## 1 Introduction

Distributed Constrained Optimization Problems (DCOPs) are a popular framework to coordinate interactions in cooperative multi-agent systems. A number of real world problems such as distributed event scheduling [maheswaran2004taking] and the distributed RLFA problem[cabon1999radio] can be modelled with this framework[rashik2020speeding, khan2018near, mahmud2019aed, choudhury2020particle, khan2018generic, khan2018speeding]

. The constraints among the participating agents in these applications, and many other besides, can be both hard and soft. In any case, the algorithms that have been proposed to solve DCOPs can be broadly classified into exact and non-exact algorithms. The former (e.g.

[modi2005adopt, petcu2005scalable]) always finds a globally optimal solution. In contrast, the latter algorithms (e.g. [farinelli2008decentralised, zhang2005distributed]) trade solution quality at the expense of reduced computation and communication costs.Among the non-exact approaches, Generalized Distributive Law based algorithms, such as Max-Sum [farinelli2008decentralised] and Bounded Max-Sum (BMS) [rogers2011bounded], have received particular attention. Specifically, Bounded Max-Sum is extremely attractive variant of Max-Sum which produces good approximate solution for DCOP problems with cycles. However, BMS does not actively consider such constraints that are hard although there is a number of real-life applications containing hard constraints. We particularly observe that the presence of hard constraints can be utilized to further improve BMS’s solution quality by removing inconsistent values from agents’ domain and thus reduce the upper bound of the global solution. It is worth noting that due to hard constraints the traditional BMS algorithm exhibits a situation where each agent may own a set of allowable assignments in place of a specific assignment. Each combination of the agent’s assignments will experience the same profit for the tree structured graphical representation that is a acyclic graph (e.g. Factor Graphs or Junction Tree) of a given DCOP, but produce different profit for that cyclic DCOP. To the best of our knowledge, there exists no method for choosing the best one. In this paper, we propose a novel approach which aims at enforcing consistency and selecting the most preferable combination of agent’s assignments.

## 2 Problem Formulation

A DCOP model can be formally expressed as a tuple A, X, D, F, where A = {} is a set of agents, X = {} is a set of variables, D = {} is a set of domains for the variables in X. F = {} is a set of constraint functions. denotes value for each possible combination of the variables of . The dependencies between the functions and variables can be graphically represented by factor graph FG. Finally, the mapping of variable node to agent is represented by : X A where one variable will be assigned to one agent.

Within this model, the main objective of DCOP algorithms such as BMS, is to find the assignment of each variable, and approximate solution by maximizing the sum of all functions that is . After removing appropriate dependencies from FG according to the phases of BMS, an acyclic graph is formed. Additionally, the maximum impact is calculated which is used for computing the upper bound on the value of the unknown optimal solution as . Here is the solution found by executing BMS on the corresponding acyclic graph. Now, the first objective of our approach is to update the domain of each variables so that the maximum impact is minimized as in (Equation 1). Here, = inconsistent domain values of variable .

(1) | |||

Due to the presence of hard constraints, the BMS algorithm experiences tie variable assignment(s) after executing Max-Sum on its acyclic graph. Our second objective is to select appropriate variable assignment so that it provides the most preferable solution for the constraint graph (Equation 2). Here, is set of tie assignments for variable .

(2) | |||

## 3 HARD CONSISTENCY ENFORCED BOUNDED Max-Sum (HBMS)

With the motivation of utilizing hard constraints, our objective is to
decrease the upper bound on the optimal solution and also increase the solution quality of BMS. Our first contribution in the upper bound is obtained by changing the maximum impact . In our first phase, consistency enforcement, we update the variable domains (Equation 1) by enforcing arc-consistency on the constraint graph. This step also speeds up the execution time of the algorithm. For example, in Figure 0(a), ^{1}^{1}1In this paper, we have considered all constraints as binary and mentioned as for illustration where are dependent on calculates its maximum impact using Equation 3. According to this equation, selects its value for maximizing and for minimizing the function. After the consistency enforcement phase, each variable’s domain gets pruned and () pair changes to () by reducing . This is true for and in the same way.

(3) |

In the second phase, we generate a spanning tree (i.e. acyclic graph) from the factor graph by removing the most suitable dependencies. We do this following the same way as the BMS algorithm. Then run Max-Sum algorithm on the acyclic factor graph. If the hard constraints are satisfied, each of them will contribute the same in profit maximization. This phenomenon increases the possibility for each variable having multiple assignment ( is the set of assignment allowed for variable ) with same profit for the acyclic graph. For example, in Figure 0(b), after executing Max-Sum, is assigned with multiple values that is . However, we need to chose the variable assignments in such a way that the profit for the main constraint graph is maximized (Equation 2). For this purpose, we utilize the removed dependencies and the set of variables that are dependent on function but are not a part of the acyclic graph . In the third phase, we model a smaller DCOP problem as such that, F = {} where and select a set of agents and variables dependent on the function accordingly. Finally, for each , will be equals to . According to Figure 0(b), we create a new DCOP and represents it as a factor graph but this time it includes the dependencies that are removed in the previous phase. It consists of and their corresponding variables () , () and (). At this phase, we execute BMS on this smaller graph and eventually get information about the priority of the tie assignments of each variables that we found in second phase. In Figure 0(c), we can see the profit for each assignment received from the third phase (e.g. for ). Finally, In the fourth phase, we use this information to execute Max-Sum on again. This step will finally select the preferable variable assignment which improves solution quality. For instance, the variable assignment is . The complexity of HBMS is twice of the BMS algorithm since we execute this algorithm two times. The computation cost for the smaller graph in the second phase is negligible. Finally, the complexity of the arc consistency enforcement phase is where is the number of edges of the constraint graph and is the average domain size.

## 4 Empirical Evaluation

In this section, we empirically evaluate the improvement in solution quality of HBMS in comparison to the Bounded Max-Sum algorithm. To benchmark the result, we run experiment on random constraint graphs. We vary the number of nodes from 5 to 30 in Figure 2, set the variable domain in [0,..,40], 30% hard constraints along with soft constraints and functions’ utility values from 0 to 500. We observe improvement in solution quality around 5-30% on average. However, we experience negative results in some instances. In the future, we would like to explore that area for observing the reasons behind this situation.

## 5 Conclusions and Future Work

The major finding of this paper is that by taking advantage of the hard constraints, we can significantly improve the solution quality of the Bounded Max-Sum algorithm. Another notable contribution is in the reduction of the upper bound. Our empirical evidence presents that it is possible to improve the solution around 10-30% than BMS. In our future work, we would to like to observe the impact of different forms of consistency enforcement, and fix the negative results observed in the evaluation. The final research direction includes extending the potential application domain.

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