purchase any of the edges in K. To see why this holds,
we let e be an edge in K. All the edges in every path
containing e must have a fraction equal to at least the
fraction of e (Max ﬂow - Min Cut theorem). So if e
is purchased, i.e., e > α, then all the other edges on
the paths containing e are purchased too, since they
have a higher fraction than e. Hence, the probability
is equal to:
∏
e∈K
(1 − f
e
) ≤ e
−
∑
e∈K
f
e
≤ 1/e
The second inequality holds since the algorithm en-
sures that
∑
e∈K
f
e
≥ 1 at this point. Thereby, the
expected cost of purchasing a path in the third step,
for all 1 ≤ i ≤ 2
d
log(kn + 1)
e
, would be less than
1/(kn)
2
· Opt, where the optimal solution cost Opt
can be used as an upper bound for the minimum-
weight path constructed by the algorithm in the third
step.
The total number of pairs the algorithm receives
is at most kn, since each of the n requests can ask for
at most k services. Summing up over all these pairs,
we conclude the expected cost of the algorithm in the
third step:
Cost
E
00
≤ 1/(kn) · Opt (4)
By combining Equations 3 and 4, we conclude the
following.
Theorem 2. (Upper Bound). There is an online ran-
domized algorithm for the non-metric Online Facility
Location with Service Installation Costs (OFL-SIC),
that has an asymptotically optimal competitive ratio
of O (log(nk)log m), where m is the number of facili-
ties, n is the number of requests, and k is the number
of services.
7 CONCLUDING REMARKS &
FUTURE WORK
In this paper, we have studied the non-metric On-
line Facility Location with Service Installation Costs
problem (OFL-SIC), which could also be called the
non-metric Online Multi-Commodity Facility Loca-
tion with linear costs problem (non-metric OMCFL
with linear costs). A next step would be to consider
the non-metric Online Multi-Commodity Facility Lo-
cation problem (OMCFL) for other facility cost func-
tions, such as the cost functions deﬁned for the metric
case in (Castenow et al., 2020). It seems like other
techniques than the ones used in this paper would be
needed to achieve results for these cost functions.
Moreover, unlike in the ofﬂine setting, for the gen-
eral facility cost function, there are no online algo-
rithms in the literature for both the metric and non-
metric cases. So there is a lot to investigate in this
direction.
Another direction is to assume facilities with ca-
pacities, for both the metric and non-metric variants.
This would reﬂect a more natural real-world facil-
ity location problem, in which the number of clients
served by each facility is limited by the resources
available at the facility (Cygan et al., 2018).
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