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7 DISCUSSION AND OPEN
PROBLEMS
In this paper, we have studied OPC-CDS in general
graphs. Connected Dominating Set problems have
been extensively studied in the offline setting. These
were motivated by real-world network applications in
which geometric graph models were used (see (Mah-
dian and Yan, 2011; Amb
¨
uhl et al., 2006)). One re-
search direction is to extend this study to the online
setting by considering such graphs for OPC-CDS and
its variants. This might yield to competitive ratios
dependent on the properties of the geometric graphs
rather than the number of nodes.
Another set of open problems would generate by
considering other online models. We have made our
study based on the oblivious adversary model. It is
interesting to consider other weaker adversary models
such as stochastic as in (Manshadi et al., 2010), or
random as in (Mahdian and Yan, 2011).
Competitive analysis offers a worst case perfor-
mance evaluation of online algorithms. It would be
interesting to also consider other models. (Cheung,
2016) performed a computational study of various on-
line algorithms for Steiner problems to reveal their
performance in average. It would be interesting to do
the same for our proposed algorithms, as in (Hamann
et al., 2018)), for the Online Connected Dominating
Set problem. Another study by (Angelopoulos, 2019)
based the analysis of online algorithms on additional
parameters of the problem, known as parameterized
analysis of online algorithms. For the Online Node-
weighted Steiner Tree problem, he showed a tight
competitive ratio that depends on the maximum node
weight, minimum node weight, and the number of ter-
minals. Investigating parameterized analysis for our
problems would initiate an interesting study.
REFERENCES
Alon, N., Awerbuch, B., and Azar, Y. (2003). The on-
line set cover problem. In Proceedings of the Thirty-
fifth Annual ACM Symposium on Theory of Comput-
ing, STOC ’03, pages 100–105, New York, NY, USA.
ACM.
Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., and
Naor, J. S. (2006). A general approach to online net-
work optimization problems. ACM Trans. Algorithms,
2(4):640–660.
Amb
¨
uhl, C., Erlebach, T., Mihal
´
ak, M., and Nunkesser, M.
(2006). Constant-factor approximation for minimum-
weight (connected) dominating sets in unit disk
graphs. In Approximation, Randomization, and Com-
binatorial Optimization. Algorithms and Techniques,
pages 3–14. Springer.
Angelopoulos, S. (2019). Parameterized analysis of the on-
line priority and node-weighted steiner tree problems.
Theory Comput. Syst., 63(6):1413–1447.
Ausiello, G., Bonifaci, V., and Laura, L. (2008). The online
prize-collecting traveling salesman problem. Informa-
tion Processing Letters, 107(6):199 – 204.
Cheung, S. S. (2016). Offline and online facility location
and network design. PhD thesis, Operations Research
and Information Engineering, Cornell University.
Demange, M. and Paschos, V. T. (2005). Online vertex-
covering. Theoretical Computer Science, 332(1):83 –
108.
Felice, M. C. S., Cheung, S.-S., Lee, O., and Williamson,
D. P. (2015). The online prize-collecting facility loca-
tion problem. Electronic Notes in Discrete Mathemat-
ics, 50:151 – 156. LAGOS’15 – VIII Latin-American
Algorithms, Graphs and Optimization Symposium.
Fotakis, D. (2007). A primal-dual algorithm for online non-
uniform facility location. Journal of Discrete Algo-
rithms, 5(1):141 – 148.
Fotakis, D. (2008). On the competitive ratio for online fa-
cility location. Algorithmica, 50(1):1–57.
Guha, S. and Khuller, S. (1998). Approximation algo-
rithms for connected dominating sets. Algorithmica,
20(4):374–387.
Hajiaghayi, M., Liaghat, V., and Panigrahi, D. (2014). Near-
optimal online algorithms for prize-collecting Steiner
problems. In Esparza, J., Fraigniaud, P., Husfeldt, T.,
and Koutsoupias, E., editors, Automata, Languages,
and Programming, pages 576–587, Berlin, Heidel-
berg. Springer Berlin Heidelberg.
Hajiaghayi, M. T., Liaghat, V., and Panigrahi, D. (2013).
Online node-weighted Steiner forest and extensions
via disk paintings. In 54th Annual IEEE Symposium
on Foundations of Computer Science, FOCS 2013, 26-
29 October, 2013, Berkeley, CA, USA, pages 558–567.
Hamann, H., Markarian, C., Meyer auf der Heide, F., and
Wahby, M. (2018). Pick, pack, & survive: Charging
robots in a modern warehouse based on online con-
nected dominating sets. In 9th International Confer-
ence on Fun with Algorithms, FUN 2018, June 13-15,
2018, La Maddalena, Italy, pages 22:1–22:13.
JunFeng, D. and JianHua, T. (2014). A factor 2-
approximation algorithm for the prize-collecting ver-
tex cover problem. Journal of Beijing University
of Chemical Technology (Natural Science Edition),
41(2):120.
Korman, S. (2005). On the use of randomization in the on-
line set cover problem. Master’s thesis, Weizmann In-
stitute of Science, Israel.
Mahdian, M. and Yan, Q. (2011). Online bipartite matching
with random arrivals: An approach based on strongly
factor-revealing lps. In Proceedings of the Forty-Third
Annual ACM Symposium on Theory of Computing,
STOC ?11, page 597?606, New York, NY, USA. As-
sociation for Computing Machinery.
Manshadi, V. H., Gharan, S. O., and Saberi, A. (2010). On-
line stochastic matching: online actions based on of-
fline statistics. Math. Oper. Res., 37:559–573.
Markarian, C. (2018). An optimal algorithm for online
prize-collecting node-weighted steiner forest. In Com-
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