![](bg8.png)
verifying whether or not there exists a feasible
process selected from a nested workflow with
additional constraints is NP-complete. We used
workflows modeling 3SAT and subset-sum
problems in the proofs and these workflows have a
structure that is not typical for real-life workflows.
Hence, there is a hope that real-life workflows can
still be verified in a reasonable time in practice. The
next step is to find such a procedure that can do
verification of nested workflows with additional
constraints efficiently in practice. Note that this
verification procedure should work with constraints
of all types mentioned in the paper together.
Figure 9: A tree representation of nested workflows.
Some of the extra constraints can be verified
easily. Notice that a Nested TNA can be represented
as a tree showing how the root task is decomposed
into sub-tasks and so on until activities are obtained
in the leafs (Figure 9). There are basically two
different locations where the binary custom
constraint can be placed in this tree. Either the
constraint connects two tasks on the same
path/branch to the root task (for example the
constraint between tasks A and B in Figure 9) or the
constraint connects the tasks from different sub-trees
with a common ancestor task (for example the
constraint between tasks C and D in Figure 9). The
constraints along the path to the root are those
constraints that are easy to verify as they are
frequently redundant (entailed by the workflow
constraints) or inconsistent. The constraints between
tasks from different sub-trees are easy to verify if
their common ancestor (task E in Figure 9)
decomposes alternatively. The only situation which
makes verification non-trivial is when task E is a
serial or parallel decomposition as in Figure 9. This
is exactly the case of the nested workflows used in
the proofs of NP-completeness in this paper. For
these cases a straightforward approach is using a
search procedure that finds for each activity a valid
process containing this activity. If no valid process is
found, the activity is reported to the user as a
problematic activity. In such a case, it is not always
clear which custom constraints cause the problem.
Providing the most accurate explanation is an
interesting open problem.
ACKNOWLEDGEMENTS
The research is supported by the Czech Science
Foundation under the contract P202/10/1188. The
author would like to thank Vladimír Rovenský for
implementing the FlowOpt Workflow Editor;
screenshots from this software are used in the paper.
REFERENCES
Bae, J., Bae, H., Kang, S.-H., Kim, Z., 2004. Automatic
Control of Workflow Processes Using ECA Rules.
IEEE Transactions on Knowledge and Data Enginee-
ring, vol. 16, no. 8, pp. 1010-1023.
Barták, R., Čepek, O., 2007. Temporal Networks with
Alternatives: Complexity and Model. In Proceedings
of the Twentieth International Florida AI Research So-
ciety Conference (FLAIRS), AAAI Press, pp. 641–646.
Barták, R., Čepek, O., 2008. Nested Temporal Networks
with Alternatives: Recognition, Tractability, and
Models. In Artificial Intelligence: Methodology, Sys-
tems, and Applications (AIMSA 2008), LNAI 5253,
Springer Verlag, pp. 235-246.
Barták, R., Čepek, O., Hejna, M., 2008. Temporal
Reasoning in Nested Temporal Networks with Alter-
natives. In Recent Advances in Constraints, LNAI
5129, Springer Verlag, pp. 17-31.
Barták, R., Cully, M., Jaška, M., Novák, L., Rovenský, V.,
Sheahan, C., Skalický, T., Thanh-Tung, D., 2011.
FlowOpt: A Set of Tools for Modeling, Optimizing,
Analyzing, and Visualizing Production Workflows. In
Proceedings of the ICAPS 2011 System Demonstra-
tions, pp. 6-9.
Beck, J. Ch., Fox, M. S., 2000. Constraint-directed
techniques for scheduling alternative activities. Arti-
ficial Intelligence, vol. 121, pp. 211-250.
Bi, H. H., Zhao, J. L., 2004. Applying Propositional Logic
to Workflow Verification. Information Technology
and Management, vol. 5, no 3-4, pp. 293-318.
Garey, M. R., Johnson, D. S., 1979. Computers and Intra-
ctability: A Guide to the Theory of NP-Completeness,
W. H. Freeman and Company.
Giro, S., 2007. Workflow Verification: A New Tower of
Babel. In AIS-CMS International Modeling and Simu-
lation Multiconference, Buenos Aires, Argentina.
Sadiq, W., Orlowska, M. E., 2000. Analyzing Process
Models using Graph Reduction Techniques.
Information Systems, vol. 25, no. 2, pp. 117–134.
A
B
C
D
E
ON COMPLEXITY OF VERIFYING NESTED WORKFLOWS WITH EXTRA CONSTRAINTS
353