H-DPOP vs NCBB: Search Space Comparison

This second set of experiments (H-DPOP vs NCBB) compares the search space explored and message exchanges in H-DPOP vs different versions of NCBB. For space comparisons we compare the logical sizes of the corresponding units (hypercubes in DPOP, CDDs in H-DPOP and total cache size in search with caching).

5.4.2.1 H-DPOP vs NCBB: N-Queens

We performed a comparison of search space using the number of cache entries in NCBB’s different versions and the number of util values in H-DPOP (excluding the size of CDD Tree for fair comparison) at each agent on the graph coloring problem. We selected a particular instance of queen graph (6×6 board,p=0.2, domain size=4, width=9). Our aim was to find a 4 coloring of the graph optimizing the costs assigned for color combinations. As the performance of any branch and bound search is cost dependent we generated 50 random instances of the same problem differing in the cost assignment to color combinations, each data-point is an average of 50 instances.

As stated in Section5.3.1.1, NCBB uses much smaller cache size for all agents (Figure5.8(a)). The reason is the non inclusion ofSepiassignments for which subtrees do not provide a solution within the bounds. The cache size inN CBB^∗is similar for most of the agents to the message size in H-DPOP.

There are a few cases (for agents 7,10,11,12,17,19,33) in whichN CBB^∗is better than H-DPOP.

The relevant part of the DFS tree (with depth 15) for this problem is shown in figure5.8(b), all nodes without any children are the leaves. A deeper look into the DFS arrangement suggests that all

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(a) Explored Search Space (b) DFS tree for NQueens

Figure 5.8:NQueen Problems: H-DPOP vs NCBB Search Space comparison

the nodes with size variations are in the lower part of the DFS tree. A search algorithm will have tighter upper bounds on the solution cost when it is expanding a high depth node, so it is natural that the effect of bounding is more pronounced for such nodes. On the contrary H-DPOP does not make use of any bounding, it prunes only the inconsistent combinations. Hence it takes more space at such nodes lower in the DFS.

An interesting result is that at the node with maximum size (Agent 6, with highest width=9) H-DPOP is much better (with size 216) as compared to cache size of 1094 inN CBB^∗. At high width regionsN CBB^∗does not provide good pruning (based only on bounding) however H-DPOP prunes many combinations based on consistency check. This is consistent with our previous results that at highly constrained regions H-DPOP provide very high pruning and almost negates the effect of in-creasing complexity.

Figure5.9compares NCBB and H-DPOP on the full range of6×6board size queen problems.

The problems are same as used in Section5.4.1.2for solving NQueen problem using graph coloring.

For each data point there are 20 randomly generated instances. As we can see from the explored search space graph (figure5.9(a)) bothN CBB^∗and H-DPOP explore nearly similar size search space.NCBB Hard Constraintexplores marginally larger search space thanN CBB^∗and as expected its search space size is very similar to H-DPOP since both H-DPOP andNCBB Hard Constraintdo pruning based only on hard constraints.

There is not a dramatic benefit of bounding on the search betweenN CBB^∗ andNCBB Hard Constraint. This further strengthens our claim that the major portion of pruning is attributed to the hard constraints which are exploited more efficiently by H-DPOP. One important advantage of H-DPOP is that it uses much less messages than NCBB orN CBB^∗(figure5.9(b)). Even for the simpler problems

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Figure 5.9:NQueen Problems (full range): H-DPOP vs NCBB comparison

(withp= 0.05),N CBBuses far more number of messages than H-DPOP which always has a constant message count (70). This advantage coupled with nearly equivalent explored search space make H-DPOP much superior to a branch and bound scheme like NCBB.

5.4.2.2 H-DPOP vs NCBB: Combinatorial Auctions

In this section we compare NCBB’s different versions and H-DPOP on two metrics: explored search space and messages exchanged. The comparisons are shown in figure5.10. The data set used is the same as in the previous section (H-DPOP vs DPOP on CA).

Notably explored search space is similar for bothN CBB^∗,NCBB Hard Constraintand H-DPOP for all bids. NCBB Originaluses smaller cache size as it does not caches all combinations. The difference betweenNCBB Hard ConstraintandNCBB Modifiedis again minimal suggesting that only the hard constraints play the vital role for pruning.

With respect to the message exchanges H-DPOP is much superior to all versions of NCBB on both paths and arbitrary distribution (figure5.10). There is a slight difference in the number of messages betweenNCBB Modifiedand NCBB original but it is small to be visible on graph. NCBB Modified uses less number of messages (by around 5%).

Interestingly on the arbitrary problems (figure5.10(d))NCBB Hard Constraintis slightly better than its other two counterparts in terms of message exchanges. We found out that this trend occurs becauseNCBB Hard Constraintbacktracks whenever it finds a single inconsistency in the partial solu-tion. However both NCBB original and NCBB Modified tolerate inconsistent solutions until they find a better one. Figuring out the upper bound (the cost of violating one hard constraint) on the consistent solution makes NCBB and NCBB Modified to exchange extra messages.

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Figure 5.10:Auctions: H-DPOP vs NCBB comparison

Once again in these set of experiments we have shown that explored search space is similar in both NCBB and H-DPOP, with H-DPOP requiring only a linear number of messages. Also the effect of bounding is negligible on pruning the search space as main pruning is provided by the hard constraints.