DEMOSTRACIÓN DE HIPÓTESIS

Next, we describe the approach that resolves multiple conflicts. As stated in Chapter 2, a valid relaxation must resolve all conflicts in an inconsistent temporal problem.

Given multiple minimal conflicts, the relaxation must suspend at least one constraint in each conflict, which makes it a covering set of all minimal conflicts. Therefore, a discrete minimal relaxation can be defined as the minimal covering set of all minimal conflicts in an inconsistent problem [10, 4].

However, it is difficult to accurately identify all minimal conflicts in a problem prior to the enumeration. Detecting all the minimal conflicts requires a lot of computation:

for example, in the case of an OCSTN, the algorithm goes through each negative cycle in each component STN. A better approach is to use an incremental search strategy that constructs candidate relaxations based on known conflicts, and updates the candidates when new conflicts are detected. This procedure includes the following steps:

• Generate candidate temporal relaxations based on known conflicts.

• If the candidate is consistent, return as a valid minimal relaxation and move to the next candidate. This is different from CD-A*, which continues extending the search tree. In addition, this minimal relaxation is added to the collection of known conflicts so that the same relaxation will not appear again.

• If the candidate is inconsistent, extract a new minimal conflict from it and

update existing candidates using the conflict (FunctionUpdateCandidates).

Through the update procedure, we can guarantee that all candidates can resolve at least known conflicts.

• If all candidates are tested to be consistent, terminate the enumeration. It indi-cates that no more conflicts can be discovered, and hence all minimal relaxations have been found.

This is very similar to an incremental repair procedure: a candidate is repeatedly improved by newly discovered conflicts until it makes the over-constrained problem consistent. CD-A* focuses minimal covering by performing expansion in best first order. In BCDR, an incremental set covering algorithm is used to generate candidate minimal relaxation sets using sequentially discovered minimal conflicts (Algorithm 2).

input : NewMinCFLT, newly discovered minimal conflict

input : PrevCandidates, previous candidate minimal relaxation sets output: UpdatedCandidates, updated candidate minimal relaxation sets // Generate constituent relaxations of the minimal conflict

1 ConstituentRelaxations← ResolveConflict(NewMinCFLT);

// Start with the cross product of previous candidates and the new conflict.

2 UpdatedCandidates← PrevCandidates ⊗ ConstituentRelaxations;

// Remove redundant (non-minimal) candidates.

3 UpdatedCandidates←

RemoveRedundantCandidates(UpdatedCandidates);

4 return UpdatedCandidates

Algorithm 2: UpdateCandidates

For example, assume that BCDR tests the first candidate and gets a minimal conflict (highlighted arcs in Figure 3-7).

This minimal conflict contains four temporal constraints, ’Drive Office Cosi’, ’dine-in Cosi’, ’Drive Cosi Home’, ’TimeConstra’dine-int’, and two decisions, ’LeaveOffice:Drive Office Cosi’, ’ArriveDD:dine-in Cosi’. To resolve it, we can suspend any one temporal

Start

Leave Office

End [0min,80min]

Arrive Cosi (Drive Office Cosi)

[40min,50min]

Arrive Quiznos (Drive Office Quiznos)

[30min,40min]

Arrive Subway (Drive Office Subway)

[25min,35min]

Leave Cosi (Dine-in Cosi)

[15min,15min]

(Take-out Cosi) [5min,5min]

Arrive Home (Drive Cosi Home)

[30min,35min]

Leave Quiznos (Dine-in Quiznos)

[25min,25min]

(Take-out Quiznos) [10min,10min]

(Drive Quiznos Home) [35min,50min]

Leave Subway (Dine-in Subway)

[35min,35min]

(Take-out Subway) [10min,10min]

(Drive Subway Home) [30min,35min]

Figure 3-7: A minimal conflict in John’s trip plan

constraint or switch one of the decisions. In total, five candidate minimal relaxations can be generated:

Suspend: ’dine-in Cosi’ or ’TimeConstraint’

Switch

choice: ’ArriveDD:take-out Cosi’ or ’LeaveOffice:Drive Office Quiznos’

or ’LeaveOffice:Drive Office Subway’.

Next, assuming that BCDR takes the last candidate, it will expand the candi-date to ’LeaveOffice:Drive Office Subway ArriveSubway:Take-out Subway’. The con-sistency check will then indicate that the candidate is still inconsistent and a new minimal conflict can be extracted (Figure 3-8).

Start

Leave Office

End [0min,80min]

Arrive Cosi (Drive Office Cosi)

[40min,50min]

Arrive Quiznos (Drive Office Quiznos)

[30min,40min]

Arrive Subway (Drive Office Subway)

[25min,35min]

Leave Cosi (Dine-in Cosi)

[15min,15min]

(Take-out Cosi) [5min,5min]

Arrive Home (Drive Cosi Home)

[30min,35min]

Leave Quiznos (Dine-in Quiznos)

[25min,25min]

(Take-out Quiznos) [10min,10min]

(Drive Quiznos Home) [35min,50min]

Leave Subway (Dine-in Subway)

[35min,35min]

(Take-out Subway) [10min,10min]

(Drive Subway Home) [30min,35min]

Figure 3-8: Another minimal conflict in John’s trip plan

With the discovery of this new minimal conflict, BCDR then updates the current list of candidate temporal relaxations by removing ineffective candidates and adding

new ones. It computes the minimal covering sets of all minimal conflicts found so far, making sure that each candidate resolves all of them. If one candidate has conflict-ing decisions, for example, ’LeaveOffice:Drive Office Subway’ and ’LeaveOffice:Drive Office Quiznos’, the candidate will be removed from the list, too.

Suspend: ’TimeConstraint’ or ’dine-in Subway’ or ’dine-in Cosi’

Assign: ’LeaveOffice:Drive Office Quiznos’ or ’ArriveSubway:take-out Subway’

or’ArriveCosi:take-out Cosi’

Note that this process is different from iterative repair algorithms, which are stochastic in that they cannot guarantee the completeness and optimality of the re-sults. Function UpdateCandidates guarantees that BCDR finds all minimal con-flicts and corresponding minimal temporal relaxations to an inconsistent OCSTN, given enough iterations. It terminates when all the candidates in CANDs are con-sistent. It indicates that all minimal conflicts in the inconsistent problem have been detected, and no more minimal relaxation can be generated.

Finally, we prove the completeness and soundness of BCDR in resolving over-constrained OCSTNs.

Theorem 3. [Completeness of BCDR] BCDR can find all minimal relaxations given an over-constrained OCSTN.

Proof. [Proof by contradiction]

Let MR be a minimal relaxation to an OCSTN P, and BCDR fails to generate it.

• Since the incremental set covering method used by BCDR is complete, if MR is not generated by BCDR, at least one minimal conflict, MinCF LT, has not been detected by BCDR yet. Otherwise all minimal relaxation sets toP must have been generated.

• IfMinCF LT is unknown to BCDR, some of the candidate relaxations generated by BCDR must be inconsistent when BCDR terminates, since not all of the candidate relaxations can resolve the unknownMinCF LT.

• However, this contradicts with the termination condition of BCDR: BCDR will only terminate if all candidates are minimal relaxations that resolve all conflicts.

Hence the assumption is faulty. BCDR is complete.

Theorem 4. [Soundness of BCDR] All the results generated by BCDR are minimal relaxations that resolve all conflicts in an OCSTN.

Proof. A minimal relaxation generated by BCDR is correct in two aspects:

1. Must be a valid relaxation: since all relaxations generated by BCDR must pass consistency check before being returned to the user, they have to resolve all the conflicts in the over-constrained OCSTN.

2. Must be minimal: this is guaranteed by the minimal set covering process. All candidates generated are the minimal covering sets of known conflicts.

Hence all minimal relaxations returned by BCDR are correct, that is, areminimal and resolve allconflicts.

In summary, we presented the candidate generation method used in BCDR. Given an inconsistent OCSTN, the method is guaranteed to find all discrete minimal relax-ations that resolve all its conflicts. We make use of the property of minimal conflicts, for which a conflict can be resolved by suspending any one constraint in it. We then compute candidate relaxations through a minimal set covering process. In addition, to improve the performance in real world applications, we developed the incremental approach UpdateCandidates that generates candidate based on known conflicts, then makes updates when new conflicts are detected. Therefore, the first minimal relaxation will be generated as early as possible, and even prior to the discovery of all conflicts.