Online Algorithms for Leasing Vertex Cover and Leasing Non-metric
Facility Location
Christine Markarian
1
and Friedhelm Meyer auf der Heide
2
1
Department of Mathematical Sciences, Haigazian University, Beirut, Lebanon
2
Heinz Nixdorf Institute and Computer Science Department, University of Paderborn, Paderborn, Germany
Keywords:
Online Algorithms, Competitive Analysis, Leasing, Parking Permit Problem, Randomized Rounding,
Primal-Dual Algorithms, Operations Research, Resource Allocation.
Abstract:
We consider leasing variants of two classical N P -hard optimization problems, Vertex Cover (VC) and non-
metric Facility Location (non-metric FL). These contain the well-known Parking Permit problem due to Mey-
erson [FOCS 2005] as a special sub-case and can be found as sub-problems in many operations research
applications. We give the first online algorithms for these two problems, evaluated using the standard notion
of competitive analysis in which the online algorithm whose input instance is revealed over time is compared
to the optimal offline algorithm which knows the entire input sequence in advance and is optimal. Our al-
gorithms have optimal and near-optimal competitive ratios for the leasing variants of VC and non-metric FL,
respectively.
1 INTRODUCTION
Many companies have stopped buying their resources
and started leasing them instead. This has reduced
their costs and provided them with flexibility not of-
fered by buying. As a result of this flexibility, more
complex decisions had to be made since more options
were now available, including different prices for a
single resource depending on the length and time of
the lease in comparison to a fixed buying price. These
options respect economy of scale such that a longer
lease costs less per unit time. The aim is to decide
which lease to buy when without making regrets in
the future, that is, while minimizing costs. Should we
know future demands in advance, we would make use
of the wide range of offline approaches in the litera-
ture that would yield optimal leasing decisions. How-
ever, since demands are not always known in advance,
leasing decisions are often made online. To better
grasp the difficulty of such scenarios and make wiser
decisions, many theoretical leasing models have been
studied, the first of which was by Meyerson in 2005
who introduced a toy example problem, the Parking
Permit problem (PP) (Meyerson, 2005). Each day,
depending on the weather, we have to either use the
car (if it is rainy) or walk (if it is sunny). In the for-
mer case, we must have a valid parking permit, which
we choose among |L| different lease types. At any
time, lease prices respect economy of scale. The goal
is to buy a set of leases in order to cover all rainy days
while minimizing the total cost of purchases and with-
out using weather forecasts. Interestingly, it turns out
that providing a provably good solution to this sim-
ple problem requires a challenging algorithmic ap-
proach due to the online nature of leasing. To mea-
sure the quality of his approach, Meyerson (Meyer-
son, 2005) used the standard notion of competitive
analysis that is commonly used to measure the per-
formance of online algorithms (Sleator and Tarjan,
1985). Given an optimization problem X, the worst
case of the ratio over all instances of X between the
cost incurred by an online algorithm Alg and the op-
timal offline algorithm is called the competitive ratio
of Alg. An online algorithm with competitive ratio
r is said to be r-competitive. Meyerson (Meyerson,
2005) gave a deterministic O(|L|)-competitive and a
randomized O(log |L|)-competitive algorithm along
with matching lower bounds for PP. A series of more
complex leasing problems generalizing PP followed
Meyeron’s work. These include the leasing variants
of classical N P -hard optimization problems found
in various application areas, such as Set Cover (Ab-
shoff et al., 2016), metric Facility Location (Abshoff
et al., 2016; Nagarajan and Williamson, 2013), and
edge-weighted Steiner Tree/ Forest (Bienkowski et al.,
2017; Meyerson, 2005). Unlike in the classical non-
Markarian, C. and der Heide, F.
Online Algorithms for Leasing Vertex Cover and Leasing Non-metric Facility Location.
DOI: 10.5220/0007369503150321
In Proceedings of the 8th International Conference on Operations Research and Enterprise Systems (ICORES 2019), pages 315-321
ISBN: 978-989-758-352-0
Copyright
c
 2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
315

leasing setting in which when a resource is chosen
into the solution, it can be used forever without in-
curring future costs, resources in the leasing setting
have duration and expire. In order to further use a
resource, it needs to be paid for. These leasing vari-
ants are natural generalizations that capture more re-
alistic application scenarios. The leasing model by
Meyerson has later seen a number of extensions, in-
cluding lease prices changing over time (Feldkord
et al., 2017), leases with length and capacity (de Lima
et al., 2017a), and leases accompanied with penal-
ties (Markarian, 2017).
In this paper, we continue the algorithmic study
of leasing problems by considering the leasing vari-
ants of two classical N P-hard optimization prob-
lems, Vertex Cover and (non-metric) Facility Loca-
tion. As far as we are aware of, no online algorithms
are known for them in the literature. In what follows,
we define these two variants and state our result for
each.
1.1 Leasing Vertex Cover
Definition 1. (Leasing Vertex Cover (LVC)) We are
given an undirected graph G = (V, E) with |V | = n
and node-weight function w : V → R
+
. There is a
set L of different lease types each characterized by
a duration and cost, and a sequence of edges of G
arriving over time. A node can be leased using lease
type l for cost c
l
and remains active for time d
l
. In
each step t, an edge (u, v) arrives and LVC asks is to
ensure that either node u or node v is active at time t,
while minimizing overall leasing costs.
A problem related to LVC is the Online Vertex
Cover problem (OVC) due to Demange et al. (De-
mange and Paschos, 2005), in which the input graph is
not known in advance and the nodes along with their
incident edges are revealed over time. The goal is to
construct a minimum weight subset S of nodes such
that each arriving edge is covered by S (i.e., is incident
to at least one node in S). Demange et al. (Demange
and Paschos, 2005) gave upper and lower bounds de-
pending on the maximum degree of the input graph
for OVC. Our model in LVC is quite different, since
edges are revealed over time, such that each edge may
appear more than once and needs to be covered only at
the current time step it arrives. LVC has deterministic
Ω(|L|) and randomized Ω(log |L|) lower bounds on
its competitive ratio resulting from lower bounds for
PP.
Result 1. We propose an O(|L|)-competitive deter-
ministic algorithm for LVC, which is asymptotically
optimal and is based on a simple primal-dual ap-
proach.
1.2 Leasing Non-metric Facility
Location
Definition 2. (Leasing Non-metric Facility Location
(LNFL)) We are given an undirected connected graph
G = (V, E) with |V | = n, node-weight function w :
V → R
+
, and unit edge-weight function w : E → 1.
There is a set L of different lease types each char-
acterized by a duration and cost, and a sequence of
nodes of G arriving over time. The distance d
uv
be-
tween nodes u and v of G is the total edge-weight of
the shortest path between u and v. There are m nodes
assigned as potential facilities. A facility node can be
leased using lease type l for cost c
l
and remains ac-
tive for time d
l
. In each step t, a node u ∈ V (client)
arrives and needs to be connected to a node of G (fa-
cility) active at time t. Connecting node u to node v
costs the distance d
uv
between them. LNFL asks to
connect all arriving nodes, while minimizing overall
leasing and connection costs.
A closely related problem is the Leasing Met-
ric Facility Location problem, the metric variant of
LNFL to which Nagarajan and Williamson (Nagara-
jan and Williamson, 2013) gave an O(|L| · logn)-
competitive algorithm. Kling et al. (Kling et al.,
2012) extended this result by giving an O(l
K
log(l
K
))-
competitive algorithm, where l
K
is the maximum
lease length. Both of these works exploit triangle in-
equality in their competitive analysis and hence do
not extend to LNFL. Furthermore, LNFL general-
izes the Online Non-metric Facility Location problem
(ONFL) to which Alon et al. (Alon et al., 2006) gave
a randomized O(lognlogm)-competitive algorithm.
The Online Set Cover problem (OSC) due to Alon
et al. (Alon et al., 2003) is a special case of ONFL
in which the facilities are sets and the connection
cost is 0 if the element belongs to the set and infin-
ity otherwise. OSC has a randomized lower bound of
Ω(logmlogn) due to Korman (Korman, 2005), where
m is the number of sets and n is the number of ele-
ments. Hence, LNFL has a randomized lower bound
of Ω(log nlogm + log |L|) on its competitive ratio re-
sulting from lower bounds for OSC and PP.
Result 2. We propose an O(log n logm +
log|L|logn)-competitive randomized algorithm
for LNFL, based on rounding online a fractional so-
lution constructed using a multiplicative incremental
approach. The latter is a common technique used in
many online covering problems, one of the first of
which is OSC due to Alon et al. (Alon et al., 2003).
Our algorithm is based on the randomized algorithm
for ONFL due to Alon et al. (Alon et al., 2006), which
adopts a similar technique.
ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems
316

Outline. The rest of the paper is organized as fol-
lows. In Section 2, we address the offline variants of
LVC and LNFL and give some preliminaries. In Sec-
tions 3 and 4, we present the online algorithms for
LVC and LNFL, respectively, and evaluate their per-
formance using competitive analysis. We conclude in
Section 5 with some open problems.
2 OFFLINE MODEL & INTERVAL
MODEL
In this section, we first address the offline variants of
LVC and LNFL and then define a simplified model
for the lease types, called the Interval Model, which
we use throughout the paper.
2.1 Offline Model
Koutris (Koutris, 2010) gave a 6-approximation algo-
rithm for the offline variant of LVC, based on a stan-
dard primal-dual approach. The latter is an offline al-
gorithm for a generalization of LVC, the Leasing Set
Cover problem, with (3d)-approximation ratio, where
d is the maximum number of subsets an element be-
longs to. As for the offline variant of LNFL, we use
the following transformation. An offline instance of
Leasing Non-metric Facility Location can be trans-
formed into an instance of the offline variant of Leas-
ing Set Cover, in which elements are represented by n
clients and subsets are formulated by taking all com-
binations formed by m facilities and |L| lease types
over n time steps. There is an O(log n) approxima-
tion for the offline variant of Leasing Set Cover due
to Anthony et al. (Anthony and Gupta, 2007), based
on reducing the latter to its corresponding multistage
stochastic optimization problem. Note that a similar
transformation between the online variants of LVC
and LNFL is possible but yields a competitive ratio
that is polynomial in m for LNFL. This can be done
by applying the O(lognlog(m· |L|))-competitive ran-
domized algorithm for Leasing Set Cover due to Ab-
schoff et al. (Abshoff et al., 2016). Moreover, Leas-
ing Set Cover has a lower bound that depends on the
number of subsets m, Ω(lognlog m + log |L|). Hence,
no online algorithm for Leasing Set Cover will yield
a better competitive ratio for LNFL using this trans-
formation.
2.2 Interval Model
We assume the following model, known as the Inter-
val model due to Meyerson (Meyerson, 2005), for the
leasing options.
Definition 3. (Interval Model) Leases in the interval
model satisfy the following two properties: (i) Lease
lengths l
k
are powers of two. (ii) Leases of the same
type do not overlap.
Meyerson (Meyerson, 2005) showed that the in-
terval model simplifies the original leasing model at
the cost of a constant factor in the competitive ra-
tio. This model has been adopted in all other leas-
ing optimization problems studied thus far (Abshoff
et al., 2016; Bienkowski et al., 2017; Nagarajan and
Williamson, 2013).
3 LEASING VERTEX COVER
In this section, we present a deterministic online al-
gorithm for LVC and show its competitive analysis.
3.1 Online Algorithm
We formulate LVC as an integer linear program (see
Figure 1). Nodes are represented as triplets, such that
node u ∈ V with lease type l at starting time step t is
denoted as (u, l, t). The collection of all these triplets
is denoted as V . x
(u,l,t)
is a primal variable indicating
whether (u, l, t) is bought (= 1) or not (= 0). c
l
is
the cost of lease type l and d
l
is its duration. w
u
is
the weight of node u. Edges are represented as pairs,
such that edge (u, v) ∈ E arriving at time t is denoted
as (uv, t). Recall that an edge may appear at more
than one time step. The collection of all these pairs
is denoted as D. The steps of the algorithm upon the
arrival of a new edge are depicted in Algorithm 1.
Algorithm 1: Online Deterministic Primal-dual Algorithm
for LVC.
When an edge (u, v) arrives at time t,
(i) increase its dual variable y
(uv,t)
until the dual
constraint of a triplet covering time step t corre-
sponding to one of its endpoints becomes tight.
(ii) set the primal variable x
(u,l,t)
corresponding to
every tight dual constraint to 1.
3.2 Competitive Analysis
Algorithm 1 constructs a feasible dual solution, since
it never violates the dual constraints - it stops increas-
ing the dual variables as soon as a constraint is tight.
Furthermore, since the algorithm makes sure the pri-
mal variable of at least one endpoint covering time
step t of each edge arriving at time t is set to 1, the
primal solution constructed is also feasible. It remains
Online Algorithms for Leasing Vertex Cover and Leasing Non-metric Facility Location
317

min
∑
(u,l,t)∈V
x
(u,l,t)
· c
l
· w
u
Subj to: ∀(uv, t
0
) ∈ D :
∑
(u,l,t)
(v,l
0
,t
00
)∈V
t
0
∈[t,t+d
l
]
t
0
∈[t
00
,t
00
+d
l
0
]
x
(u,l,t)
+ x
(v,l
0
,t
00
)
≥ 1
∀(u, l,t) ∈ V : x
(u,l,t)
∈ {0, 1}
max
∑
(uv,t
0
)∈D
y
(uv,t
0
)
Subj to: ∀(u, l, t) ∈ V :
∑
v∈V
(uv,t
0
)∈D
t
0
∈[t,t+d
l
]
y
(uv,t
0
)
≤ c
l
· w
u
∀(uv,t
0
) ∈ D : y
(uv,t
0
)
≥ 0
Figure 1: ILP Formulation of Leasing Vertex Cover
to show a relationship between the cost of the primal
solution and that of the dual solution. Let P ⊆ V de-
note the primal solution constructed by the algorithm.
Since the dual constraint is tight for each (u, l,t) ∈ P,
we have
c
l
· w
u
=
∑
v∈V
(uv,t
0
)∈D
t
0
∈[t,t+d
l
]
y
(uv,t
0
)
Summing up over all triplets in P, we get
∑
(u,l,t)∈P
c
l
· w
u
=
∑
(u,l,t)∈P
∑
v∈V
(uv,t
0
)∈D
t
0
∈[t,t+d
l
]
y
(uv,t
0
)
=
∑
(uv,t
0
)∈D
y
(uv,t
0
)
∑
(u,l,t)∈P
(v,l
0
,t
00
)∈P
t
0
∈[t,t+d
l
]
t
0
∈[t
00
,t
00
+d
l
0
]
1
Assuming the interval model, we have that on
each day t, there are exactly |L| leases covering time
step t. Since each edge can be covered by at most two
nodes, the algorithm can only set the primal variables
corresponding to at most 2 · |L| triplets to 1. Thus,
∑
(u,l,t)∈P
(v,l
0
,t
00
)∈P
t
0
∈[t,t+d
l
]
t
0
∈[t
00
,t
00
+d
l
0
]
1 ≤ 2 · |L|
By Weak Duality theorem, we have that any feasi-
ble dual solution is a lower bound to any feasible pri-
mal solution, including the optimal primal solution.
This completes the analysis and the theorem below
follows.
Theorem 1. There is an O(|L)|-competitive deter-
ministic algorithm for Leasing Vertex Cover.
4 LEASING NON-METRIC
FACILITY LOCATION
In this section, we present a randomized online algo-
rithm for LNFL and show its competitive analysis.
4.1 Online Algorithm
We formulate LNFL as follows. We are given a bipar-
tite graph G = (A∪B, E), where A contains the set of
n nodes and B contains the set of m potential facilities
of the input graph G, and a root node r. Nodes have
no weights in G. The edge set E is weighted and
formed as follows. There is an edge between each
node in A and each node in B. An edge (u, v) between
node u ∈ A and node v ∈ B has cost equal to the dis-
tance d
uv
between nodes u and v in G (cost of edge
(u, u) is set to 0). There are |L| edges between each
node u in B and root node r, corresponding to each
of the lease types. Leases are structured according to
the Interval model described earlier. There is a cost
associated with edge (u, r) of lease type l, equal to
(c
l
· w
u
), where c
l
is the cost of lease type l and w
u
is
the weight of node u in G. In each step, a node from
A arrives and needs to be connected to r. The online
algorithm needs to ensure that there is a path between
each arriving node to r at the time of arrival, while
minimizing the total costs.
We denote by c
e
the cost of edge e. The algorithm
assigns to each edge e of G associated with a time
interval and lease type, a fraction f
e
, initially set to
0. We define the maximum flow between two nodes u
and v to be the smallest total fraction of edges which
if removed would disconnect u from v. These edges
form a minimum cut. When a node arrives at time t,
the algorithm disregards:
• all edges whose lease does not cover time t
• all edges between nodes in A that previously ar-
rived and all nodes in B w
A random variable µ is chosen as the minimum
among 2
d
log(n + 1)
e
independent random variables,
distributed uniformly in the interval [0, 1]. All loga-
rithms are to the base 2. We say a node u arriving
at time t is connected to r if there is a path of leased
edges covering step t from u to r. The steps of the
algorithm upon the arrival of a new node are depicted
in Algorithm 2 below.
4.2 Competitive Analysis
Algorithm 2 ensures a feasible integral solution upon
its termination, due to Step (iii). It also constructs a
feasible fractional solution in Step (i). We manage to
ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems
318

Algorithm 2: Online Randomized Algorithm for LNFL.
Input. Graph G and node u arriving at time t
Output. Set of leased edges covering step t
When u arrives,
(i) If u is connected to r, do nothing.
Else, while the maximum flow between u and r
is less than 1,
- compute a minimum cut C between u and r
- increment the fraction f
e
of each edge e ∈ C
with the equation:
f
e
← f
e
(1 +
1
c
e
) +
1
|C| · c
e
(ii) Lease edge e if w
e
≥ µ.
(iii) If u is not connected to r, lease the edges of a
path from u to r of cheapest cost.
bound the cost of the fractional solution constructed
in terms of the cost of the optimal integral solution
(Lemma 1). We also show that the cost of our in-
tegral solution constructed can be bounded in terms
of the cost of our fractional solution (Lemma 2) and
conclude the competitive ratio of the algorithm.
Let E
0
be the collection of all leases the algorithm
can purchase, that is, all edges associated with all time
intervals and all lease types. We refer to edges in E
0
as edge-intervals.
Lemma 1. The cost of the algorithm’s fractional so-
lution C
f rac
is at most O(log(m · |L|)) · Opt, where
Opt is the cost of the optimal integral solution.
Proof. The fractional solution constructed has cost:
∑
e∈E
0
c
e
· f
e
Consider an iteration in which the algorithm is
given a node u that is not connected to r. We show
that every increment the algorithm performs can be
upper bounded by 2. Notice that the algorithm per-
forms an increment only if the maximum flow is less
than 1. Hence the edges of every minimum cut have a
total fraction of less 1 before an increment. Let C be
a minimum cut between u and r. The total fractional
cost added by the edge-intervals of C is at most:
∑
e∈C
c
e
· (
f
e
c
e
+
1
|
C
|
· c
e
) ≤ 2
Now we need an upper bound on the total num-
ber of increments the algorithm performs. Any opti-
mal integral solution must contain at least one of the
edge-intervals of each minimum cut, by the definition
of minimum cut. Let us fix such an edge-interval e
0
in a minimum cut C
e
0
. The total number of incre-
ments needed to make the fraction f
e
0
1 can be upper
bounded by (c
e
0
· log
|
C
e
0
|
). As soon as this fraction
gets 1, its edge-interval can’t be in any future min-
imum cut. Adding up over the set E
0
opt
⊆ E
0
of all
edge-intervals of the optimal integral solution, we get:
∑
e
0
∈E
0
opt
c
e
0
· log
|
C
e
0
|
Now we upper bound the number of edge-
intervals in any minimum cut by (m · |L|). Fix a fa-
cility node v ∈ B. A minimum cut between a node
u to r through v will either contain the edge-interval
corresponding to edge (u, v) or the |L| edge-intervals
corresponding to the edges from v to r. Since there are
m edge-intervals from u to all other facility nodes and
each of the m facility nodes has |L| edge-intervals to
r, t |C
e
0
| ≤ (m · |L|) for all e
0
∈ E
0
opt
. Adding up over
all edge-intervals of the optimal integral solution, we
get:
∑
e∈E
0
c
e
· f
e
≤ O(log(m · |L|)) · Opt
Lemma 2. The cost of the algorithm’s integral solu-
tion is at most O(log n) ·C
f rac
, where C
f rac
is the cost
of the algorithm’s fractional solution.
Proof. The algorithm purchases its edge-intervals in
Steps (ii) and (iii) only. We measure the expected cost
of the integral solution in each step. Let X
e
(i) be the
random variable indicating the event that f
e
> µ for
1 ≤ i ≤ 2
d
log(n + 1)
e
. The expected cost E(
∑
e∈E
0
c
e
)
of all edge-intervals is at most:
∑
e∈E
0
2
d
log(n+1)
e
∑
i=1
c
e
· E(X
e
(i)) =
∑
e∈E
0
2
d
log(n+1)
e
∑
i=1
c
e
· f
e
≤ 2
d
log(n + 1)
e
·C
f rac
This bounds the integral cost of the algorithm in
Step (ii). As for Step (iii), the algorithm performs it
only if Step (ii) fails to connect the given node. Fix a
node u that was not connected to r in Step (ii). The
flow of a path is the minimum of all weights of the
edge-integrals of the path. At the end of Step (i), the
algorithm ensures that the sum of flows of all paths
between u and r is at least 1. Hence, the probabil-
ity that the algorithm fails to connect node u is upper
bounded by the probability that µ exceeds the flow of
each path from u to r. Fix a minimum cut C con-
structed at the end of Step (i). The probability that u
is not connected is thus:
Online Algorithms for Leasing Vertex Cover and Leasing Non-metric Facility Location
319

∏
e∈C
(1 − f
e
) ≤ e
−
∑
e∈C
f
e
≤
1
e
Hence, for all 1 ≤ i ≤ 2
d
log(n + 1)
e
, the probabil-
ity that u is not connected is at most
1
n
2
. In Step (iii),
the algorithm purchases edge-intervals of a cheapest
path from u to r, which is a lower bound on the cost of
the optimal solution Opt. This yields a negligible ex-
pected cost of n·
Opt
n
2
, for the at most n arriving nodes,
and concludes the proof.
Lemma 1 and Lemma 2 ultimately lead to the fol-
lowing theorem.
Theorem 2. There is an O(lognlogm+log|L|logn)-
competitive randomized algorithm for Leasing Non-
metric Facility Location.
5 DISCUSSION & OPEN
PROBLEMS
We have presented the first online algorithms, with
optimal/near-optimal competitive ratios, for the leas-
ing variants of two classical optimization problems.
Below is a summary of currently known results as
well as our results (in bold) for LVC and LNFL, re-
spectively.
LVC
• Online Lower Bound: Ω(|L|) (Meyerson, 2005)
(deterministic)
• Offline Lower Bound: 2 (Khot and Regev, 2008)
(deterministic)
• Online Upper Bound: O(|L|)-competitive (de-
terministic)
• Offline Upper Bound: 6-approximation (Koutris,
2010) (deterministic)
LNFL
• Online Lower Bound: Ω(lognlogm +
log|L|) (Korman, 2005; Meyerson, 2005)
(randomized)
• Offline Lower Bound: Ω(logn) (Slavik, 1997)
(deterministic)
• Online Upper Bound: O(log n log(m · |L|))-
competitive (randomized)
• Offline Upper Bound: O(logn)-
approximation (Anthony and Gupta, 2007)
(deterministic)
An interesting open problem is to close the gap
between the offline lower and upper bounds for LVC.
Another direction is to consider a deterministic al-
gorithm for LNFL. While it seems hard to design
a deterministic algorithm with a non-trivial compet-
itive ratio for the latter since the algorithm for the on-
line non-leasing variant of the problem as well those
for most algorithms for related problems are random-
ized, it remains an interesting open question to figure
whether it is possible at all.
Furthermore, most online (leasing) problems for
variants of Facility Location assume an underlying
metric and use the associated metric properties in the
design and/or analysis of the algorithms (de Lima
et al., 2017b; Felice et al., 2015). It would be in-
teresting to look at some of these variants such as
Connected Facility Location, in the online (leasing)
model, without considering an underlying metric.
REFERENCES
Abshoff, S., Kling, P., Markarian, C., Meyer auf der Heide,
F., and Pietrzyk, P. (2016). Towards the price of leas-
ing online. J. Comb. Optim., 32(4):1197–1216.
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.
Anthony, B. M. and Gupta, A. (2007). Infrastructure
leasing problems. In Fischetti, M. and Williamson,
D. P., editors, Integer Programming and Combinato-
rial Optimization, pages 424–438, Berlin, Heidelberg.
Springer Berlin Heidelberg.
Bienkowski, M., Kraska, A., and Schmidt, P. (2017). A de-
terministic algorithm for online steiner tree leasing. In
Ellen, F., Kolokolova, A., and Sack, J.-R., editors, Al-
gorithms and Data Structures, pages 169–180, Cham.
Springer International Publishing.
de Lima, M., Felice, M. S., and Lee, O. (2017a). On gener-
alizations of the parking permit problem and network
leasing problems. Electronic Notes in Discrete Mathe-
matics, 62:225 – 230. LAGOS’17 IX Latin and Amer-
ican Algorithms, Graphs and Optimization.
de Lima, M. S., Felice, M. C. S., and Lee, O. (2017b).
Connected facility leasing problems. In Joint Pro-
ceedings of the 18th Italian Conference on Theoretical
Computer Science and the 32nd Italian Conference on
Computational Logic co-located with the 2017 IEEE
International Workshop on Measurements and Net-
working (2017 IEEE M&N), Naples, Italy, September
26-28, 2017., pages 162–173.
ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems
320

Demange, M. and Paschos, V. T. (2005). Online vertex-
covering. Theoretical Computer Science, 332(1):83 –
108.
Feldkord, B., Markarian, C., and Meyer auf der Heide, F.
(2017). Price fluctuation in online leasing. In Com-
binatorial Optimization and Applications - 11th Inter-
national Conference, COCOA 2017, Shanghai, China,
December 16-18, 2017, Proceedings, Part II, pages
17–31.
Felice, M. C. S., Cheung, S., Lee, O., and Williamson, D. P.
(2015). The online prize-collecting facility location
problem. Electronic Notes in Discrete Mathematics,
50:151–156.
Khot, S. and Regev, O. (2008). Vertex cover might be hard
to approximate to within 2 - epsilon. Journal of Com-
puter and System Sciences, 74(3):335 – 349.
Kling, P., Meyer auf der Heide, F., and Pietrzyk, P.
(2012). An algorithm for online facility leasing. In
Structural Information and Communication Complex-
ity - 19th International Colloquium, SIROCCO 2012,
Reykjavik, Iceland, June 30-July 2, 2012, Revised Se-
lected Papers, pages 61–72.
Korman, S. (2005). On the use of randomization in the on-
line set cover problem. Master’s thesis, Weizmann In-
stitute of Science, Israel.
Koutris, P. (2010). Infrastructure Leasing Problems. Mas-
ter’s thesis, University of Athens, Greece.
Markarian, C. (2017). Leasing with uncertainty. In Opera-
tions Research Proceedings 2017, Selected Papers of
the Annual International Conference of the German
Operations Research Society (GOR), Freie Universi
¨
at
Berlin, Germany, September 6-8, 2017., pages 429–
434.
Meyerson, A. (2005). The parking permit problem. In 46th
Annual IEEE Symposium on Foundations of Computer
Science (FOCS 2005), 23-25 October 2005, Pitts-
burgh, PA, USA, Proceedings, pages 274–284.
Nagarajan, C. and Williamson, D. P. (2013). Offline
and online facility leasing. Discrete Optimization,
10(4):361–370.
Slavik, P. (1997). A tight analysis of the greedy algorithm
for set cover. Journal of Algorithms, 25(2):237 – 254.
Sleator, D. D. and Tarjan, R. E. (1985). Amortized effi-
ciency of list update and paging rules. Commun. ACM,
28(2):202–208.
Online Algorithms for Leasing Vertex Cover and Leasing Non-metric Facility Location
321