Proportional Fair FrequencyDomain Packet Scheduling for 3GPP LTE Uplink.
ABSTRACT With the power consumption issue of mobile handset taken into account, Singlecarrier FDMA (SCFDMA) has been selected for 3GPP LongTerm Evolution (LTE) uplink multiple access scheme. Like in OFDMA downlink, it enables multiple users to be served simultaneously in uplink as well. However, its single carrier property requires that all the subcarriers allocated to a single user must be contiguous in frequency within each time slot. This contiguous allocation constraint limits the sch eduling flexibility, and frequencydomain packet scheduling algor ithms in such system need to incorporate this constraint while trying to maximize their own scheduling objectives. In this paper we explore this fundamental problem of LTE SCFDMA uplink scheduling by adopting the conventional time domain Proportional Fair algorithm to maximize its objective (i.e. proportional fair criteria) in the frequencydomain setting. We show the NPhardness of the frequencydomain scheduling problem under this contiguous allocation constraint and present a set of practical algorithms fine tuned to this problem. We demonstrate that competitive performance can be achieved in terms of system throughput as well as fairness perspective, which is evaluated using 3GPP LTE system model simulations.

Conference Paper: Adaptive frame size control based on resource allocation for MachinetoMachine services in LTE wireless networks
[Show abstract] [Hide abstract]
ABSTRACT: As MachinetoMachine (M2M) services get their widely application, its own characteristic was observed and discussed. Some of the characteristics such as small data communication and the huge amount of devices are different from the traditional HumantoHuman (H2H) Services, whose radio resource allocation mechanism is designed for tens of users and large amount of data. To solve this problems for the existing LTE wireless networks, an adaptive frame size control strategy is proposed in this paper. After a M2M device receives the resources from the eNodeB, it compares the resources with the data in the transmit buffer to see if the resources can be able to fill with some copies of the data. Using the Send Multiple Copies of Data Scheme (SMCDS), the utilization of the resources allocated is improved. Simulation results show that with this strategy, system throughput and transmission success rate both increase.Wireless Communications & Signal Processing (WCSP), 2012 International Conference on; 01/2012 
Conference Paper: Opportunistic and efficient resource block allocation algorithms for LTE uplink networks
[Show abstract] [Hide abstract]
ABSTRACT: This work proposes two new Resource Block (RB) allocation algorithms for the LTE uplink. They take into account the RB adjacency constraint and update the allocation metric. Two different heuristics are proposed: an Opportunistic and Efficient RB allocation Algorithm (OEA) and a Quality of Service based Opportunistic and Efficient RB allocation Algorithm (QoS based OEA). Both algorithms seek to maximize the aggregate throughput and avoid RB wastage unlike other algorithms in the literature. The complexity of the proposed algorithms are also computed analytically and compared to other well known heuristics.Wireless Communications and Networking Conference (WCNC), 2013 IEEE; 01/2013  SourceAvailable from: Dario Vieira[Show abstract] [Hide abstract]
ABSTRACT: he Long Term Evolution (LTE) standard plays an important role in the development of MachinetoMachine (M2M) communication. However, the M2M communication has several different characteristics regarding to HumantoHuman (H2H) communication. Therefore, the shortage of radio resources, satisfaction of the Quality of Service (QoS) requirements and the reduction of the H2H traffic performance are important issues to be addressed when introducing the M2M communication in the network. In this article, we present a scheduler that may dynamically adjust to the level of congestion of the network based on the current traffic information of each device. The main goals of our approach are (i) satisfy the QoS requirements (ii) ensure fair allocation of resources and (iii) control the impact of H2H traffic performance. The simulation results demonstrated that the proposed scheduling has good performance according to the three objectives aforementioned.PIMRC 2014; 09/2014
Page 1
UCLA CSD TECHNICAL REPORT: TR0900011
Proportional Fair FrequencyDomain Packet
Scheduling for 3GPP LTE Uplink
SukBok Lee∗
Ioannis Pefkianakis∗
Adam Meyerson∗
Shugong Xu†
Songwu Lu∗
∗Computer Science Department
UCLA, CA 90095
†Sharp Laboratories of America
Camas, WA 98607
Abstract—With the power consumption issue of mobile handset
taken into account, Singlecarrier FDMA (SCFDMA) has been
selected for 3GPP LongTerm Evolution (LTE) uplink multiple
access scheme. Like in OFDMA downlink, it enables multiple
users to be served simultaneously in uplink as well. However, its
single carrier property requires that all the subcarriers allocated
to a single user must be contiguous in frequency within each time
slot. This contiguous allocation constraint limits the scheduling
flexibility, and frequencydomain packet scheduling algorithms
in such system need to incorporate this constraint while trying
to maximize their own scheduling objectives.
In this paper we explore this fundamental problem of LTE
SCFDMA uplink scheduling by adopting the conventional time
domain Proportional Fair algorithm to maximize its objective
(i.e. proportional fair criteria) in the frequencydomain setting.
We show the NPhardness of the frequencydomain scheduling
problem under this contiguous allocation constraint and present
a set of practical algorithms fine tuned to this problem. We
demonstrate that competitive performance can be achieved in
terms of system throughput as well as fairness perspective, which
is evaluated using 3GPP LTE system model simulations.
I. INTRODUCTION
In recent years Orthogonal Frequency Division Multiple
Access (OFDMA) has been considered as a strong candidate
for the broadband air interface for its robustness to multipath
fading, higher spectral efficiency and bandwidth scalability,
and it has been selected for 3GPP LongTerm Evolution (LTE)
downlink (DL) radio access technology. However, one major
disadvantage of OFDMA is that the instantaneous transmitted
RF power can vary dramatically within a single OFDM
symbol. Such an undesirable high peaktoaverage power ratio
(PAPR) is a serious concern for the uplink (UL), since power
consumption is a key consideration for the mobile handsets.
As a result of seeking an alternative to OFDMA, Single
carrier FDMA (SCFDMA) has been selected for LTE uplink
multiple access scheme. While keeping most of the advantages
of OFDMA (e.g. the same degree of multipath protection), SC
FDMA has significantly lower PAPR, since the underlying
waveform is essentially singlecarrier. Thus, lower PAPR of
SCFDMA greatly benefits the mobile terminal in terms of
transmit power efficiency.
As in DL OFDMA, multiple access in UL SCFDMA
is achieved by assigning different frequency portions of the
system bandwidth to individual users based on their channel
conditions. Such simultaneous frequencydomain multiplexing
of users (inherently in concert with timedomain scheduling)
is performed by frequency domain packet scheduling (FDPS).
In LTE UL, the system bandwidth is divided into multiple
subbands (i.e. groups of subcarriers) denoted as physical
resource blocks (RBs). In order to achieve large gain from
multiuser frequency diversity, a scheduler needs to know the
instantaneous radio channel conditions across all users and all
RBs, which are fed as input for the frequencydomainadaptive
usertoRB allocation. For example, in LTE UL each user
transmits a Sounding Reference Signal (SRS) to the scheduling
node (i.e. base station) [1], which is used as channel quality
indicator (CQI). With CQIs across all users and all RBs, a base
station performs RBtouser assignment at each time slot (e.g.
in LTE every 1ms) according to the selected scheduling policy.
Thus, in the timefrequency domain, an RB is considered as
a minimum scheduling resolution, and also a minimum unit
of the datarate adaptation by adaptive modulation and coding
(AMC) with a granularity of one subframe.
Most of the DL FDPS algorithms proposed so far adopt
the wellknown timedomain Proportional Fair (PF) algorithm
as a basic scheduling principle and apply the PF algorithm
directly over each RB onebyone independently. However,
such scheduling strategies cannot be employed in the UL SC
FDMA. Due to its single carrier property, SCFDMA requires
that all the RBs allocated to a single user must be contiguous
in frequency within each time slot (i.e. subframe) [5], [6].
Thus, LTE UL FDPS algorithms should respect this constraint
while trying to maximize their own scheduling objectives.
In this paper we study this fundamental problem of UL
frequencydomain packet scheduling under contiguous RB
allocation constraint. We analyze this problem by adopting
the widely employed PF algorithm to maximize its objective
(i.e. proportional fair criteria) in the frequencydomain setting.
The main goal of this paper is to investigate how to adapt the
timedomain PF algorithm to this problem framework.
A. The Model
We consider a cellular network whose UL system bandwidth
is divided into m RBs, and we have a single base station and
n active wireless users. The base station can allocate m RBs
to a set of n users. At each time slot multiple RBs (with the
contiguity constraint) can be assigned to a single user, each
RB however can be assigned to at most one user. In this paper
we shall work in an infinitely backlogged model in which for
each user there is always data available for service. Thus, the
base station can schedule all the m RBs every time slot.
We define the indicator variable xc
or not RB c is assigned to user i at time slot t. We assume
that channel conditions vary across RBs as well as users.
The channel conditions typically depends on the channel
frequency, so they may be different for different channels;
i(t) to indicate whether
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UCLA CSD TECHNICAL REPORT: TR0900012
moreover, they also depends on the user location and the time
slot. Therefore, each RB has userdependent and timevarying
channel condition. We use rc
channel rate for user i on RB c at time t. This channel rates
are estimated from the CQIs extracted from the UL channel
sounding. Thus, if xc
size rc
B. Problem Formulation
In the timedomain context, the well known Proportional
Fair (PF) algorithm aims to maximize, over all feasible
scheduling rules, the utility function?
as proportional fair criteria. Maximizing?
being completely starved since log0 = −∞. In order to
maximize?
that the PF algorithm achieves high throughput and maintains
proportional fairness among all users by giving priority to
users with a highquality channel rate (ri(t)) and a low current
average service rate (Ri(t)).
We now adapt this timedomain PF metric to the frequency
domain setting with the utility function
objective. Let λc
that user i has on RB c at time slot t. As justified in [10], we
can establish a FDPS version of PF objective function when
scheduling time slot t as follows:
i(t) to denote the instantaneous
i(t) = 1, then user i can transmit data of
i(t) on RB c at time slot t.
ilogRi, where Ri is
the longterm service rate of user i. This objective is known
ilogRinot only
improves overall throughput but also prevents any user from
ilogRi, we should serve the user who maximizes
ri(t)/Ri(t) at each time slot t (proven in [7], [17], [22]). Note
?
ilogRi as our
i(t) = rc
i(t)/Ri(t) be the PF metric value
max
?
i
?
c
xc
i(t)λc
i(t)
(1)
Objective (1) above is indeed analogous to the PF algorithm
which maximizes?
ity function?
scheduling algorithms apply the PF algorithm directly over
each RB onebyone, i.e. for RB c the PF algorithm selects the
best user who maximizes rc
for LTE UL we need to add the contiguous RB constraint into
this objective (1) due to the physical layer requirement of SC
FDMA. Accordingly, we can rewrite the objective (1) more
precisely as the following optimization problem:
ixi(t) · ri(t)/Ri(t) in the timedomain
setting. Hence, optimizing the objective (1) makes the util
ilogRi maximized in the frequencydomain
setting. For this reason, most of the proposed DL FDPS
i(t)/Ri(t) at time slot t. However,
max
?
?
?
b
?
xc
i∈ {0,1}
i
?
xc
i≤ 1,
c
xc
iλc
i
(1)
subject to
i
∀c
(2)
i
?
xc
i= b − a + 1,
c
xc
i≤ m
(3)
c=a
∀i,xa
i= xb
i= 1
(4)
(5)
To simplify notation, the dependence on time t is omitted.
Constraint (2) states that each RB can be assigned to at
most one user, and constraint (3) just tells that the system
has the total of m RBs. The only added is constraint (4),
1 8
1 1
1 6
1 3
1 7
1 7
1 8
1 6
1 4
1 8
1 6
1 1
1 6
1 5
1 6
1 5
1 8
1 5
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1 3
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1 2
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1 7
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1 3
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1 6
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1 3
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1 5
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1 4
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1 8
1 6
1 2
1 6
1 7
1 2
1 7
1 7
1 6
1 6
1 8
1 8
1 1
1 5
1 5
1 6
usercarrier
A
B
C
D
E
w/o contiguous requirement
Max = 85
1 8
1 1
1 6
1 3
1 7
1 7
1 8
1 6
1 4
1 8
1 6
1 1
1 6
1 5
1 6
1 5
1 8
1 5
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1 3
1 4
1 2
1 5
1 7
1 6
1 3
1 8
1 6
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1 4
1 4
1 3
1 4
1 9
1 5
1 5
1 8
1 4
1 8
1 8
1 6
1 2
1 6
1 7
1 2
1 7
1 7
1 6
1 6
1 8
1 8
1 1
1 5
1 5
1 6
usercarrier
A
B
C
D
E
w/ contiguous requirement
Max = 83
Fig. 1 Maximizing the PF objective. The numbers denote the PF
metric values λc
maximizing the objective with/without the contiguity constraint.
i. Darkcolored RBs represent assignment strategies
which enforces the contiguous RB allocation. Now we need
to optimize the objective (1) with keeping to those constraints
(i.e. choose the value xc
(1)). One crucial difference is that we now cannot apply the
PF algorithm on each RB onebyone in isolation. In other
words, the isolated local optimization of each RB hardly
optimizes the objective (1). Figure 1 exemplifies the case.
With the contiguity constraint we may need to serve users
with suboptimal PF metric value λc
optimize the PF objective (1).
Seeking to maximize the PF objective (1) under this contigu
ity constraint, we present five variations of PFFDPS algorithm
(Alg1 through Alg5). In this paper we explore the fundamental
nature of this scheduling problem by investigating how well
each of these five algorithms fits into the problem framework.
i(t) to maximize the PF objective
ifor some RBs so as to
C. Related Work
The Proportional Fair (PF) algorithm was introduced by
[15], [22], extensively studied in the research community (e.g.
delay [9], [18], instability [7], [8]), and it is widely used as
a standard scheduling algorithm in the current singlecarrier
wireless systems such as CDMA 2000 1xEVDO [11], [15].
The area of FDPS scheduling is new, and most of studies
directly adapt the timedomain PF algorithm into frequency
domain context. Their results show the potential gains of up
to 4060% average system capacity improvement over time
domain only scheduling [19], and moreover [24] shows that
the frequency selectivity of FDPS indeed helps significantly
improve the shortterm fairness. Andrews et al. [10] have
proposed the FDPSversion of MaxWeight algorithm1, and
addressed the resource wastage problem induced by small
queue condition in DL FDPS context. The objective of the
MaxWeight algorithm is the system stability, and the authors
have presented the performance from the queue perspective.
Cohen et al. [13] recently studied the DL OFDMA schedul
ing problem somewhat related to this contiguous allocation
requirement in WiMAX. They present several heuristic al
gorithms for constructing the OFDMA frame matrix as a
collection of rectangles which fit into a single matrix. The
algorithms, however, assume that 1) at each time slot the base
station somehow knows the scheduled data size for each user
in advance; 2) the same channel rate is across all RBs as
well as all users. In the WLAN context, Yuan et al. [25]
have considered a contiguous channel assignment problem to
1MaxWeight algorithm always serves the user that maximizes Qs
where Qs
user i, respectively.
i(t)r(i, t),
i(t) and r(i,t) are the queue size and the instantaneous data rate of
Page 3
UCLA CSD TECHNICAL REPORT: TR0900013
dynamically allocate the variablewidth channel to each access
point (AP). The key difference from our problem is that no
channel diversity (i.e. they assume the achievable data rate is
linear to the available bandwidth) is considered in their WLAN
context. That is, an AP with the fixed bandwidth will attain the
same throughput regardless of its central frequency assigned,
which makes their problem as a special case of ours.
In summary the contiguous RB allocation constraint is a
crucial requirement for the LTE UL scheduling algorithms,
yet no previous work has been devoted to this fundamental
issue of SCFDMA.
II. HARDNESS RESULT
In this section we first show that unfortunately we cannot
hope for an efficient algorithm that optimizes the objective (1)
under the contiguous RB restriction unless P = NP. We then
demonstrate that it is still computationally intractable in the
practical systems.
A. Hardness of objective (1)
Theorem 1: LTE UL PFFDPS problem (i.e. maximization
of the PF objective (1) under the contiguous RB allocation
constraint) is NPhard.
Proof: We use a reduction from Hamiltonian Path Prob
lem. Given a directed graph G = (V,E), we say that a path
P in G is a hamiltonian path if it contains each vertex in
V exactly once. The problem asks whether a directed graph
G contains a hamiltonian path, and this is NPcomplete [16].
As a preprocessing for our reduction, we can transform any
given directed graph G into a bipartite graph G′, by splitting
each node v in G into two nodes vland vr(say, left and right)
in G′; All the incoming/outgoing edges to/from v are attached
to vland vr, respectively, with adding an edge from vlto vr.
(See Figure 2.) It is clear that G′contains a hamiltonian path
if and only if G contains a hamiltonian path.
We now show that this hamiltonian path problem in bipar
tite graph (HAMPATHBG) is reducible to our problem. A
decision version of our problem is to determine whether for
a given frequencydomain status S (i.e. a collection of value
λc
allocation strategy with resulting aggregate value at least k.
Consider an arbitrary instance of HAMPATHBG, with 2n
nodes (n left nodes vl,1,...,vl,n ∈ V′
vr,1,...,vr,n ∈ V′
status instance S as follows. A user in S corresponds to each
node in G′. For each left node vl,i and right node vr,i, we
have user ul,i∈ Uland ur,i∈ Ur, respectively. Thus, we have
Ul + Ur = n + n = 2n users. We partition the RBs into
three classes Cl, Ct, and Cr(i.e. left, transit, right). We take
a quantity T to be somewhat sufficiently larger than n; say,
T = n2. We arrange the RBs such that T contiguous RBs of
Cland Cralternate with each other via n+2 contiguous RBs
of Ct. Such a pattern (i.e. Cl→ Ct→ Cr) repeats for n times
in the frequencydomain, so we have T ×2n+(n+2)(2n−1)
RBs. (See Figure 3.)
We first assign the scheduling metric value λc
Cl∪Crsuch that the intermediate construction has n! different
contiguous allocation strategies that correspond naturally to
the n! possible hamiltonian paths (in the case of a complete
iacross all users and all RBs), there exists a contiguous
land n right nodes
r). We construct our frequencydomain
ifor RBs ∈
A
C
D
B
Hampath [A,B,D,C] in G
Al
Br
Bl
Ar
Cl
Dr
Dl
Cr
Hampath [Al,Ar,Bl,Br,Dl,Dr,Cl,Cr] in G’
Fig. 2 Equivalence between hamiltonian paths in a given directed
graph G and its corresponding bipartite graph G′
graph G). For each user i ∈ Ul for RB c, we set the value
λc
user i ∈ Urfor RB c, we set λc
c ∈ Cl. (See Figure 3.) At this point, it seems clearly beneficial
to allocate RBs ∈ Clto users ∈ Ul, and assign RBs ∈ Crto
users ∈ Ur. It implies that, in order to get as high aggregate
value as possible, 1) a user ∈ Ul and a user ∈ Ur need to
be assigned alternately in the frequencydomain due to the
alternate RB placement of Cland Cr in our construction; 2)
every user must be served in the end, since our contiguous
allocation constraint prevents onceassigned users from being
reassigned discontiguous RBs.
Now we set the values for RBs ∈ Ctto model the constraint
imposed by the directed edges in G′. Each chunk of RBs
∈ Ctconsists of n + 2 contiguous RBs, and we denote those
RBs as Ct(0,l→r),Ct(1,l→r),...,Ct(n+1,l→r) in sequence if
the chunk is for transition from Cl to Cr (in opposite, we
denote as Ct(0,r→l),...,Ct(n+1,r→l)). For each user ul,i∈ Ul
on RB c ∈ Ct(j,l→r), we set λc
λc
c ∈ Ct(j,r→l), we set λc
i ?= j. We now encode connectivity among nodes in G′into
our construction by examining each node’s incoming edges.
For each user ur,i∈ Urfor RB c ∈ Ct(j,l→r), we first check
whether its corresponding node vr,ihas incoming edges from
any node vl,g, and sort, if any, them by g in decreasing order
(say, vl,g1,vl,g2,...). Then for c ∈ Ct(g+1,l→r)we set λc
n − g + 1 if g = g1 (i.e. the largest index), and if g ?= g1,
we set λc
g (e.g. if g = g2 then g′= g1). Lastly, we set λc
c ∈ Ct(j,l→r)if j ?= g+1. Similarly, the values λc
∈ Ulon RB c ∈ Ct(j,r→l)are set in this way. Finally, we set
the target aggregate value k = T ×2n+(n+2)(2n−1), which
is the total number of RBs. This completes the construction
of the frequencydomain status S.
i= 1 if c ∈ Cl, and λc
i= 0 if c ∈ Cr. Similarly, for each
i= 1 if c ∈ Cr, and λc
i= 0 if
ul,i= i + 1 if i = j, and
ul,i= 0 if i ?= j. Similarly, for each user ur,i∈ Ur on RB
ur,i= i + 1 if i = j, and λc
ur,i= 0 if
ur,i=
ur,i= g − g′where g′is the next larger index than
ur,i= 0 for
ul,ifor users
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
...
0
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1
1
0
0
1
1
Vl,1
Vr,1
Vl,2
Vr,2
usercarrier
T = n2
n + 2
2n*T + (2n1)(n+2) RBs
leftright transitleft
right
transit transit
Fig. 3 The intermediate construction reduced from an example
HAMPATHBG instance G′, where G′is of 4 nodes (i.e. n = 2).
Page 4
UCLA CSD TECHNICAL REPORT: TR0900014
left
right
transit
Vl,1
Vr,2
Vl,2
Vr,1
Hampath [Vl,1,Vr,1,Vl,2,Vr,2] in G’
k = max aggregate value = # RBs = 2n*T + (2n1)(n+2)
1
1 1 1 1
0
0
1
Ul,1
Ur,1
Ul,2
Ur,2
usercarrier
T = n2
n + 2
1
0
0
1
1
0
0
1
1
0
0
1
0
0
0
0
0
2
2
0
0
0
0
1
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2
3
0
1
0
0
0
0
1
0
1 1 1 1 0
0
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1
0
0
0
1
0
0
0
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0
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2
0
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1 1 1 1 0
1
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0
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0
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0
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0
0
1
2
3
3
0
1
1
0
0
0
0
1 1 1 1
1
0
0
0
1
0
0
0
1
0
0
0
1
0
left
right
transit transit
Fig. 4 The reduction from an example HAMPATHBG instance
G′, where G′consists of 4 nodes (i.e. n = 2). Darkcolored RBs
represent a satisfiable contiguity strategy with aggregate value k.
We claim that our resulting construction S has a feasible
allocation strategy if and only if G′contains a hamiltonian
path. Indeed, suppose there is a hamiltonian path in G′. The
allocation of the contiguous RB chunks to users in order of
the sequence of nodes traversing a hamiltonian path achieves
exactly the target aggregate value k, since the aggregate value
for each “transit” region can be n + 2 only when there exists
a directed edge untraversed. Such an allocation also conforms
to the contiguity constraint, so it is a feasible strategy for
S. Conversely, suppose that there is a contiguous allocation
strategy C in S. In order to achieve the target value k, every
user must be assigned in the end without being reassigned
discontiguous RBs, which forms a hamiltonian path in G.
B. Computational intractability in practice
Since we have proved in Theorem 1 that optimizing the
objective (1) is NPhard, now our last hope for optimizing
the objective (1) is probably “bruteforce” search in the sense
that it may work fine on the relatively smallsized input with
help from high computing power. That is, even though this
problem itself is NPhard, we may solve the problem by trying
all the possibilities if the size of the typical instance is small
in practice. To examine whether or not bruteforce search is
practicable, we first evaluate the running time of bruteforce
search on this problem.
Lemma 1: The running time of bruteforce search for op
timizing the objective (1) under the contiguity constraint is
O(n!) if n < m, and O(nm) if n ≥ m. (n users, m RBs)
The proof is given in the Appendix.
Unfortunately both numbers n,m are somewhat large in
practice. For example, 3GPP LTE UL is planning to support
a scalable bandwidth of 5, 10, 20 and possibly 15 MHz, each
corresponds to 25, 50, 100, and 75 RBs, respectively [3], [4].
Moreover, we may have at least several tens of active users
in a cell. Even in a sparse cell (say n = 10), it takes about
4 secs to complete the search (1 oper. ≈ 1 µs), which is too
slow to schedule data every 1 ms in the real systems. Thus,
we cannot optimize the objective (1) in practice either.
C. Upper bound of objective (1)
We conclude this section with a natural result on the upper
bound of objective (1). Let Z and Z∗be algorithms to obtain
the optimum OPT and OPT∗for the objective (1) under the
contiguity constraint and without the constraint, respectively.
Let u(c) and u′(c) be users assigned RB c by Z and Z∗,
respectively.
Lemma 2: OPT∗≥ OPT
Proof:
Since λc
OPT∗=?
contiguity constraint is at most the optimum OPT∗without
the constraint.
u′(c)≥ λc
u′(c)≥ OPT =?
u(c)for all c
cλc
cλc
u(c)
Therefore, the optimum OPT for the objective (1) under the
III. APPROXIMATION ALGORITHM
In this section we first present Alg5 to obtain 1/2
approximation for this FDPS problem under contiguous RB
constraint. This randomized approximation algorithm is how
ever too complex to be used in the practical FDPS, but we
present it here since it may give us an implication of the
approximable limits of this problem.
We let xab
i
= 1 if all the RBs between RB a and b (i.e.
contiguous RBs from a to b) are assigned to user i, and xab
0 otherwise. We then could optimize our scheduling problem
by solving the following integer program:
i =
max
?
?
?
xab
i ∈ {0,1}
i
?
?
?
a
?
xab
i ≤ 1
b≥a
?
t∈[a,b]
xab
iλt
i
subject to
a
b≥a
∀i
i
a≤t
?
b≥t
xab
i ≤ 1
∀t
∀(i,a,b) triples
We cannot solve this integer programming directly, since we
proved in Theorem 1 that optimizing our objective is NP
hard, which means this integer program is NPhard as well.
So algorithm Alg5 finds an approximation solution by using
a linear relaxation of the integer programming as follows. We
first relax the integrality constraint to read 0 ≤ xab
then we can solve the resulting linear program. This gives us
fractional values xab
i
and guarantees that the objective is at
least the integer optimum OPT:
i
≤ 1,
?
i
?
a
?
b≥a
?
t∈[a,b]
xab
iλt
i≥ OPT
We will now devise a rounding scheme to obtain integer
values for the variables, which we call ˆ xi
should satisfy all the constraints and also obtain closeto
optimum value. Suppose we have a small positive real number
ǫ. We will do the following:
1) Solve the linear relaxation of the integer program, ob
taining variables xi
2) For each i,t pair initialize Ci
3) Sort the (i,a,b) triples for which xi
order of a.
4) For each (i,a,b) triple:
a) Define ρi
ab
b) Let Pi
consider (i,a,b), we have already selected an
ab. These values
ab.
t← 0
ab> 0 in increasing
ab← αxi
abbe the probability that by the time we
Page 5
UCLA CSD TECHNICAL REPORT: TR0900015
interval2which shares the same i value or overlaps
[a,b].
c) If we have not yet selected any interval for user
i nor any interval which overlaps [a,b] then with
probability ρi
In order to bound the expected value of this rounding, we
need to bound the probability of selecting interval (i,a,b). We
will do this via the next two lemmata.
Lemma 3: Provided that 1−Pi
consider (i,a,b), the overall probability of selecting (i,a,b)
will be exactly ρi
Proof: The probability of selecting (i,a,b) is the condi
tional probability that we select (i,a,b) given that we have
not yet selected an interval which shares the same i value
or overlaps [a,b] times the probability that we have not yet
selected an interval which shares the same i value or overlaps
[a,b]. The former probability is
than or equal to one, and the latter is 1 − Pi
completes the proof.
Lemma 4: As long as α ≤
(i,a,b) we will have 1 − Pi
Proof: Let (i,a,b) be the first triple considered for which
this is not true. Since for every previously considered triple
the lemma held, all previously considered (i′,a′,b′) had selec
tion probability exactly ρi′
intervals in order of a value, any overlapping previous interval
must include a. So the probability of previously selecting an
interval with the same i or an interval overlapping [a,b] will
be bounded by:
ab/(1 − Pi
ab) select interval (i,a,b).
ab≥ ρi
abat the time we first
ab.
ρi
ab
1−Pi
abprovided this is less
ab. Multiplying
1
2, when we consider interval
ab≥ ρi
ab.
a′b′. In addition, since we consider
Pi
ab≤
?
(i,a′,b′):a′<aρi
≤ (α − ρi
The second line follows from the fact that all intervals for
i have sum of xi
including a have sum of xi
if α ≤1
a′b′ +
?
(i′,a′,b′):a′<a≤b′ρi′
a′b′
ab) + (α − ρi
ab)
abat most one (and similarly all intervals
abat most one). We conclude that
2then:
1 − Pi
ab≥ 1 − 2α + 2ρi
ab≥ 2ρi
ab
We can now bound the overall expected value of the
solution.
Theorem 2: Alg5 is a1
(1).
Proof: Assuming α =
(i,a,b) is at least
all i gives an expected value of half the linear program value.
2approximation for the PF objective
1
2, the probability of selecting
aband summing this probability over
1
2xi
IV. HEURISTIC ALGORITHMS
Although Alg5 guarantees theoretically worstcase perfor
mance bound, due to its high complexity Alg5 is impractical
for wireless scheduling in the real systems. In this section we
present a set of greedy heuristic algorithms for the objective
2Here we refer to an interval as a chunk of contiguous RBs.
L
1
L0LLLL ...L
00001100000
0
000000 ...000
userRB
A
B
OPT = L*(m1)
Alg1 = 2
˜
˜
0
Fig. 5 Bad example (2 users, m RBs) for Alg1. Darkcolored RBs
represent a resulting assignment by Alg1. L is a very large number.
(1) under contiguous RB constraint. Our greedy heuristics do
not give guaranteed error bound, and moreover we believe that
no practical greedy algorithms can give an approximation to
this particular problem (we will show it by giving bad ex
amples3). We however note that the approximation guarantee
only reflects the performance of the algorithm on the most
pathological instance which is generally not common in prac
tice. Our heuristics finetuned to the typical instances of the
problem might not perform well in their worst case scenarios,
yet their overall performanceis very good in practice, as shown
in Section V
A. Alg1: carrierbycarrier in turn
As a starter, our first greedy heuristic Alg1 is a very
natural yet coarse adaptation of algorithm Z∗that optimizes
objective (1) without the contiguity constraint (i.e. Z∗applies
PF over each RB onebyonein isolation). Emulating Z∗, Alg1
schedules data from RB1 to RBm in sequence, and for each
RB c it assigns the best user i who 1) has the maximum PF
metric value λc
ion c and 2) satisfies the contiguity constraint.
Algorithm 1 : Carrierbycarrier in turn
1: Let U be the set of schedulable users
2: Let A[m] be RBtouser assignment status
3: for RB c = 1 to m do
4:
pick the best user i ∈ U with largest value λc
5:
assign RB c to user i (i.e. A[c] ← i)
6:
Let I be RBs already assigned to user i
7:
if I = ∅ then
8:
U = U − {A[c − 1]}
9:
end if
10: end for
i
Since Alg1 schedules data from one end side RB, it is not
likely to even have a chance to try users’ high metric value
frequency portions. Figure 5 shows such a undesirable case
(this also demonstrates Alg1 cannot give an approximation).
Assignment of user B to RB2 prevents user A from being
scheduled on subsequent RBs, which would otherwise greatly
improve the result. We note that although such an extremely
bad case above is not realistic (or might not exist in practice),
this approach gives poor performance in general.
B. Alg2: largestmetricvalueRBfirst
We have shown from Alg1 that scheduling RBs in sequence
from one end side does not much help the problem. So,
viewing this scheduling problem as simply a packing problem,
adhering to its rule of thumb “pack large items first” may help
3In this section we mean by a “bad” example a problem instance where
the heuristic will lead to very bad results.