![](bg8.png)
E = 14 are Hamiltonian. Now the number of graphs
is equivalent to the number of options for placing the
E edges between V vertices:
1
/2 ·V · (V − 1)
E
(5)
which for V = 8 and E = 14, amounts to 40,116,600
graphs. Of these, approximately 61%, or 24,274,846
gaphs are Hamiltonian, the remaining 39%, or
15,841,754 graphs, are non-Hamiltonian. Summing
these results over all possible values of E for a given
V gives us the number (or percentage) of Hamiltonian
graphs in the entire edge-independent search space
(Table 3).
As it turns out, the number of Hamiltonian in-
stances ever more outweigh the number of non-
Hamiltonian instances as graphs get larger, possi-
bly making the state space harder to navigate for
our evolutionary algorithms, which have identical
numbers of evaluations for all V . This observation
might also account for the slightly deminishing re-
turns in both experiments for both algorithms as V
increases. But contrarily, these numbers do not ac-
count for graph isomorphism which might be quite
influential, but whose detection is a notorious prob-
lem in itself (McKay and Piperno, 2014). It is an
interesting and non-trivial question to see whether
other (meta)heuristic algorithms such as a properly
parameterized simulated annealing (Kirkpatrick et al.,
1983)(Dahmani et al., 2020) or genetic algorithms
(B
¨
ack et al., 1997) do better for this problem. It is also
plausible that metaheuristic parameter tuning and/or
control might set some serious sods to the dyke, as the
problem space clearly changes rapidly as V increases.
On a final note, these graphs might be difficult for
Vacul’s solving algorithm because its efficiency heav-
ily depends on pruning off edges that cannot be in
any Hamilton cycle, which only occurs when a vertex
is connected by two required edges. Because of the
compact structure of the wall-and-clique graph, this
will only happen near the full depth of the search tree,
when all but two vertices of the maximum clique are
included in a partial solution. But just the ubiquity of
pruning techniques throughout history doesn’t spell
much good for other exact algorithms either when
it comes to these graphs. The non-Hamiltonian in-
stances in this study might thereby actually be the
hardest around, but more evidence, or perhaps even
a proof, is needed to soldify this conjecture.
ACKNOWLEDGEMENTS
ECTA’s reviewers
7
really made an effort to under-
stand this somewhat pioneering approach to instance
hardness. A special hats off goes to Reviewer #4, who
took the time to hand-correct a host of typo’s and sup-
ply an annotated pdf along with the report. Thank you
all, you have done this paper a great favour.
REFERENCES
Aguirre, A. S. M. and Vardi, M. (2001). Random 3-sat and
bdds: The plot thickens further. In International Con-
ference on Principles and Practice of Constraint Pro-
gramming, pages 121–136. Springer.
B
¨
ack, T., Fogel, D. B., and Michalewicz, Z. (1997). Hand-
book of evolutionary computation. Release, 97(1):B1.
Br
´
elaz, D. (1979). New methods to color the vertices of a
graph. Communications of the ACM, 22(4):251–256.
Cheeseman, P., Kanefsky, B., and Taylor, W. M. (1991).
Where the really hard problems are. In Proceedings of
the 12th International Joint Conference on Artificial
Intelligence - Volume 1, IJCAI’91, pages 331–337,
San Francisco, CA, USA. Morgan Kaufmann Publish-
ers Inc.
Coarfa, C., Demopoulos, D. D., Aguirre, A. S. M., Sub-
ramanian, D., and Vardi, M. Y. (2000). Random 3-
sat: The plot thickens. In International Conference on
Principles and Practice of Constraint Programming,
pages 143–159. Springer.
Cook, S. A. (1971). The complexity of theorem-proving
procedures. In Proceedings of the Third Annual
ACM Symposium on Theory of Computing, STOC ’71,
pages 151–158, New York, NY, USA. ACM.
Culberson, J. C. and Vandegriend, B. (2011). The gn,m
phase transition is not hard for the hamiltonian cycle
problem. CoRR, abs/1105.5443.
Dahmani, R., Boogmans, S., Meijs, A., and van den Berg,
D. (2020). Paintings-from-polygons: simulated an-
nealing. In International Conference on Computa-
tional Creativity (ICCC’20).
de Jonge, M. and van den Berg, D. (2020). Plant propa-
gation parameterization: Offspring & population size.
Evo* 2020, page 19.
Garey, M. R. and Johnson, D. S. (1990). Computers
and Intractability; A Guide to the Theory of NP-
Completeness. W. H. Freeman & Co., New York, NY,
USA.
Geleijn, R., van der Meer, M., van der Post, Q., and van den
Berg, D. (2019). The plant propagation algorithm on
timetables: First results. EVO* 2019, page 2.
Held, M. and Karp, R. M. (1962). A dynamic program-
ming approach to sequencing problems. Journal of
the Society for Industrial and Applied mathematics,
10(1):196–210.
7
ECTA 2020 is part of the larger conference IJCCI 2020,
see http://www.ecta.ijcci.org/.
Looking for the Hardest Hamiltonian Cycle Problem Instances
47