DPOP vs H-DPOP: Message Size

These experiments mainly focus on the space savings provided by H-DPOP by pruning the search space. We have performed 4 sets of experiments: query placement problems, graph coloring problems, n-queens problems, and combinatorial auctions problems.

5.4.1.1 Optimal query placement in an overlay network

For experiments the problem is made deliberately very constrained by assuming that each server is able to execute the complete set of services. For simplicity’s sake, each server can execute only a single service at a time. The objective of the DCOP algorithm is to maximize the overall utility.

We generated random problems of different sizes, with a random number of soft constraints among variables. All-different hard constraints are introduced thus making the constraint graph fully con-nected. The size of the hypercube is the number of entries in the hypercube, the size of the CDD Message is the number of entries in the Util array combined with the logical size of CDDTree (each en-try in the CDDNode corresponds to 1 unit in the space measurement, links to children are also counted as 1 unit).

Figure5.5(a)shows the maximal/total message size in H-DPOP versus DPOP. Problem size is denoted bym∗nimplyingmvariables each having the same domain of sizen. Results show that H-DPOP is much superior to DPOP for all problem sizes, culminating with the largest problems (9 servers × 9 services) where H-DPOP produces 3 orders of magnitude smaller messages and smaller total message.

Figure5.5(b)shows the effect of problem size on space savings provided by CDDs. We count the percentage of unfeasible assignments carried in the UTIL messages, and we plot this as “wasted space in DPOP”. We see that the space wasted by DPOP is above 90%, and for larger and more difficult problems, close to 100%. In contrast, CDDs enable H-DPOP to avoid this problem. Even though CDDs introduce the overhead of representing the CDDTree explicitely, overall the space savings they provide by not sending infeasible combinations more than compensates: savings start around 48% for small problems (5*5), and increase with problem size, up to 99% for 9*9 problems.

5.4.1.2 Random Graph Coloring Problems

We performed experiments on randomly generated distributed graph coloring problems. In our setup each node in the graph is assigned an agent (or a variable in DCOP terms). The constraints among agents define the cost of having a particular color combination. The cost of two neighboring agents choosing the same color is kept very high (10000) to disallow such combinations. Thedomainof each agent is the set of available colors. The mutual task of all the agents is to find an optimal coloring

100

Number of servers * Number of services Max Message Size (Hypercube Vs CDD)

Space savings of H-DPOP vs DPOP Space wasted by DPOP

(b) Space Savings increase with problem size

Figure 5.5:Query placement problems: H-DPOP vs DPOP performance

assignment to their respective nodes.

For generating these graphs we have two parameters-number of agents and the constraint density.

We keep the number of agents fixed to 10. We start with a fully connected graph and remove the edges successively until we reach the desired constraint density and the problem is still connected.

Figure5.6shows the results on a 10 nodes randomly generated problem for a range of constraint densities (0.2-0.89). The problems within densities 0.2-0.5 were 4-colorable (implying domain size 4).

The problems from 0.5-0.9 were 6-colorable. For statistically sound results for each constraint density we generated 50 random problems and the results shown are the average of 50 runs.

Figure5.6(a)shows the full spectrum of performance of H-DPOP vs DPOP in terms of the max-imum/total message size. For accounting the message size we take into account the number of util values in the hypercube for DPOP, for the H-DPOP we count the length of the UTIL array and the (logical) size of the CDD tree in the CDD Message. As can be seen, H-DPOP is better for most of the regions (density 0.4-0.89) except for densities from 0.2-0.4. To understand the characteristics of H-DPOP we divide the densities into three regions- low density (0.2-0.4), medium density (0.4-0.7) and high density (0.7-0.9).

For thelow densityregion (Figure5.6(b)) DPOP performs better than H-DPOP. The explanation is the same as in Section5.4.1.1: at low density the size of the hypercube is small. CDDs at low density do not provide sufficient pruning to overcome the overhead introduced by the size of the CDDTree in the CDDMessage.

For themedium densityregion (Figure5.6(c)) H-DPOP is much better than DPOP. The sizes of both hypercubes and CDDMessages increase with density. This is the expected behavior as with the

1

Figure 5.6:Graph Coloring: H-DPOP vs DPOP performance

ing constraint density the width increases leading to exponential increase in message size. However with CDDs we still get much space savings.

Thehigh densityregion (Figure5.6(d)) provides interesting results for the H-DPOP. In DPOP, as expected, the maximum message size increases with the density. However we see an opposite trend in H-DPOP, instead of CDDMessage size increasing with the density, it starts decreasing. The reason is that at high constraint density the extent of pruning done by CDDs is also very high. So although the problem becomes more complex with high connectivity, the increased pruning by the CDDs overcome this increase and at very high densities the pruning dominates the increase in problem complexity.

5.4.1.3 NQueens problems using graph coloring

For ann×nchessboard a queen graph containsn²nodes, each corresponding to a square of the board.

Two nodes are connected by an edge if the corresponding squares are in the same row, column, or diagonal. The intuition behind using graph coloring on a queen graph is that we can placensets ofn queens on the board so that no two queens of the same set attack each other if the chromatic number of the graph is at leastn.

For our experiments we took the problems from Stanford Graphbase ( [113]). For a 5-colorable 5×5queen graph (width 19, 25 agents, density 0.53) DPOP was unable to execute (maximum message size19073486328125). H-DPOP successfully executed in 15 seconds with a maximum message size of9465, achieved through the high pruning power of the CDDs.

However, for board sizes6×6(7 colorable with width 31, density 0.46) and above H-DPOP was also unable to execute due to increased width and domain size. Relaxing a highly constrained problem is a well known technique in CSP literature. We adopt this technique into generating queen graphs so that the inclusion of any edge in the graph is done with a probabilityp. If this probability is 1 we get the complete queen graph.

We experimented by varying this probability from 0.05 to 0.25 for6×6board with the resulting graphs being 4-colorable. For each datapoint we took the average of 50 randomly generated problems.

Graph remains 4-colorable untilp=0.25 and increasing thepbeyond increases the coloring number.

A direct implication of this fact is that atp=0.25 the graph is highly constrainedwith respect tothe coloring number 4. This observation lead us to believe that the nature of H-DPOP and DPOP should be similar to the random coloring experiments.

Figure5.6(e)shows the result for maximum and total message size against the probabilityp. The dotted vertical line at x=0.14 divides the graph into two regions. For the first region both H-DPOP and DPOP increase in the maximum message size. However as the density (which is directly related to p) increases we see the same trend as in random problems. DPOP continues to increase in maximum size but the size in H-DPOP remains constant (p∈[0.14, 0.20]), and it starts decreasing in the region

100

Figure 5.7:Combinatorial Auctions: H-DPOP vs DPOP comparison

[0.20,0.25].

5.4.1.4 Winner Determination in Combinatorial Auctions

Combinatorial Auctions (CA) provide an efficient means to allocate resources to multiple agents. In CA bidders can bid on a bundle of goods in addition to single item bidding. This provides for comple-mentarity and substitutability among the goods. In our experimental setting there is a single seller and multiple buyers (agents). The agents are distributed (geographically or logically) and have information aboutonly thoseagents with whom their bids overlap. The mutual task of agents is to find a solution (assign winning or losing to bids) which maximizes the seller’s revenue providing a feasible solution (no overlap among winning bids).

In our formulation we do search through the constraint network of bids presented (rather than con-sidering all possible bids). Such a formulation has been shown to be very effective in CABOB [185, 183] and BOB [184]. However we do not intend to compare with these approaches as they are both centralized and use linear programming to augment the search method (not feasible in distributed set-ting).

Thevariablesin our setting are thebidspresented by the agents. Each agent is responsible for the bid it presents. Thedomainof each variable is the set{wining, losing}. Hard constraints are for-mulated between bids sharing one or more goods, disallowing several of them to be assignedwinning.

The value of each bid is modeled as an unary constraint on the associated variable.

We generated random problems using CATS (Combinatorial Auctions Test Suite [122]) using the pathsandArbitrarydistributions. For the paths distribution the number of bids was varied for a fixed number of goods (100). Each agent is allowed to present only one bid. Inpathsdistribution goods are

the edges in the network of cities. Agents place bids on a path from one city to other based on their utilities. In our setting we fixed the number of cities to 100 with initial connection 2 (link density).

Since the city network structure is fixed, as the number of bids increases we expect a higher number of bids to overlap with each other and increase the problem complexity. For theArbitrarydistribution we use all the default CATS parameters. The number of goods is 50, and the number of bids varies from 25 to 50 increasing the complexity of the problem. Each data point is obtained as the average of 20 instances.

Figure5.7(a)shows a comparison of DPOP with H-DPOP (average of 20 problems for each data-point) onpathsdistribution. DPOP as expected increases in message complexity with the number of bids. The pruning provided by H-DPOP is very high (around 99% of hypercubes) and increases with number of bids. Because of such high pruning H-DPOP runs on problms with very high width (35, bids=70) where memory requirements for DPOP are prohibitively expensive. We see a similar trend for thearbitrarydistribution (figure5.7(b)). H-DPOP is much superior to DPOP and provides very high pruning.