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Applying Max-sum to asymmetric distributed constraint optimization problems

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Abstract

We study the adjustment and use of the Max-sum algorithm for solving Asymmetric Distributed Constraint Optimization Problems (ADCOPs). First, we formalize asymmetric factor-graphs and apply the different versions of Max-sum to them. Apparently, in contrast to local search algorithms, most Max-sum versions perform similarly when solving symmetric and asymmetric problems and some even perform better on asymmetric problems. Second, we prove that the convergence properties of Max-sum_ADVP (an algorithm that was previously found to outperform standard Max-sum and Bounded Max-sum) and the quality of the solutions it produces, are dependent on the order between nodes involved in each constraint, i.e., the inner constraint order (ICO). A standard ICO allows to reproduce the properties achieved for symmetric problems. Third, we demonstrate that a non-standard ICO can be used to balance exploration and exploitation. Our results indicate that Max-sum_ADVP with non-standard ICO and Damped Max-sum, when solving asymmetric problems, both outperform other versions of Max-sum, as well as local search algorithms specifically designed for solving ADCOPs.

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Notes

  1. For additional examples, including supply chain and smart grid applications, see [15].

  2. We thank an anonymous reviewer of a previous version of this paper that indicated that this redundancy can be derived from [20, 30] and allowed us to omit our own redundancy proof.

  3. Following [9] we use the terms “variable-node” and “function-node” to refer to nodes in the factor graph corresponding to variables and constraints, respectively.

  4. In order to make the example easier to follow, we do not normalize the messages sent by reducing \(\alpha\).

  5. We describe Max-sum_AD and Max-sum_ADVP more formally than Bounded Max-sum and in more detail, since we address it in our theoretical analysis.

  6. For asymmetric problems, other alternatives will be discussed.

  7. We assume that only variable-nodes perform damping as we do in our implementation.

  8. Again, we ignore orders where both variable-nodes come before or after both function-nodes (as discussed for symmetric problems).

  9. All algorithms we consider are deterministic, thus, executions of the same algorithm on the same problem are identical.

  10. By sum of messages we mean the vector of costs and not the value selection added for VP of course.

  11. The assumption on the equality of these costs in both executions does not cause a loss of generality. The proof holds when they are different as well, just that the value of \(\beta\) will be different.

  12. The source code for all our experiments can be downloaded from https://github.com/liel-cohen/ADCOP_Max-sum.

  13. In order to avoid density we omitted the DCOP results of Max-sum and Max-sum_VP, which are similar to the ADCOP results produced by these algorithms.

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Authors and Affiliations

  1. Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Roie Zivan, Tomer Parash, Liel Cohen-Lavi & Yarden Naveh

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  1. Roie Zivan
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Correspondence to Roie Zivan.

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This paper is an extension of our IJCAI paper [57]. Besides an extended description, examples and complete proofs, it includes comparisons of the proposed algorithms with versions that include damping [6], which were not included in the IJCAI version.

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Zivan, R., Parash, T., Cohen-Lavi, L. et al. Applying Max-sum to asymmetric distributed constraint optimization problems. Auton Agent Multi-Agent Syst 34, 13 (2020). https://doi.org/10.1007/s10458-019-09436-8

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