The Treewidth of Induced Graphs of Conditional Preference Networks Is Small
Abstract
:1. Introduction
- 1
- We explore the treewidth problem about the induced graphs of CP-nets— as far as we know, there is no literature availlable to study the treewidth characterization of induced graphs of CP-nets.
- 2
- We design a more efficient algorithm to solve the treewidth of induced graphs of CP-nets utilizing a Bucket Elimination approach; this approach uses the randomness characteristics of input order to speed search.
- 3
- We find that the treewidth of induced graphs of CP-nets is very small; this interesting discovery may lay the foundation for designing an algorithm for solving tractable reasoning tasks, such as dominance queries, in the future.
2. Background on Treewidth and CP-nets
2.1. Treewidth
- (i)
- . That is, each graph vertex is contained in at least one tree node.
- (ii)
- If vertices v and w both are connected in a graph G, then v and w are contained in at least one subset .
- (iii)
- Considering the three nodes in the tree, when j exists in the path of i and k,, that is, common vertices of i and k must appear in the j.
2.2. CP-nets
3. Related Work
4. Treewidth of the Induced Graphs of CP-nets
4.1. Bucket Elimination Technique
4.2. Algorithm for the Treewidth of an Induced Graph
Algorithm 1: solving treewidth of included graphs of CP-nets N′. | |
Input: An adjacency matrix A[i, j] of induced graph | |
Output: The treewidth of N′ | |
1 | treewidth ← 0; |
2 | Mdec ← 0; |
3 | mdec ← 0; |
// Calculate the total number of vertex elimination order sum. | |
|
4.3. A Motivate Example
π | width | |
4.4. Time Complexity Analysis
5. Experimental Evaluation
5.1. Experimental Environment
5.2. Experimental Results on Simulated Data
n | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 |
|E| | 12 | 32 | 80 | 192 | 448 | 1024 | 2304 | 5120 |
tw | 1 | 2 | 6 | 7 | 8 | 9 | 9 | 10 |
time(s) | 0.016 | 0.031 | 0.094 | 0.313 | 1.103 | 4.064 | 15.102 | 61.057 |
V | n | ||
---|---|---|---|
2 | 4 | 4 | 6 |
3 | 8 | 12 | 28 |
4 | 16 | 32 | 128 |
5 | 32 | 80 | 496 |
6 | 64 | 192 | 2016 |
7 | 128 | 448 | 8128 |
6. Conclusions and Future Work
Acknowledgments
Author Contributions
Conflicts of Interest
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Liu, J.; Liu, J. The Treewidth of Induced Graphs of Conditional Preference Networks Is Small. Information 2016, 7, 5. https://doi.org/10.3390/info7010005
Liu J, Liu J. The Treewidth of Induced Graphs of Conditional Preference Networks Is Small. Information. 2016; 7(1):5. https://doi.org/10.3390/info7010005
Chicago/Turabian StyleLiu, Jie, and Jinglei Liu. 2016. "The Treewidth of Induced Graphs of Conditional Preference Networks Is Small" Information 7, no. 1: 5. https://doi.org/10.3390/info7010005