Improvement of β MH on min { β H , β S , β P } 

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... E 1 and the set B ([0 0 0]). Applying the exclusivity relations again for each b ∈ B ( a ), and discarding JAs already included in a or B ( a ), we generate a set B ( b ) = ∪ E ∈ E k f ( b , E ) which contains all JAs that potentially exclude b from being k -optimal. In Figure 6, we apply E 1 to find B ( b ) for all b ∈ B ( a ) = { [1 0 0] , [0 1 0] , [0 0 1] , [1 0 1] } where the grayed out JAs are those discarded for being in { a }∪ B ( a ). To ensure that the region that a claims is disjoint from the regions claimed by other k -optima, a should only claim a fraction of each b ∈ B ( a ). This can be achieved if a shares each b equally with all other k -optima that might exclude b . These additional k -optima are contained within B ( b ). However, not all b ∈ B ( b ) can actually be k -optimal as they might exclude each other. If we construct a graph H k ( b ) with nodes for all b ∈ B ( b ) and edges formed using E k , and we find M b , the size of the MIS, then a can safely claim 1 / (1 + M b ) of b . We again use clique partitioning to safely estimate M b . In Figure 6, for b = [0 1 0] , B ([0 1 0]) leads to a three-node, three-edge exclusivity graph H k ([0 1 0]). By adding the values of 1 / (1 + M b ) for all b ∈ B ( a ) (plus one for itself), we obtain that a can safely claim a region of size 3, which implies β S RP = 2 3 / 3 = 2. Algorithm 1’s runtime is polynomial in the number of possible JAs, which is a comparatively small cost for a bound that applies to every possible instantiation of rewards to actions. An exhaustive search for the MIS of H k would be exponential in this number (doubly exponential in the number of agents). We performed five evaluations in addition to the experiment described in Section 2 . The first evaluates the impact of k -optimality for higher values of k . For each of the three DCOP graphs from Figure 2(a-c), Figure 7(a-c) shows key properties for 1-, 2- and 3- optima. The first column of each table shows | A ̃ | , the size of the neighborhood containing all JAs within a distance of k from a k - optimal JA a , and hence of lower reward than a . For example, in the joint patrol domain described in Section 2, Figure 7(a) shows that, if agents are arranged as in the DCOP graph from Figure 2 (a), any 1-optimal joint patrol must have a higher reward than at least 10 other joint patrols. We see that as k increases, the k -optimal set contains JAs that each individually dominate a larger and larger neighborhood. The second column shows, for each of the three graphs, the average reward of each k -optimal JA set found over 20 problem instances, generated by assigning rewards to the links from a uniform random distribution. We define the reward of a k -optimal JA set as the mean reward of all k -optimal JAs that exist for a particular problem instance; each figure in the second column is therefore a mean of means. As k was increased, leading to a larger neighborhood of dominated JAs, the average reward of the k -optimal JA sets show a significant increase (T-tests showed the increase in average reward as k increased was significant within 5%.) However as k increases, the number of possible k -optimal JAs decreases, and hence the next four evaluations explore the e ff ec- tiveness of the di ff erent bounds on the number of k -optima. For the three DCOP graphs shown in Figure 2, Figure 8 provides a concrete demonstration of the gains in resource allocation due to the tighter bounds made possible with graph-based analysis. The x axis in Figure 8 shows k , and the y axis shows the β HS P and β S RP bounds on the number of k -optima that can exist. To understand the impli- cations of these results on resource allocation, consider a patrolling problem where the constraints between agents are shown in the 10- agent DCOP graph from Figure 2(a), and all agents consume one unit of fuel for each JA taken. Suppose that k = 2 has been chosen, and so at runtime, the agents will use MGM-2 [9], repeatedly, to find and execute a set of 2-optimal JAs. We must allocate enough fuel to the agents a priori so they can execute up to all possible 2-optimal JAs. Figure 8(a) shows that if β HS P is used, the agents would be loaded with 93 units of fuel to ensure enough for all 2- optimal JAs. However, β S RP reveals that only 18 units of fuel are su ffi cient, a five-fold savings. (For clarity we note that on all three graphs, both bounds are 1 when k = I and 2 when I − 3 ≤ k < I .) To systematically investigate the impact of graph structure on bounds, we generated a large number of DCOP graphs of varying size and density. We started with complete binary graphs (all pairs of agents are connected) where each node (agent) had a unique ID. Edges were repeatedly removed according to the following two- step process: (1) Find the lowest-ID node that has more than one incident edge. (2) If such a node exists, find the lowest-ID node that shares an edge with it, and remove this edge. Figure 9 shows the β HS P and β S RP bounds for k -optima for k ∈ { 1 , 2 , 3 , 4 } and I ∈ { 7 , 8 , 9 , 10 } . For each of the 16 plots shown, the y axis shows the bounds and the x -axis shows the number of links removed from the graph according to the above method. While β HS P < β S RP for very dense graphs, β S RP provides significant gains for the vast ma- jority of cases. For example, for the graph with 10 agents, and 24 links removed, and a fixed k = 1, β HS P implies that we must equip the agents with 512 resources to ensure that all resources are not exhausted before all 1-optimal actions are executed. However, β S RP indicates a that a 15-fold reduction to 34 resources will suf- fice, yielding a savings of 478 due to the use of graph structure when computing bounds. A fourth experiment compared β HS P and β S RP to the bound obtained by applying F CLIQUE , β FCLIQUE to DCOP graphs from the previous experiment. Selected results are shown in Figure 10 for graphs of 8 and 9 agents. While β FCLIQUE is marginally better for k = 1, β S RP has clear gains for k = 4. Identifying the relative e ff ec- tiveness of various algorithms that exploit our exclusivity relation sets is clearly an area for future work. Finally, Figure 11 compares the constant-time-computable graph- independent bounds from Section 3, in particular, showing the improvement of β MH over min { β H , β S , β P } for selected odd values of k , given three possible actions for each agent ( q = 3). The x axis shows I , the number of agents and the y -axis show s 100 · (min { β H , β S , β P } − β MH ) / min { β H , β S , β P } . For odd values of k > 1, as I increased, β provided a tighter bound on the number of k ...