By Albert Nijenhuis, Herbert S. Wilf
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Additional info for Combinatorial Algorithms for Computers and Calculators, Second Edition (Computer Science and Applied Mathematics)
If such a path is eligible up to infinite depth, then it is easy to see that MOA ** will not terminate. Taking all these aspects into account, the general conditions of admissibility of MOA** may be stated as follows. Admissibility conditions of MOA ** MOA** terminates with all non-dominated solutions iff: 1. There is no infinite path which is eligible up to infinite depth, and 2. Every non-dominated solution path is eligible, and 3. The pmax-ordering of the solution paths describe the same sequence as the representative cost vector of the paths based on K-ordering.
There is no infinite path which is eligible up to infinite depth, and 2. Every non-dominated solution path is eligible, and 3. The pmax-ordering of the solution paths describe the same sequence as the representative cost vector of the paths based on K-ordering. The third condition may be relaxed by modifying step 5 of MOA ** as follows: 5. 1 Put n in SOLUTION_GOALS and its cost in SOLUTION_COSTS. 2 Remove dominated solutions (if any) from SOLUTION_COSTS. 3 GoTo [Step 2]. 2 becomes necessary because if the third condition is relaxed then the solution nodes may not arrive in the K-ordered sequence of their cost vectors and it is possible that some solution node entered in SOLUTION_GOALS is dominated by the cost vector of some solution found later.
Proof: We prove the sufficiency condition first. If a node n and all its ancestors have one or 30 3 Multiobjective State Space Search more non-dominated cost ,vectors then step 2 of MOA" shows that the node is a candidate for expansion. However, if every non-dominated cost vector of n (or of one of its ancestors) equals the cost vector of other solution paths, then it is possible that those solution paths are found earlier and n is never expanded. Otherwise, it is easy to see that the node n will be expanded by MOA··.