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Balancing Asymmetry in Max-sum Using Split Constraint Factor Graphs

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Principles and Practice of Constraint Programming (CP 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11008))

Abstract

Max-sum is a version of Belief Propagation, used for solving DCOPs. On tree-structured problems, Max-sum converges to the optimal solution in linear time. When the constraint graph representing the problem includes multiple cycles, Max-sum might not converge and explore low quality solutions. Damping is a method that increases the chances that Max-sum will converge. Damped Max-sum (DMS) was recently found to produce high quality solutions for DCOP when combined with an anytime framework.

We propose a novel method for adjusting the level of asymmetry in the factor graph, in order to achieve a balance between exploitation and exploration, when using Max-sum for solving DCOPs. By converting a standard factor graph to an equivalent split constraint factor graph (SCFG), in which each function-node is split to two function-nodes, we can control the level of asymmetry for each constraint. Our empirical results demonstrate that by applying DMS to SCFGs with a minor level of asymmetry we can find high quality solutions in a small number of iterations, even without using an anytime framework. As part of our investigation of this success, we prove that for a factor-graph with a single constraint, if this constraint is split symmetrically, Max-sum applied to the resulting cycle is guaranteed to converge to the optimal solution and demonstrate that for an asymmetric split, convergence is not guaranteed.

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Notes

  1. 1.

    A similar factor graph was used in [31] for representing asymmetric DCOPs.

  2. 2.

    The inference algorithm for minimization problems is actually Min-sum. However, we will continue to refer to it as Max-sum since this name is widely accepted.

  3. 3.

    We say that a variable is involved in a constraint if it is one of the variables the constraint refers to.

  4. 4.

    For an example of the need to break ties in the factor-graph see [30].

  5. 5.

    For lack of space we describe the algorithm briefly and refer the reader to more detailed descriptions in [4, 17, 30].

  6. 6.

    Although Lemmas 1 and 2 can be implied from Lemma 3, for simplicity of presentation we enclose all three, provide complete proof for Lemma 1, and intuitive explanations how to generalize the proof so it will apply to Lemmas 2 and 3.

  7. 7.

    Recall that we assumed in Sect. 3.1 that there are no ties, so such a cost is unique.

  8. 8.

    Without loss of generality, we assume that k is odd. If it was even, then the assumption was that the cost is \(\frac{k}{2}c\).

  9. 9.

    further insights on the relation between the success of our empirical results and the properties presented in this section are detailed in Sect. 6.2.

  10. 10.

    This range was selected so that the numbers do not become too small and due to precision, generate distorted SCFGs. Obviously, if the input costs are between 0 and 100, adding 100 to each cost can be the first step of the splitting method.

  11. 11.

    For lack of space we do not present convergence graphs for the other problems. As expected the meeting scheduling convergence results were similar to graph coloring while the results for the other problem types were similar to the convergence results of the sparse uniform random problems.

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Authors and Affiliations

  1. Ben Gurion University of the Negev, Beer Sheva, Israel

    Liel Cohen & Roie Zivan

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  1. Liel Cohen
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  2. Roie Zivan
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Correspondence to Roie Zivan .

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  1. Carnegie Mellon University, Pittsburgh, PA, USA

    John Hooker

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Cohen, L., Zivan, R. (2018). Balancing Asymmetry in Max-sum Using Split Constraint Factor Graphs. In: Hooker, J. (eds) Principles and Practice of Constraint Programming. CP 2018. Lecture Notes in Computer Science(), vol 11008. Springer, Cham. https://doi.org/10.1007/978-3-319-98334-9_43

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  • DOI: https://doi.org/10.1007/978-3-319-98334-9_43

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