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UCLA CSD TECHNICAL REPORT: TR-0900011

Proportional Fair Frequency-Domain Packet

Scheduling for 3GPP LTE Uplink

Suk-Bok 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, Single-carrier FDMA (SC-FDMA) has been

selected for 3GPP Long-Term 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 frequency-domain 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

SC-FDMA uplink scheduling by adopting the conventional time-

domain Proportional Fair algorithm to maximize its objective

(i.e. proportional fair criteria) in the frequency-domain setting.

We show the NP-hardness of the frequency-domain 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 Long-Term 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 peak-to-average 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 (SC-FDMA) 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 single-carrier. Thus, lower PAPR of

SC-FDMA greatly benefits the mobile terminal in terms of

transmit power efficiency.

As in DL OFDMA, multiple access in UL SC-FDMA

is achieved by assigning different frequency portions of the

system bandwidth to individual users based on their channel

conditions. Such simultaneous frequency-domain multiplexing

of users (inherently in concert with time-domain 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 frequency-domainadaptive

user-to-RB 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 RB-to-user assignment at each time slot (e.g.

in LTE every 1ms) according to the selected scheduling policy.

Thus, in the time-frequency domain, an RB is considered as

a minimum scheduling resolution, and also a minimum unit

of the data-rate adaptation by adaptive modulation and coding

(AMC) with a granularity of one sub-frame.

Most of the DL FDPS algorithms proposed so far adopt

the well-known time-domain Proportional Fair (PF) algorithm

as a basic scheduling principle and apply the PF algorithm

directly over each RB one-by-one independently. However,

such scheduling strategies cannot be employed in the UL SC-

FDMA. Due to its single carrier property, SC-FDMA requires

that all the RBs allocated to a single user must be contiguous

in frequency within each time slot (i.e. sub-frame) [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

frequency-domain 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 frequency-domain setting.

The main goal of this paper is to investigate how to adapt the

time-domain 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: TR-0900012

moreover, they also depends on the user location and the time

slot. Therefore, each RB has user-dependent and time-varying

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 time-domain 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 high-quality channel rate (ri(t)) and a low current

average service rate (Ri(t)).

We now adapt this time-domain 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 long-term 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 one-by-one, 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 time-domain

setting. Hence, optimizing the objective (1) makes the util-

ilogRi maximized in the frequency-domain

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

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1 4

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1 1

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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

1 6

1 3

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1 2

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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. Dark-colored 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 one-by-one 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 PF-FDPS 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 single-carrier

wireless systems such as CDMA 2000 1xEV-DO [11], [15].

The area of FDPS scheduling is new, and most of studies

directly adapt the time-domain PF algorithm into frequency-

domain context. Their results show the potential gains of up

to 40-60% 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 short-term fairness. Andrews et al. [10] have

proposed the FDPS-version 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

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UCLA CSD TECHNICAL REPORT: TR-0900013

dynamically allocate the variable-width 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 SC-FDMA.

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 PF-FDPS problem (i.e. maximization

of the PF objective (1) under the contiguous RB allocation

constraint) is NP-hard.

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 NP-complete [16].

As a pre-processing 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 (HAM-PATH-BG) is reducible to our problem. A

decision version of our problem is to determine whether for

a given frequency-domain status S (i.e. a collection of value

λc

allocation strategy with resulting aggregate value at least k.

Consider an arbitrary instance of HAM-PATH-BG, 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 frequency-domain, 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 frequency-domain

ifor RBs ∈

A

C

D

B

Ham-path [A,B,D,C] in G

Al

Br

Bl

Ar

Cl

Dr

Dl

Cr

Ham-path [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 frequency-domain 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 once-assigned users from being

re-assigned 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 frequency-domain 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

0

1

1

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1

...

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...

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

Vl,1

Vr,1

Vl,2

Vr,2

usercarrier

T = n2

n + 2

2n*T + (2n-1)(n+2) RBs

leftrighttransitleft

right

transittransit

Fig. 3 The intermediate construction reduced from an example

HAM-PATH-BG instance G′, where G′is of 4 nodes (i.e. n = 2).

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UCLA CSD TECHNICAL REPORT: TR-0900014

left

right

transit

Vl,1

Vr,2

Vl,2

Vr,1

Ham-path [Vl,1,Vr,1,Vl,2,Vr,2] in G’

k = max aggregate value = # RBs = 2n*T + (2n-1)(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

2

2

3

0

1

0

0

0

0

1

0

1 1 1 1 0

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3

3

0

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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 HAM-PATH-BG instance

G′, where G′consists of 4 nodes (i.e. n = 2). Dark-colored 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 re-assigned

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 NP-hard, now our last hope for optimizing

the objective (1) is probably “brute-force” search in the sense

that it may work fine on the relatively small-sized input with

help from high computing power. That is, even though this

problem itself is NP-hard, 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 brute-force search is

practicable, we first evaluate the running time of brute-force

search on this problem.

Lemma 1: The running time of brute-force 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 NP-hard 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 close-to-

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

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UCLA CSD TECHNICAL REPORT: TR-0900015

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.

2-approximation for the PF objective

1

2, the probability of selecting

aband summing this probability over

1

2xi

IV. HEURISTIC ALGORITHMS

Although Alg5 guarantees theoretically worst-case 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*(m-1)

Alg1 = 2

˜

˜

0

Fig. 5 Bad example (2 users, m RBs) for Alg1. Dark-colored 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 fine-tuned 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: carrier-by-carrier 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 one-by-onein 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 : Carrier-by-carrier in turn

1: Let U be the set of schedulable users

2: Let A[m] be RB-to-user 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: largest-metric-value-RB-first

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.