rules ≤ 230, Maxino had the most outstanding performance. However, when #rules = 240, Maxino failed on two instances while MaxHS and LMHS became more efficient. In addition, when #rules reaches 300, there is only one solver that managed to solve all the 100 problem instances, i.e. MaxHS. Since solving problems with larger #rules is of greater significance, we employ MaxHS as the solver to evaluate AWPM.

Note that multiple versions of Maxino and QMaxSAT were evaluated, and we found that the runtime required for different versions of the same solver is quite similar. Therefore, in Table 6 and 7 we only list results obtained by one version of each solver while omitting others.

Appendix B: Theoretical Analysis on d

This part explains the reason why d is quite large in our experiments by theoretical derivation, described as follows.

Let \(m \in \mathbb {Z}^{+}\) and \(\overline m \in \mathbb {Z}^{+}\) be the total number of agents in an MC-net and the average number of agents in a rule, respectively. If \(\overline m \le m/2\), we have

$$d = \frac{{C_{m}^{\overline m}C_{m - \overline m}^{\overline m}}}{{C_{m}^{\overline m}C_{m}^{\overline m}}} = \frac{{C_{m - \overline m}^{\overline m}}}{{C_{m}^{\overline m}}}, $$

where \(C_{m}^{\overline m}\) is the number of \(\overline m\)-combinations of m.

Next, we compute \(\overline m\). Let random variable X represent the number of agents contained in a rule, X = {1, 2,⋯ , m}, where m is the number of agents in an MC-net. According to the method of generating instances illustrated in [17, 24, 34], each condition is created with one random agent, then a new random agent is repeatedly added with the probability of α until an agent is not added or the rule includes all agents. Based on this method, the probability that X = t is represented as follows.

$$P\left( X \right) = \left\{ {\begin{array}{*{20}{c}} {{\alpha^{t - 1}}\left( {1 - \alpha } \right)}&{{\text{if}}\begin{array}{*{20}{l}} \end{array}t < m,}\\ {{\alpha^{m - 1}}}&{{\mathrm{otherwise.}}} \end{array}} \right. $$

Then the expectation of X is \(E\left (X \right ) = \sum \limits _{t = 1}^{m - 1} {t{\alpha ^{t - 1}}\left ({1 - \alpha } \right )} + m{\alpha ^{m - 1}}\). By simplifying the formula, we have

$$E\left( X \right) = \frac{{1 - {\alpha^{m}}}}{{1 - \alpha }} $$

Thus, \(\overline m\) can be estimated by the expected value of X, i.e., \(\overline m = {\frac {{1 - {\alpha ^{m}}}}{{1 - \alpha }}}\). Footnote 7 In our experiments, α = 0.55, and m ranges from 100 to 300. With this experimental setting, it is immediate that d ≈ 0.96. If a smaller d is required, then we can make the value of α larger. Specifically, when α reaches 0.75, \(\overline m \approx 4\) and d ≈ 0.85.

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Liao, X., Koshimura, M., Nomoto, K. et al. Improved WPM encoding for coalition structure generation under MC-nets. Constraints 24, 25–55 (2019). https://doi.org/10.1007/s10601-018-9295-4

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