We consider the problem of pricing edges of a line graph so as to maximize the profit made from selling intervals to single-minded customers. An instance is given by a set
E
of
n
edges with a limited supply for each edge, and a set of
m
clients, where each client specifies one interval of
E
she is interested in and a budget
B
j
which is the maximum price she is willing to pay for that interval. An
envy-free
pricing is one in which every customer is allocated an (possibly empty) interval maximizing her utility. Grandoni and Rothvoss (SIAM J. Comput. 2016) proposed a
polynomial-time approximation scheme
(
PTAS
) for the unlimited supply case with running time (
nm
)
O
((1/ɛ)
1/ɛ
)
, which was extended to the limited supply case by Grandoni and Wiese (ESA 2019). By utilizing the known
hierarchical decomposition
of
doubling metrics
, we give a PTAS with running time (
nm
)
O
(1/ ɛ
2
)
for the unlimited supply case. We then consider the limited supply case, and the notion of ɛ-envy-free pricing in which a customer gets an allocation maximizing her utility within an
additive error
of ɛ. For this case, we develop an approximation scheme with running time (
nm
)
O
(log
5/2
max
e
H
e
/ɛ
3
)
, where
H
e
=
B
max
(
e
)/
B
min
(
e
) is the maximum ratio of the budgets of any two customers demanding edge
e
. This yields a PTAS in the
uniform budget
case, and a quasi-PTAS for the general case. The best approximation known, in both cases, for the exact envy-free pricing version is
O
(log
c
max
), where
c
max
is the maximum item supply. Our method is based on the known hierarchical decomposition of doubling metrics, and can be applied to other problems, such as the
maximum feasible subsystem
problem with interval matrices.
Lower Bounds for On-line Graph Colorings