DJAO: A Communication-Constrained DCOP algorithm that combines features of ADOPT and Action-GDL

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1 JAO: A ommunication-onstrained OP algorithm that combines features of AOPT and Action-GL Yoonheui Kim and Victor Lesser University of Massachusetts at Amherst MA 3, USA {ykim,lesser}@cs.umass.edu Abstract In this paper we propose a novel OP algorithm, called JAO, that is able to efficiently find a solution with low communication overhead; this algorithm can be used for optimal and bounded approximate solutions by appropriately setting the error bounds. Our approach builds on distributed junction trees used in Action-GL to represent independence relations among variables. We construct an / search space based on these junction trees. This new type of search space results in higher degrees for each node, consequently yielding a more efficient search graph in the distributed settings. JAO uses a branch-and-bound search algorithm to distributedly find solutions within this search graph. We introduce heuristics to compute the upper and lower bound estimates that the search starts with, which is integral to our approach for reducing communication overhead. We empirically evaluate our approach in various settings. Introduction In this paper, we formulate a new algorithm for distributed constraint optimization problems (OPs), called JAO, which works on a distributed junction tree (Paskin, Guestrin, and McFadden 25). JAO operates in two phases. In the first phase, heuristic upper and lower bounds for variable value configurations are created using a bottom-up propagation scheme similar in character to Action-GL (Vinyals, Rodriguez-Aguilar, and erquides 2). Except that instead of transmitting values for all configurations, we transmit only the filtered upper and lower bounds of configuration values. The next phase using these heuristics conducts an AOPT-like (Modi et al. 25; Yeoh, Felner, and Koenig 28) search on / search graph based on the junction tree, which we call / search junction graph, to find a solution with desired precision. This two-phase strategy reduces overall communication significantly. / search tree and context-minimal / search graph An / search space (Marinescu and echter 25) is introduced to exploit independencies encoded by the graph- opyright c 24, Association for the Advancement of Artificial Intelligence ( All rights reserved. ical model upon which OP algorithms such as Action- GL and Max-Sum (Farinelli et al. 28) are constructed. An / search tree is a search space with additive nodes whose subtrees denote disjoint search spaces under different variables in addition to nodes in traditional search trees whose subtrees denote disjoint search spaces under values of variables. nodes decompose the search space in their subtrees under Generalized istributive Law framework (Aji and McEliece 2). It reduces the size of OP search space from O(exp(n)) to O(n exp(m)), where m is the depth of the pseudo-tree (Koller and Friedman 29) and n is the number of variables. OP algorithms such as AOPT (Modi et al. 25) and n- AOPT (Yeoh, Felner, and Koenig 28) can be viewed as distributed search algorithms on this / search space. efinition (/ search tree) Given a OP instance P, its primal graph G and a pseudotree T of G, the associated / search tree S T (P ) has alternating levels of nodes and nodes. The nodes are labeled X i and correspond to variables. The nodes are labeled X i, a and correspond to value assignments in the domains of variables. The root of the / search tree is an node, labeled with the root of T. The children of an node X i are nodes labeled with assignments X i, a, consistent along the path from the root. The children of an node X i, a are nodes labeled with the children of variable X i in T. The path of a node n S T, denoted P ath ST (n), is the path from the root of S T to n, and corresponds to a partial value assignment to all variables along the path. An example of / search tree is given in Fig. a. ecause nodes decompose the problem into separate subproblems, variables in different subtrees of an node n are considered independently given the value assignment along the path to n. The arcs in S T are annotated by appropriate labels of the cost functions. efinition 2 (label) The label l(x i, X i, a ) is defined as the sum of all the cost function values for which variable X i is contained in their scope and whose scope is fully assigned along the path from root to n. efinition 3 (value) The value v(n) of a node n S T, is defined recursively as follows: (i) if n = X i, a is a terminal node then v(n) = l(x i, X i, a ); (ii) if n =

2 X i, a is an internal node then v(n)=(x i, X i, a ) + n succ(n) v(n ); (iii) if n = X i is an internal node then v(n) = max n succ(n) v(n ), where succ(n) are the children of n in S T. In (echter and Mateescu 27), the / search graph shown in Fig. c was introduced to reduce the size of the search tree in Fig a by merging two nodes that root identical subtrees. A ontext-based merge operation is defined as either i) merging two nodes that share the same variable assignments on the ancestors of these nodes, which have connections in G to these nodes or their descendants, or ii) merging two nodes that share assignments on these nodes and ancestors of nodes, which have connections in G to these nodes descendants. Example For the primal graph G in Fig. b and a search tree in Fig. a for G, nodes for that shares assignments for can be merged as is connected to in G. In contrast, nodes for cannot be merged as the assignments for A and should match. nodes at the lowest level can be merged if these nodes share the assignments as these nodes do not have any descendant. A A A (b) (d) (a) A (c) Figure : an / search graph efinition 4 (context minimal / graph) The / search graph of G that is closed under contextbased merge operator is called a context minimal / search graph. Junction Tree and OP A distributed constraint optimization problem (OP) instance P = A, X,, F is formally defined by the following parameters: A set of variables X = {X,..., X r }, where each variable has a finite domain (maximum size N) of possible values that it can be assigned. A A set of constraint functions F = (F,..., F k ), where each constraint function, F j : X j R, takes as input any setting of the variables X j X and provides a real valued utility. In OP, we assume that each variable x i is owned by an agent a i A and that an agent only knows about the constraint functions in which it is involved. The OP can be represented using a constraint network, where there is a node corresponding to each variable x i and where there is an edge (hyper-edge) for each constraint F j that connects all variables that are involved in the function F j. The objective in the OP is to find the complete variable configuration x that maximizes F F j F j(x j ). The dual constraint graph (Koller and Friedman 29) is a transformation of a non-binary network into a special type of binary network. It contains cliques (or c-variables) domains of which ranges over all possible value combinations permitted by the corresponding constraint functions, and shared variables in any two adjacent cliques have same values. A junction tree (or join tree) (owell et al. 27) T is a subgraph of the dual graph which is a tree and satisfies the condition that cliques associated with a variable x form a connected subset of T. A Junction tree is represented as a tuple X,, S, F. where X is a set of variables, is a set of cliques, where each clique i is a subset of variables i X; S is a set of separators, where each separator is an arc between two adjacent cliques containing their intersection; and F is a set of potentials, where each potential in F is assigned to each clique in. A distributed junction tree (Paskin, Guestrin, and McFadden 25) decomposes a OP into a series of subproblems, some of which can be solved in parallel. A subproblem represented as a clique c i can be solved independently given the local constraint functions f i F and the values from neighbors on separator s i S. Separators S specify which values will be used in the neighboring cliques in order to compute the solution for its local subproblem. JAO(k) First Phase: Heuristics Generation Preprocessing techniques to supply the search with heuristic values has successfully been used to enhance both centralized and decentralized search methods. (Ali, Koenig, and Tambe 25; Wallace 996). In this section we describe a scheme for generating initial heuristic estimates h U and h L used in JAO(k), based on a new function filtering technique, which we call Soft Filtering, described here. (Pujol-Gonzalez et al. 2; rito and Meseguer 2) used the Function Filtering technique (rito and Meseguer 2) on OPs to prune variable configurations of local nodes, that do not yield the optimal solution. We use the soft function filtering technique to generate heuristics that maintains the tuples that potentially yield the optimal solution while summarizing the rest with upper and lower bounds. Unlike the heuristics in (Ali, Koenig, and Tambe 25; Wallace 996) which are generated by solving lower complexity problems than the original, JAO solves the original

3 problem and focuses on reducing communication by filtering tuples that are unlikely to be part of the optimal solution. The Soft Filtering technique used in JAO summarizes constraint functions to reduce communication required for transmitting such function. A simple difference from Function Filtering is that the Soft Filtering technique provides summarized lower and upper bounds on filtered configurations. Let the variable configuration S in message m be divided into two sets filtered configurations S F, and nonfiltered configurations S NF. Let U be the upper bounds of values on variable configurations and L the lower bounds. The values U m and L m in the messages are filtered as follows. { U(v) if v SNF U m (v) = max SF U(v) if v S F L m is similarly defined with max replaced with min. Filtered configurations are summarized as a filtered tuple with a single upper and lower bounds, therefore reducing the number of items in each message from 2S to 2 S NF +2. The optimal strategy is guaranteed to remain in the search space as no solution is completely dropped. This summarization builds a basis for the next phase where an AOPT-like search finds a solution within a desired accuracy. Among many ways to select which items to filter, we select items in the bottom ( d)% of the function range. Second Phase: Search on / junction graph On the pseudo-tree based search graph, functions are evaluated only when their scope is fully assigned along the path. The search backtracks to evaluate different variable assignment which occurs among the domain of a single variable at each level. This complete decentralization in value selection in the distributed setting results in the exponential number of messages in AOPT. Instead, we introduce / search graph on a junction tree where each level is associated with each clique in the junction tree in a OP (See Fig. 2), consequently yielding a more compact search graph with a lower number of nodes. This search graph is a contextminimal / search graph upon construction. A Figure 2: an / search graph based on a junction tree efinition 5 (/ search junction graph) Given a OP instance P and its junction tree T, the associated / search junction graph S T (P ) has alternating levels of nodes and nodes. The nodes are labeled i : S i, a, N i where a are variables assignments in the domains of variables in separators S i whose value are propagated from ancestors and newly appeared variables N i in clique i. These nodes correspond to the cliques with partial assignment. The nodes are labeled S ij, b and correspond to value assignments in the domains of the separator between clique i and its child j. The root of the / search graph is an node, labeled with the root of T. The children of an node S ij, b are nodes who are labeled with j : S j, b, N j with the same assignment on variables in separators S ij and S j. Example onsider the graphical model in Fig. b describing a graph coloring problem over domains {,}. An / search graph based on a possible pseudo-tree is given in Fig c and an / search junction graph in Fig d is given in Fig. 2. Observe that the function evaluation on l({a, }, a) occurs at the expansion of nodes at level 3 in Fig. c instead of at level in Fig. 2. On Fig. c, the search is unguided until the third expansion. It also elongates the backtrack path leading to an increase in the number of visited nodes to evaluate a single function. Theorem Given a OP instance P and a junction tree T, its / search junction graph is sound and complete. It contains all and only solutions. The solution space in / search junction graph is identical to the junction tree, thus it contains all and only solutions. onsequently, any search algorithm that traverses the / search junction graph in a depth-first manner is guaranteed to have a time complexity equal to the time complexity of Action-GL (Vinyals, Rodriguez-Aguilar, and erquides 2) on the same junction tree which is exponential in the tree width. Theorem 2 The size of search junction graph has exactly same size as the total complexity of junction tree as no subtree is redundant. The depth of the graph does not exceed the number of agents. The search result for its subtree is stored at each node, therefore no identical subtree is explored twice and a value assignment on a cost function is never repeated. The arcs in S T are annotated by appropriate labels of the cost functions. The nodes in S T are associated with a value, accumulating the result of the computation resulted from the subtree below. labels on the arcs are determined by values from neighboring nodes unlike the one defined in ef. 3. efinition 6 label: The label l( i : S i, a, N i, S ij, b ) of the arc from the node to the node S ij, b is defined as the cost function values contained in the clique i whose scope is fully assigned with values from the parent node and its child node. The value v(n) of a node n S T (P ) is computed in the same way as in ef. 3. Likewise, the value of each node can be recursively computed from leaves to root. We can show that: Proposition Given an / search graph S T (P ) of a OP instance, the value function v(n) is the maximum cost solution to the subproblem rooted at n, subject to the current variable instantiation along the path from root to n. If n is the root of S T, then v(n) is the maximum cost solution to P.

4 We prove optimality by verifying the value of nodes are identical to the values produced during the execution of Action-GL. The value of an node is identical to the value of corresponding assignments in the messages from the corresponding clique of Action-GL given the context along the search path to these nodes. Valuation of nodes is the combination of local utility functions and values of its child nodes and corresponds to the maximum achievable value of variable assignments given the variable assignments along the search path. Proposition 2 / search junction graph S T (P ) is context-minimal upon construction The separators S i and S ij contain all and only variables that build a context for each node and a single node is created for each value assignment in the separator, thus it is contextminimal. Search in distributed settings Each agent in the system distributedly conducts its share of search for the nodes on S T (P ) it owns. Agents are responsible for valuation of owned nodes and path determination. efinition 7 Agent ownership: Each node in S T (P ) is owned by an agent. Agent A i owns all nodes associated with its own clique i : nodes i : S i, a, N i and child nodes of these are assigned to A i. For example, suppose clique A, A and in Fig d are owned by agent A, A 2, and A 3 respectively. nodes of clique are 3 :,,, 3 :,,. These nodes and child nodes of these are assigned to A 3. Search procedure between nodes belonging to different agents incurs communication. When an agent A j chooses to expand a child node i : S i, a, N i, A j transmits partial assignments a to an agent A i who owns the child nodes. Updated function values are sent back to A j when the search backs up. Search paths that incur communication are displayed as dotted lines in Fig. 2. JAO on / search junction graph If each node n S T (P ) is assigned a heuristic lower-bound estimate L(n) and heuristic upper-bound estimate UP(n), then we can calculate the lower and upper bound estimates of assignments and dominated search space can be pruned. ounds on Partial Solution Similarly to (Marinescu and echter 25), a partially expanded search graph, denoted as PSG, contains the root node, will have a frontier containing all the nodes that were generated but not expanded. Each expansion of a leaf node on PSG updates the lower and upper bound estimates on / search graph. An active partial subtree AP T (n) rooted at a node n ending at a tip node t contains the path between n and t, and all children of nodes on the path. A dynamic heuristic function of a node n relative to the current PSG given the initial heuristic functions h U and h L can be computed. efinition 8 (ynamic Lower and Upper ound) Given an active partial tree AP T (n), the dynamic heuristic estimate of upper and lower bound function, U(n) and L(n), is defined recursively as follows: (i) if there is a single node n in AP T (n) and is evaluated, then U(n) = v(n) = L(n) else if n is a single node in AP T U(n) = h U (n) and L(n) = h L (n); (ii) n = S ij, b is an node, having children m,..., m k, and label = l( i : S i, a, N i, S ij, b ), then U(n) = min(h U (n), label + k i= U(m i)) L(n) = max(h L (n), label + k i= L(m)) ; (iii) if n = i : P i, a, N i where n is an node, having an child m, then U(n) = min(h(n), U(m)) and L(n) = max(h(n), L(m i )). Theorem 3 L(n) is a lower bound on the optimal solution to the subproblem rooted at n, namely L(n) v(n), and also by definition L(n) h L (n). Also, U(n) v(n) and U(n) h L (n). Proof: We will prove by induction assuming the correctness of heuristics that v(n) h U(n), v(n) h L(n). asis: At leaf nodes of / junction search graph, it is trivial that v(n) = U(n) = L(n) as v(n) is computed using local constraints and does not involve any heuristic function. Induction step: At any node having children m,..., m k, v(n) = label + i v(mi) label + i U(mi), where v(m i) U(m i). v(n) min(h U(n), label + i U(mi)) = U(n), where v(n) h U(n). At any node having child m = argmax i v(m i), v(n) = v(m) U(m) v(n) min(h U(n), U(m)) = U(n). L(n) v(n) can be proved similarly. Therefore, L(n) v(n) U(n). Also, U (n) and L(n) provides tighter bounds than the initial heuristic functions. Proposition 3 (Pruning rule) For any node n and its sibling m, if U(n) < L(m) or U(n) = L(n) then subtree below n can be pruned. JAO(k) We now set up a JAO search on S T (P ) whose nodes are assigned to agents. Starting from the root agent given the initial heuristic upper and lower bound functions h U and h L, the objective is to search one of the solution that satisfies the termination condition while pruning dominated candidate solutions. JAO agents use three types of messages: VALUE, OST, and TERMINATE. At the start, the root agent expands the nodes from its node and selects the best branch in the subtree and sends VALUE messages containing variable values on the chosen branch to its child nodes. Upon receipt of a VALUE message, an agent evaluates whether the back-up condition is satisfied for the given value assignments b in the message. If the back-up condition is satisfied, the agent backs up with updated values by sending a OST message to its parent. Otherwise, it expands the nodes compatible with b and selects the best branch and sends a VALUE message to that child node.

5 Algorithm : JAO(k)() procedure Init() wait ; // number of waited messages k i, k c ; // own and child s k value m b nil ; // node in par(a i) to backtrack to m, m ; // node with max, second max U n c; // node context-compatible with m b procedure RootRun() Init(); UpdateM(); if (hecktermination()) then Send(TERMINATE) to c succ(a i); terminate; end k i U(m ) max(u(m ) k, L(m )); wait succ(a i) ; Send(VALUE, m, k i); loop forever while (message queue is not empty) do pop msg off message queue; When Received(msg); if (hecktermination()) then Send(TERMINATE) to c succ(a i); terminate; else if ( wait==) then UpdateM(); k i = U(m ) max(u(m ) k, L(m )); Send(VALUE, m, k i) to c succ(a i); end procedure Run() Init(); loop forever while (message queue is not empty) do pop msg off message queue; When Received(msg); if (ecide ackup() && wait==) then Send(OST, m b, U(m b ), L(m b )) to par(a i) else if (wait==) then UpdateM(); Send(VALUE, m, k c), to c succ(a i) end Upon receipt of a OST message containing the updated lower and upper bounds on the chosen expanded nodes from all child nodes, it recalculates the lower and upper bounds of its node. It then re-evaluates the back-up condition for the received VALUE message. Unless it satisfies the back-up condition, then the question of which branch to select is re-examined and the agents sends another VALUE message to its children. These steps are repeated until a termination condition in Prop 4 holds for the root agent. It then sends a TERMINATE message to each of its children and terminate. Upon receipt of a TERMINATE message, each agent does the same. Algorithm 2: JAO(k)(2) procedure UpdateM() m = argmax U(m), for m succ(n r); m = argmax U(m), for m succ(n r) \ m ; procedure When Received(OST, m, v U, v L) wait wait, U(m) v U, L(m) v L; U (n) U(n), where n = par(m) ; U(n) max(u(n), U(m)) ; L(n) max(l(n), L(m)); procedure When Received(VALUE, m, k) k i k, m b m, wait succ(a i) ; n c context compatible(m),; procedure ecide ackup() if (U(n c) U (n c) k i) then return true; else k c (k i (U(n) U (n)))/ succ(a i) ; return false; end procedure When Received(TERMINATE) Send(TERMINATE) to c succ(a i); terminate; procedure heck Termination() if ( U(n r) == L(n r)) then return true; else if for m succ(n r)\m, U(m) L(m ) then return true; return false; Proposition 4 Given an node n and nodes m,..., m k at the root agent, JAO(k) is terminated if U and L satisfies the condition U(n) = L(n) or i, U(m j ) L(m i ) for j, i j. Each agent stores the lower and upper bounds of expanded nodes and updates these values upon each OST message arrival. The memory requirement for each agent does not exceed O(n d ) where n is the size of variable domain and d is the induced width of the junction tree. Among many different search strategies which determines the back-up condition for solving OP and OP, bestfirst search and depth-first branch-and-bound search have been primarily studied (Marinescu and echter 25; 27; Modi et al. 25; Yeoh, Felner, and Koenig 28). est-first search always follows the best item found and in the distributed setting whenever there is an update, agents propagate it to all ancestors whose best items may change. On the other hand, depth-first search retains the current path until it is certainly dominated or the true value of node v(n) is found. In (Gutierrez, Meseguer, and Yeoh 2), AOP T (k) provides a trade-off between these two extremes, where the search keeps the current path until the distance between the best solution on the current path and the best solution found so far becomes greater than a given constant k. Similarly, we developed JAO(k), which subsumes both depth-first and best-first search strategy on / search junction graph. It performs depth-first when k =, best-

6 first when k = ɛ, and a hybrid when ɛ < k <, where k is the distance between the best found solution U(m ) and the next best solution U(m ) found so far. Unlike AOPT(k) in which each agent determines k using the best solutions based on its subproblems provided by the node s subtree, each agent in JAO instead uses a measurement that considers a global perspective of the entire problem on the current best solution. The search backtracks when U(m ) max(u(m ) k, L(m )), which occurs as soon as the best solution is dominated by the second best with the best-first strategy with k = ɛ, and when the true value for m is found (Thus, U(m ) = L(m )) with the depth-first strategy with k =. Algorithm and 2 shows the pseudocode of JAO(k), where a i is a generic agent, par(a i ) its parent agent, succ(a i ) its set of child agents, par(n) the parent node of the node n in the search graph, succ(n) the set of node n s child nodes, and n r the node of the root agent. The root agent runs RootRun() which contains search initiation whereas all other agents run Run(). The pseudo-code uses a predicate context compatible(m) to select a node whose variable value assignment matches that of the node m. Example of JAO On the junction tree in Fig. d, let there be three agents, A, A 2 and A 3 for the cliques A, A and respectively. The constraint function f(a, ) are assigned to clique A, f(a, ) and f(, ) to clique A, f(, ) to. A f(a, ) : 4 5 A f(a, ) : 2 3 f(, ) : 4 2 f(, ) : Phase : The first phase starts by the agent A 2 computing the local potential b by merging f(a, ) and f(, ) in the clique A as well as A 3. A 6 5 b(a,, ) : Each agent who does not own the root node generates a filtered message once they have received from all the child agents (agents who own the child nodes). The message from A 2 and A 3 to A in Action-GL would be as follows. M A3 A : 4 A 6 M A2 A : Filtered messages are created and sent with the filtering rate l = 8. FS denotes the filtered set of variable configurations. For the message M A2 A, the function range is (8 4), thus items with upper bound equal or less than (4+ 4*.8) are filtered except (A=, =). Messages in JAO are: M A3 A : h L h U 4 4 FS A h L h U M A2 A : 8 8 FS 4 6 Once messages received, A calculates potential b as the total sum of received messages and local functions. A L U 5 7 b(a) : Phase 2: A checks the termination condition on the possible solution with the highest upper bound (A=, =). Since L(A=, =) does not dominate U(A=, =), the search starts. A computes the distance k between maximum and the second maximum upper bounds. The distance k = 4 3 =. The search backtracks when the upper bound decreases by equal or more than min(k, k ). The upper and lower bound gap originates only from M A2 A. Therefore, A sends a VALUE message with a variables configuration (A=, =) and min(k, k ) to A 2. A 2 receives this VALUE message. It then sends a OST message L(, )=5, U(, )=5 as it is a leaf node. Upon receipt of the OST message, the root node updates its potential. A L U 6 7 b(a, ) : Since the lower bound of (A=, =) dominates upper bounds of all other configurations, the termination condition is satisfied and the search terminates. Approximate JAO(k) An approximate version of the algorithm can be obtained by relaxing the constraint on upper and lower bound gap similar to search-based OP algorithms (Modi et al. 25; Yeoh, Sun, and Koenig 29). Approximate JAO with an error bound e terminates when the solution contains no more than error e such that that value of the found solution n is no worse than v(n ) e, where n is the optimal solution. The corresponding termination condition is L(n) U(n) e or i, U(m j ) L(m i )+e for j, i j. Also, the search backtracks when U(m ) max(u(m ) k, L(m ) + e). Empirical Evaluation In this section we evaluate the performance of JAO search. For each experiment, we report the communication costs, Ns(non-concurrent constraint check), and solution quality for approximate solutions with respect to optimal solutions. We used a JAO that sends VALUE messages to at most 25 nodes when there are ties. We evaluate and compare our approach with Action-GL and AOPT(k) with k = 4 which was reasonable among 4, 4 and 4. ommunication costs are measured as the number of bytes sent during execution and the message count of both A single variable value is 4bytes, and cost 8bytes.

7 c Algorithm Total ytes Ns Msgs JAO(k = ɛ) JAO(k = ) JAO(k = ) JAO(k = 5) Action-GL AOPT(K=4) JAO(k = ɛ) JAO(k = ) JAO(k = ) JAO(k = 5) Action-GL AOPT(K=4) JAO(k = ɛ) JAO(k = ) JAO(k = ) JAO(k = 5) Action-GL AOPT(K=4) Table : Performance of Optimal JAO(k) on Random inary OP Instances Savings WRT GL Solution Quality Overall ommunication ost l=.8 l=.7 l= Accuracy Loss(%) (a) ommunication Savings.97 Solution Quality.94 l=.8.9 l=.7.88 l=.6 Solution ound Accuracy Loss(%) (c) Solution Quality onstraint hecks WRT GL Message ount WRT GL Overall omputation ost l=.8 l=.7 l= Accuracy Loss (b) omputation time Message ount l=.8 l=.7 l= Accuracy Loss (d) Message ount Figure 3: Performance of Approximate JAO(k=) Algorithm Total ytes Ns Msgs A JAO(k = ɛ) 63,36 22,27,3 35 JAO(k =,, ) 55,2 25,43, Action-GL 3,624,86 9,95,92 26 AOPT(K=3,,),2,364 24,68,428 2,5,732 JAO(JAOk = ɛ) 278,68 3,44, JAO(k =,, ) 238,696 2,998, Action-GL 4,274, AOPT(K=3,,) 54,735,4 66,542,75 9,869,28 JAO(k = ɛ) 27,8 9,379, JAO(k =,, ) 2,56 7,59,783 9 Action-GL,294,382 6,49,23 78 AOPT(K=3,,),9,24 2,625,36 78,3 JAO(k = ɛ) 78,528 3,98, JAO(k =,, ) 482,654 43,36, Action-GL,32,229 58,754, AOPT(K=3,,) 2,573,856 66,347,439 3,82,54 Table 2: Sensor Network Instances UTIL and VALUE messages. For approximate results, we show the true utility of found solution which is often much higher than the estimated lower bound. We experimented on random binary OP instances with variables of domain size. The function cost are randomly generated over the range,..., 2. We varied the number of constraint functions c over 2, 25 and 3 and averaged our results over 2 instances for each value of c. The total communication amount is largely determined by the structure of junction tree. Thus, five junction trees were constructed for each problem instance. Initial heuristics were generated using soft filtering where the bottom 7% is filtered. The JAO with k = ɛ(i.e., conducts best-first search) consistently performed well in these experiments. Tab. 2 shows the results on sensor network instances from a public repository (Yin 28) with a heuristic which filters 9% (l =.9). Tab. and 2 show JAO requires less communication than both AOPT 2 and Action-GL. ompared to the random OP instances, function ranges in sensor network instances are wider and a large set of clearly dominated variable configurations results in significant communication savings with JAO. JAO with high k values performed better in terms of communication on these lower connectivity graphs compared to the random OP instances. Lastly, in an experiment on the random binary OP instances, we measured the methods trend as the solution quality guarantee changes. We evaluated 2 instances for each quality loss ranging from loss = % to loss = 8% using a heuristic which filters 8%(l=.8) and 7%(l=.7). Results in Fig. 3 show that JAO gains significant savings in communication as the error bound increases. With an 8% heuristic, it transmits about 8 times less information than that of Action-GL when loss = 8% while it uses 3 times more computation (Ns) and about 3 messages per agent. onclusions We addressed the problem of solving OP exactly and with precise approximation bounds by developing a new distributed algorithm called JAO. There are three novel ideas in JAO. Firstly, it uses an / junction graph representation, which builds a basis for efficient search in the distributed settings. The second is a two phase search strategy that combines characteristics of AOPT and Action-GL. The third is a soft filtering technique to significantly reduce communication without losing any accuracy. 2 Results from (Gutierrez, Meseguer, and Yeoh 2)

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