Distributed constraint optimization

The main ingredients of a DCOP problem are agents and variables. Importantly, each variable is owned by an agent; this is what makes the problem distributed. Formally, a DCOP is a tuple ${\displaystyle \langle A,V,{\mathfrak {D}},f,\alpha ,\eta \rangle }$, where:

• ${\displaystyle A}$ is the set of agents, ${\displaystyle \{a_{1},\dots ,a_{|A|}\}}$.
• ${\displaystyle V}$ is the set of variables, ${\displaystyle \{v_{1},v_{2},\dots ,v_{|V|}\}}$.
• ${\displaystyle {\mathfrak {D}}}$ is the set of variable-domains, ${\displaystyle \{D_{1},D_{2},\dots ,D_{|V|}\}}$, where each ${\displaystyle D_{j}\in {\mathfrak {D}}}$ is a finite set containing the possible values of variable ${\displaystyle v_{j}}$.
  • If ${\displaystyle D_{j}\in {\mathfrak {D}}}$ contains only two values (e.g. 0 or 1), then ${\displaystyle v_{j}}$ is called a binary variable.
• ${\displaystyle f}$ is the cost function. It is a function[1]
  $${\displaystyle f:\bigcup _{S\subseteq V}\times _{v_{j}\in S}D_{j}\to \mathbb {R} }$$

  
  that maps every possible partial assignment to a cost. Usually, only few values of ${\displaystyle f}$ are non-zero, and it is represented as a list of the tuples that are assigned a non-zero value. Each such tuple is called a constraint. Each constraint ${\displaystyle C}$ in this set is a function ${\displaystyle f_{C}:D_{1}\times \cdots \times D_{k}\to \mathbb {R} }$ assigning a real value to each possible assignment of the variables. Some special kinds of constraints are:
  • Unary constraints - constraints on a single variable, i.e., ${\displaystyle f_{C}:D_{j}\to \mathbb {R} }$ for some ${\displaystyle v_{j}\in V}$.
  • Binary constraints - constraints on two variables, i.e, ${\displaystyle f_{C}:D_{j_{1}}\times D_{j_{2}}\to \mathbb {R} }$ for some ${\displaystyle v_{j_{1}},v_{j_{2}}\in V}$.
• ${\displaystyle \alpha }$ is the ownership function. It is a function ${\displaystyle \alpha :V\to A}$ mapping each variable to its associated agent. ${\displaystyle \alpha (v_{j})\mapsto a_{i}}$ means that variable ${\displaystyle v_{j}}$ "belongs" to agent ${\displaystyle a_{i}}$. This implies that it is agent ${\displaystyle a_{i}}$'s responsibility to assign the value of variable ${\displaystyle v_{j}}$. Note that ${\displaystyle \alpha }$ is not necessarily an injection, i.e., one agent may own more than one variables. It is also not necessarily a surjection, i.e., some agents may own no variables.
• ${\displaystyle \eta }$ is the objective function. It is an operator that aggregates all of the individual ${\displaystyle f}$ costs for all possible variable assignments. This is usually accomplished through summation:
  $${\displaystyle \eta (f)\mapsto \sum _{s\in \bigcup _{S\subseteq V}\times _{v_{j}\in S}D_{j}}f(s).}$$

  

The objective of a DCOP is to have each agent assign values to its associated variables in order to either minimize or maximize ${\displaystyle \eta (f)}$ for a given assignment of the variables.

A value assignment is a pair ${\displaystyle (v_{j},d_{j})}$ where ${\displaystyle d_{j}}$ is an element of the domain ${\displaystyle D_{j}}$.

A partial assignment is a set of value-assignments where each ${\displaystyle v_{j}}$ appears at most once. It is also called a context. This can be thought of as a function mapping variables in the DCOP to their current values:

$${\displaystyle t:V\to (D\in {\mathfrak {D}})\cup \{\emptyset \}.}$$

Note that a context is essentially a partial solution and need not contain values for every variable in the problem; therefore, ${\displaystyle t(v_{i})\mapsto \emptyset }$ implies that the agent ${\displaystyle \alpha (v_{i})}$ has not yet assigned a value to variable ${\displaystyle v_{i}}$. Given this representation, the "domain" (that is, the set of input values) of the function f can be thought of as the set of all possible contexts for the DCOP. Therefore, in the remainder of this article we may use the notion of a context (i.e., the ${\displaystyle t}$ function) as an input to the ${\displaystyle f}$ function.

A full assignment is an assignment in which each ${\displaystyle v_{j}}$ appears exactly once, that is, all variables are assigned. It is also called a solution to the DCOP.

An optimal solution is a full assignment in which the objective function ${\displaystyle \eta (f)}$ is optimized (i.e., maximized or minimized, depending on the type of problem).

The graph coloring problem is as follows: given a graph ${\displaystyle G=\langle N,E\rangle }$ and a set of colors ${\displaystyle C}$, assign each vertex, ${\displaystyle n\subset N}$, a color, ${\displaystyle c\leq C}$, such that the number of adjacent vertices with the same color is minimized.

As a DCOP, there is one agent per vertex that is assigned to decide the associated color. Each agent has a single variable whose associated domain is of cardinality ${\displaystyle |C|}$ (there is one domain value for each possible color). For each vertex ${\displaystyle n_{i}\leq N}$, there is a variable ${\displaystyle v_{i}\in V}$ with domain ${\displaystyle D_{i}=C}$. For each pair of adjacent vertices ${\displaystyle \langle n_{i},n_{j}\rangle \in E}$, there is a constraint of cost 1 if both of the associated variables are assigned the same color:

$${\displaystyle (\forall c\subseteq C:f(\langle v_{i},c\rangle ,\langle v_{j},c\rangle )\mapsto 1).}$$

The objective, then, is to minimize ${\displaystyle \eta (f)}$.

The distributed multiple- variant of the knapsack problem is as follows: given a set of items of varying volume and a set of knapsacks of varying capacity, assign each item to a knapsack such that the amount of overflow is minimized. Let ${\displaystyle I}$ be the set of items, ${\displaystyle K}$ be the set of knapsacks, ${\displaystyle s:I\to \mathbb {N} }$ be a function mapping items to their volume, and ${\displaystyle c:K\to \mathbb {N} }$ be a function mapping knapsacks to their capacities.

To encode this problem as a DCOP, for each ${\displaystyle i\in I}$ create one variable ${\displaystyle v_{i}\in V}$ with associated domain ${\displaystyle D_{i}=K}$. Then for all possible contexts ${\displaystyle t}$:

$${\displaystyle f(t)\mapsto \sum _{k\in K}{\begin{cases}0&r(t,k)\leq c(k),\\r(t,k)-c(k)&{\text{otherwise}},\end{cases}}}$$

where ${\displaystyle r(t,k)}$ represents the total weight assigned by context ${\displaystyle t}$ to knapsack ${\displaystyle k}$:

$${\displaystyle r(t,k)=\sum _{v_{i}\in t^{-1}(k)}s(i).}$$

The item allocation problem is as follows. There are several items that have to be divided among several agents. Each agent has a different valuation for the items. The goal is to optimize some global goal, such as maximizing the sum of utilities or minimizing the envy. The item allocation problem can be formulated as a DCOP as follows.[2]

• Add a binary variable v_ij for each agent i and item j. The variable value is "1" if the agent gets the item, and "0" otherwise. The variable is owned by agent i.
• To express the constraint that each item is given to at most one agent, add binary constraints for each two different variables related to the same item, with an infinite cost if the two variables are simultaneously "1", and a zero cost otherwise.
• To express the constraint that all items must be allocated, add an n-ary constraint for each item (where n is the number of agents), with an infinite cost if no variable related to this item is "1".

DCOP was applied to other problems, such as:

• coordinating mobile sensors;
• meeting and task scheduling.

A simple way for solving an ADCOP is to replace each constraint ${\displaystyle f_{C}:D_{1}\times \cdots \times D_{k}\to \mathbb {R} ^{k}}$ with a constraint ${\displaystyle f_{C}':D_{1}\times \cdots \times D_{k}\to \mathbb {R} }$, which equals the sum of the functions ${\displaystyle f_{C}^{1}+\cdots +f_{C}^{k}}$. However, this solution requires the agents to reveal their cost functions. Often, this is not desired due to privacy considerations.[14][15][16]

Another approach is called Private Events as Variables (PEAV).[17] In this approach, each variable owns, in addition to his own variables, also "mirror variables" of all the variables owned by his neighbors in the constraint network. There are additional constraints (with a cost of infinity) that guarantee that the mirror variables equal the original variables. The disadvantage of this method is that the number of variables and constraints is much larger than the original, which leads to a higher run-time.

A third approach is to adapt existing algorithms, developed for DCOPs, to the ADCOP framework. This has been done for both complete-search algorithms and local-search algorithms.[13]

The structure of an ADCOP problem is similar to the game-theoretic concept of a simultaneous game. In both cases, there are agents who control variables (in game theory, the variables are the agents' possible actions or strategies). In both cases, each choice of variables by the different agents result in a different payoff to each agent. However, there is a fundamental difference:[13]

• In a simultaneous game, the agents are selfish - each of them wants to maximize his/her own utility (or minimize his/her own cost). Therefore, the best outcome that can be sought for in such setting is an equilibrium - a situation in which no agent can unilaterally increase his/her own gain.
• In an ADCOP, the agents are considered cooperative: they act according to the protocol even if it decreases their own utility. Therefore, the goal is more challenging: we would like to maximize the sum of utilities (or minimize the sum of costs). A Nash equilibrium roughly corresponds to a local optimum of this problem, while we are looking for a global optimum.

There are some intermediate models in which the agents are partially-cooperative: they are willing to decrease their utility to help the global goal, but only if their own cost is not too high. An example of partially-cooperative agents are employees in a firm. On one hand, each employee wants to maximize their own utility; on the other hand, they also want to contribute to the success of the firm. Therefore, they are willing to help others or do some other time-consuming tasks that help the firm, as long as it is not too burdensome on them. Some models for partially-cooperative agents are:[18]

• Guaranteed personal benefit: the agents agree to act for the global good if their own utility is at least as high as in the non-cooperative setting (i.e., the final outcome must be a Pareto improvement of the original state).
• Lambda-cooperation: there is a parameter ${\displaystyle \lambda \in [0,1]}$. The agents agree to act for the global good if their own utility is at least as high as ${\displaystyle (1-\lambda )}$ times their non-cooperative utility.

Solving such partial-coopreation ADCOPs requires adaptations of ADCOP algorithms.[18]