Fairness in Communication for Omniscience

0.16 MB·

by  Ni Ding

1
Fairness in Communication for Omniscience
Ni Ding∗, Chung Chan‡, Qiaoqiao Zhou‡, Rodney A. Kennedy∗ and Parastoo Sadeghi∗
Abstract—Weconsidertheproblemofhow tofairlydistribute battery usage of some users. Fairness has been considered in
the minimum sum-rate among the users in communication for [8]–[10]forCCDEandin[11]forCO.However,themethods
omniscience (CO). We formulate a problem of minimizing a
proposed in [8], [9] can only determine an integer valued
weightedquadraticfunctionoverasubmodularbasepolyhedron
fairest minimum sum-rate strategy in terms of Jain’s fairness
which contains all achievable rate vectors, or transmission
strategies, for CO that have the same sum-rate. By solving index [12], which may not be applied to general CO systems
6 it, we can determine the rate vector that optimizes the Jain’s where the transmission rates could be fractional; the method
1 fairness measure, a more commonly used fairness index than in [10] gives a fractional optimizer but relies on building a
0 the Shapley value in communications engineering. We show multi-layer hypergraph model which can be quite complex
2 that the optimizer is a lexicographically optimal (lex-optimal)
for large scale systems; the Shapley value proposed in [11]
base and can be determined by a decomposition algorithm
n (DA) that is based on submodular function minimization (SFM) incurs exponentialcomplexity and does not coincide with the
a algorithm and completes in strongly polynomial time. We prove Jain’sfairestsolution,the commonlyusedfairnessmeasurein
J
that the lex-optimal minimum sum-rate strategy for CO can be communications engineering. We will give example to show
7 determined by finding the lex-optimal base in each user subset
the difference between Jain’s fairest solution and Shapley
2 in thefundamentalpartition and thecomplexity can bereduced
value in this paper.
accordingly.

] The main purpose of this paper is to determine the fairest
T
solution in the minimum sum-rate strategy set for CO. We
I I. INTRODUCTION
. consider the problem of minimizing a weighted quadratic
s Communication for omniscience (CO) is a problem pro- function on a constraint set. The constraint set contains all
c
[ posed in [1]. It is assumed that there is a group of users achievable transmission strategies that have the same sum-
in the system and each of them observes a component of a rate which is greater than or equal to the minimal one for
1
discrete memorylessmultiple source in private. The users can CO. This problem is equivalent to determining the Jain’s
v
5 exchangetheir informationin order to attain omniscience, the fairest transmission strategy when all users are assigned the
8 state that each user obtains the total information in the entire same weight. We show that the constraint set of this problem
2 multiple source in the system. A typicalexampleis the coded is a submodular base polyhedron and the minimizer is the
7 cooperativedataexchange(CCDE)problem[2]whereagroup lexicographicallyoptimal(lex-optimal)base.Theproblemcan
0
ofgeographicallycloseuserscommunicatewitheachothervia be reduced to an SFM problem, and the lex-optimal base
.
1 error-free broadcast channels in order to recover a packet set. can be determined by a decomposition algorithm (DA) in
0 The fundamental problem in CO and CCDE is how to O(|V|2·SFM(|V|)) time, where |V| is the cardinality of the
6
achieve omniscience with minimum total transmission rate,
1 user set and SFM is the complexityof minimizing a submod-
or minimum sum-rate. The studies in [3], [4] show that the
: ularfunctionwhichisstronglypolynomial.Wealsoshowthat
v minimumsum-ratecanbeobtainedbysolvinganoptimization
thelex-optimalminimumsum-ratestrategycanbedetermined
i
X problem over the partitions of user set, while the authors in by obtaining the lex-optimal base for each user subset in the
[5],[6]proposepolynomialtimealgorithmsthatdeterminethe
r fundamentalpartition [3], the optimalpartition corresponding
a minimumsum-rateandacorrespondingstrategybasedonsub-
to the minimum sum-rate. The task of determining the lex-
modular function minimization (SFM) techniques. However,
optimal base in a user subset is less complex than in the
the minimum sum-rate strategy is not unique in general, and
ground/entire user set and can be completed in parallel in
thealgorithmsin[5],[6]utilizingEdmond’sgreedyalgorithm
distributed systems.
[7]necessarilyreturnanextremalpoint,oranunfairminimum
sum-rate strategy.
On the other hand, fairness is an important factor in CO. II. SYSTEMMODEL
For example, in CCDE, the users are considered as peers in Let V = {1,2,...} be a finite set. We assume that there
wirelesscommunications.Afairtransmissionstrategyencour- are |V| > 1 users in the system. Let Z = (Z : i ∈ V)
V i
ages them to take part in CO and helps prevent driving the be a vector of discrete random variables indexed by V. For
∗Ni Ding, Rodney A. Kennedy and Parastoo Sadeghi are with the each i∈V, user i can privately observe an n-sequenceZni of
Research School of Engineering, College of Engineering and Com- the random source Zi that is i.i.d. generated according to the
puter Science, the Australian National University (email: {ni.ding, rod- joint distribution PZ . We allow users exchange their sources
ney.kennedy, parastoo.sadeghi}@anu.edu.au). The work of Ni Ding (email: V
directly so as to let all of them recover the source sequence
[email protected]) has been done when she was a junior research
assistantwiththeInstituteofNetworkCodingatChineseUniversityofHong ZnV. Let rV = (ri : i ∈ V) be a rate vector indexed by V.
KongfromNov23,2015toFeb6,2016. We callr anachievableratevector,ortransmissionstrategy,
‡ChungChan(email:[email protected]) andQiaoqiaoZhou(email: V
if omniscience is possible by letting users communicate with
[email protected]) arewiththeInstitute ofNetwork CodingatChinese
University ofHongKong. the rates designated by rV. Let r be the function associated
2
withrV suchthatr(X)= i∈Xri,∀X ⊆V.ForX,Y ⊆V, B(fˆ4#,≤)
let H(ZX) be the amount of randomnessin ZX measured by 2 P(fˆ#,≤)
Shannon entropy and H(ZPX|ZY) be the conditional entropy Jain’4sfairestrate
of Z given Z . It is shown in [1] that an achievable rate Shapleyvalueˆr
X Y V
vector must satisfy the Slepian-Wolf constraints: r(X) ≥ r3 1 (λ,λ,λ)
H(Z |Z ),∀X ⊂V. Let α∈R and
X V\X +
H(Z |Z ) X ⊂V
f (X)= X V\X . 0
α 2
(α X =V
1 3
The polyhedron and base polyhedron of f are respectively 2
α r
2 1 r
P(f ,≥)={r |r(X)≥f (X),∀X ⊆V}, 0 0 1
α V α
B(f ,≥)={r |r ∈P(f ,≥),r(V)=α}, Fig. 1: In the CO system in Example II.1, the Shapley value
α V V α
ˆr =(8,2,2) is the gravity center of B(fˆ#,≤). The Jain’s
where B(fα,≥) contains all achievable rate vectors that have faVirest r3ate3 v3ector is r∗ = (2,1,1) 6= ˆr 4. Instead, r∗ =
sum-rate equal α. B(fα,≥) = ∅ means that there is no (2,1,1)=argmax{r(VV)|r ∈P(fˆ#,≤)V,r ≤(2,2,2V)}.
achievable rate vector that has sum-rate α. Let f#(X) = V 4 V
α
f (V) − f (V \ X),∀X ⊆ V be the dual set function of
α α
fα. We have B(fα#,≤) = B(fα,≥) [13].1 Denote Π(V) the It can be shown that B(fˆ#,≤)=B(f#,≤)=B(f ,≥) is a
set of all partitions of V. It is shown in [3], [4], [11] that f# 4 4 4
α convex region, as shown in Fig. 1, instead of a singleton.
is intersecting submodular and B(f ,≥) 6= ∅ if α is no less
α
than the minimum sum-rate In this paper, we assume that α ≥ α∗(V). We will show
H(Z |Z ) thatthis assumptionholdsfor all elementsin the fundamental
α∗(V)= max V\C C . (1) partitionP∗ inSectionIV-B.SinceB(fˆ#,≤)=B(f#,≤)=
P∈Π(V):|P|>1 |P|−1 α α
CX∈P B(fα,≥) when α≥α∗(V), we mainly present the results in
We call the maximizer of (1) the fundamental partition and terms of fˆ#. In Section VI, we will show how to solve the
α
denote by P∗. If α≥α∗(V), B(fˆ#,≤)=B(f#,≤) where fairness problem by the oracle calls of f#.2
α α α
fˆ#(X)= min f#(C), ∀X ⊆V
α α
P∈Π(X)C∈P III. MOTIVATION:JAIN’S FAIRNESS VS. SHAPLEYVALUE
X
is the Dilworth truncation of fα# and fˆα# is submodular. It is shown in [11] that a fair rate vector in B(fˆα#,≤) can
Usually, B(fα,≥) is not a singleton when it is nonempty. be determinedby the Shapley valueˆrV [14]. Each tuple rˆi of
So, a natural question that follows is to find a rate vector in ˆrV quantifies the average marginal contribution of user i in
B(fα,≥) that distributes the transmission load as evenly as CO, and ˆrV is the gravity center of B(fˆα#,≤) [14]. But, the
possible among the users. Jain’s fairest3 rate vector in B(fˆα#,≤) is the minimizer of
Example II.1. Consider the user set V = {1,2,3} and let min{ r2 |r ∈B(fˆ#,≤)} (2)
i V α
W be an independent uniformly distributed random bit. The
i i∈V
X
three users observe respectively
that does not usually coincide with the Shapley value.
Z =(W ,W ,W ,W ,W ),
1 a b c d e Example III.1. In the system in Example II.1, in B(fˆ#,≤),
Z2 =(Wa,Wb,Wf), the Shapley value is rˆV = (83,23,23), while the Jain’s4fairest
Z =(W ,W ,W ). rate vector is (2,1,1). See Fig. 1.
3 c d f
Inthissystem,itcanbeshownthatα∗(V)=3.5andB(f#,≤ It can be easily seen based on the definition of Shapley
α
) = B(f ,≥) = ∅ if α < 3.5. For α = 3.5, B(f# ,≤) = valuein[14]thatˆr isusuallybiasedsothattheuserhasmore
α 3.5 V
B(f ,≥)={(2.5,0.5,0.5)} is singleton. However, for α= informationonZ transmitsmore.Forexample,inthesystem
3.5 V
4, we have f# as in Example II.1, we have H(Z ) greater than H(Z ) and
4 1 2
f#(∅)=0,f#({1})=3,f#({2})=1,f#({3})=1, H(Z3).ThethreevertexpointsofB(fˆ4#,≤),(2,1,1),(3,0,1)
4 4 4 4 and (3,1,0), all have r greater than the other entries, and so
1
f4#({1,2})=4,f4#({1,3})=4,f4#({2,3})=3, does the gravity center ˆrV. However, in communications or
f#({1,2,3})=4, networkengineering,we wantto distributethe ratesas evenly
4
as possible without considering the users’ prior knowledge
and its Dilworth truncation as
about the sources. So, the ideal rate vector is in the form of
fˆ4#(∅)=0,fˆ4#({1})=3,fˆ4#({2})=1,fˆ4#({3})=1, λ·χV, where χX is the characteristic or incidence vector of
fˆ4#({1,2})=4,fˆ4#({1,3})=4,fˆ4#({2,3})=2, X ⊆V.IfsucharatevectordoesnotbelongtoB(fˆα#,≤),we
fˆ#({1,2,3})=4.
4 2Theoracletakes X ⊆V asaninputandoutputs fα#(X).
of1thTehedumaaljsoertitfyunstcutidoinesfoα#nCanOdaitrsebbaasseedpoolnyhtheder(oinnteBrs(efcα#tin,g≤))s,ueb.mg.,od[3u]l–a[r6it]y. to31Ja(ifna’isrefsat)irn[1e2ss].index of rV is |(VP|iP∈Vi∈rVi)r2i2 ranges from |V1| (unfairest)
3
need to at least find one that is as close as possible to it, e.g. whichdep(r∗ ,1)={1,2,3},dep(r∗,2)={2},dep(r∗ ,3)=
V V V
the rate vector (2,1,1) in Fig. 1. In this sense, the minimizer {1,2,3}andLemmaIV.1holds.Forexample,fori=1,j =2,
of (2) provides a better solution than Shapley value, and it is we have j ∈ dep(r∗,i)\{i}, ri∗ = 0.6 ≥ rj∗ = 0.5. Also,
worth discussing on how to solve problem (2) efficiently. TwV(r∗V) = (0.5,0.V6,0.6)≥lex Twwi V(rV) forwajny other rV ∈
B(fˆ#,≤). For example, for r = (2.2,1,0.8) ∈ B(fˆ#,≤),
4 V 4
IV. MAIN RESULTS we have TwV(rV) = (0.5,0.55,0.8) and TwV(r∗V) ≥lex
LetwV bea positiveweightvector.We consideraproblem TwV(rV) because the 1st entries of TwV(rV) and TwV(r∗V)
that is more general than (2): are equal while the 2nd entry of TwV(r∗V) is greater than
r2 that of TwV(rV). If we change wV to (1,1,1), we have
min{ i |r ∈B(fˆ#,≤)}. (3) r∗ = (2,1,1). It also can be shown that Lemma IV.1 holds
w V α V
i∈V i and (2,1,1) is the lex-optimal base in B(fˆ#,≤).
X 4
The objectivefunctioncan be consideredas a weightedJain’s
fairness measure in that (3) reduces to (2) when wV = B. Lex-optimal minimum sum-rate strategy
χV = (1,...,1). The main purpose of this section is to Let the fundamental partition be P∗ = {C ,...,C }
1 |P∗|
show that the fairest minimum sum-rate strategy for CO can
where ∅6=C ⊂V,∀m∈{1,...,|P∗|}. We have [3]
be determined based on the fundamental partition P∗. We m
start this section by presenting some existing results on the α∗(V)= f# (C )= fˆ# (C ).
α∗(V) m α∗(V) m
minimization problem (3) in Section IV-A, where we show CmX∈P∗ CmX∈P∗
that the minimizer of (3) is a lexicographically optimal (lex- Also, for each C ∈P∗, we have [8]
optimal)baseinB(fˆ#,≤).Basedontheseresults,wepresent m
our main result in Sαection IV-B: The lex-optimal minimum fα#∗(V)(Cm)=fˆα#∗(V)(Cm)≥α∗(Cm).
sum-ratestrategyisadirectmergeofthethelex-optimalbases
Alternatively speaking, in order to achieve omniscience with
of the user subsets in the fundamental partition P∗.
the minimum sum-rate α∗(V), each user subset C in the
m
fundamental partition P∗ must transmit exactly fˆ# (C )
α∗(V) m
A. Lex-optimal Base timeswhichis greaterthanor equalto theminimumsum-rate
A set X is called rV-tight if r(X) = fˆα#(X).4 Let for achieving the omniscience in Cm itself. Let hCm(X) =
dep(rV,i) be the smallest rV-tight set that includes i. For fˆα#∗(V)(X),∀X ⊆ Cm. For Cm,Cm′ ∈ P∗ such that
r ∈B(fˆ#,≤), dep(r ,i) can be expressed as [13] C 6= C , denote r ⊕ r = (r : i ∈ C ⊔ C )
V α V thme directm′merging ofCmr aCnmd′ r .i For exammple, mfo′r
dep(rV,i)={j |∃ξ >0: rV +ξ(χi−χj)∈B(fˆα#,≤)}, r = (r ,r ,r ) = (2,3C,m4) and rCm′ = (r ,r ) = (4,1),
where ξ(χi − χj) is called elementary transformation. It rCCmm ⊕ rC1m′ 4= 5(r1,...,r5) = (2,4C,m1′,3,4).2Th3e following
is shown in [13] that the local optimality with respect to theoremstatesthatthelex-optimalminimumsum-ratestrategy
the direction χi − χj implies the global optimality: r∗V is w.r.t.wV isadirectmergeofthelex-optimalbaseineachuser
the minimizer of (3) if i∈V (rwi∗i)2 ≤ i∈V wri2i,∀rV = subset Cm w.r.t. wCm in the fundamental partition P∗.
r∗V + ξ(χi − χj) ∈ B(fˆα#P,≤). The condiPtion is equivalent Theorem IV.3. The lex-optimal base in B(fˆα#∗(V)) w.r.t. wV
to (4) in the lemma below. is r∗ = r∗ ⊕ ... ⊕ r∗ , where P∗ is the fundamental
Lemma IV.1 ([13,Theorem8.1]). r∗V istheminimizerof (3) wpartVit.ion anCd1r∗Cm is theCle|Px∗-o|ptimalbase in B(hCm,≤) w.r.t.
iff Cm
wri∗i ≥ wrj∗j, ∀j ∈dep(r∗V,i)\{i}. (4) fˆ# Pr(oCof:) F=orhany(CrV),∀C∈ ∈B(Pfˆα∗#∗,(Vi.e).,,≤C), ris(Crm)-tig=ht
α∗(V) m Cm m m m V
Tor(d4L,e2er,ti1n)g(T(w2oV,f3(u,rs1Ve)r))in==dic(0e(s.wr5σsσ,(u(111c),)h1,..t5.h).a,twwrwrhσσσσ(e(|(|(Vr1V1e))||))σ≤)(1.w).h.=e≤re1wr,σσσ(σ((||ViV()|2|))),is=e.ga3n., Bfro′V(rh=aCllmr∗Cu,≤s1e⊕)r w.su..r.b.⊕ts.erwt∗CCC|Pmm∗.|Tiwnhhetehnre,erfr′Vu∗Cnm∈daiBms(etfhnˆαe#ta∗ll(eVxp)-a,or≤tpitt)iimonneaclPebs∗as.saerLioleyft
and σ(3) = 2. It is shown in [15] that, for r∗V and any other [13,Lemma3.1].Also,foralli∈Cm ∈P∗,dep(r′V,i)⊆Cm
rdwrVoσσm((ii∈′′i))nBaft(oefsrˆα#Ts,ow≤mV)e(,rwVi′e). ahIntadviesdewcrnσ∗aσo(l(ilit)e)edd=bTywrwσσTV((iiw)()rV,∗V∀(ri)∗V>l)exi≥′iclaeonxgdTrawwrpVσ∗σh(((iiic′r′))aVl>l)y. acbnoadnsediwrwti′ii.orn.≥t.(4wwr)j′Vji,n.∀Lje∈mmdeapI(Vr.∗V1.,iT)h\er{eif}o,rew,rh′Vichisistheeqluexiv-aolpetnimt taol
So, r∗ is called the lex-optimal base in B(fˆ#,≤) w.r.t. w . Example IV.4. Let the user set be V ={1,2,3,4}. The four
V α V
users observe respectively
Example IV.2. In the system in Example II.1, for w =
V
(4,2,1), we have lex-optimal base r∗ = (2.4,1,0.6), for Z =(W ,W ,W ,W ,W ),
V 1 c d f g h
Z =(W ,W ,W ,W ),
du4eAtontrhVe-ctiognhsttrsaeitntXr(mXe)an≤srfaˆαt#e(sXatu),raftoiornξi>ne0n,trrieVs+riξfo·rχaill∈/i∈P(Xfˆα#,i,.e≤., Z32 =(Wca,Wdd,Weg,Wfh,Wg,Wh),
),∀i ∈X. The example in Section V shows how tight set is related to the Z =(W ,W ,W ).
minimizerr∗ of(3). 4 a b f
V
4
pathtor∗ pathtor∗
V V
directionλ χ directionλ w
1 V 1 V
directionλ χ directionλ w
r∗ 2 V\S1 r∗ 2 V\S1
V V
1 B(fˆ#,≤) 1 B(fˆ#,≤)
4 4
r3 P(fˆ#,≤) r3 P(fˆ#,≤)
4 4
0 0
1 1
2 2
r2 0 0 1 r1 r2 0 0 1 r1
(a)path(0,0,0)→(0,1,1)→(2,1,1)w.r.t.wV =(1,1,1) (b)path(0,0,0)→(0,1,0)→(2.4,1,0.6)w.r.t.wV =(4,2,1)
Fig. 2: The path towards the lex-optimal r∗ w.r.t. w in the system in Example II.1.
V V
Inthissystem, we haveminimumsum-rateforCO α∗(V)=6 Example V.1. We explain the path (0,0,0) → (0,1,1) →
and the fundamental partition P∗ = {{1,2,3},{4}}. For (2,1,1) towards the lex-optimal base r∗ = (2,1,1) w.r.t.
V
C = {1,2,3}, we have h (X) = fˆ#(X),∀X ⊆ C and w =(1,1,1)inFig.2(a)byusing(5)asfollows.Weinitiate
1 C1 6 1 V
B(h ,≤) 6= ∅ because h (C ) = 5 > α∗(C ) = 4. r∗ =(0,0,0) and the direction λ·w =λ·χ =(λ,λ,λ).
C1 C1 1 1 V V V
Consider determining the lex-optimal base in B(fˆ#,≤) w.r.t. We increase λ from 0 to move along the direction χ until
6 V
w =χ =(1,1,1,1)byTheoremIV.3.Thelex-optimalbase at λ = 1 we reach a boundary of P(fˆ#,≤) that corre-
V V 4
w.r.t. w = (1,1,1) in B(h ,≤) is r∗ = (r∗,r∗,r∗) = sponds to the inequality r({2,3}) ≤ fˆ#({2,3}) = 2. Here,
(5,5,5C).1ForC ={4},r∗ =C1r∗ =1.TCh1edirect1me2rgi3ngof the boundary point (1,1,1) is necessa4rily the maximizer of
r3 3an3dr is r2∗ ⊕r∗ =C2(5,54,5,1). Itcan be shown that max{r(V) | r ∈ P(fˆ#,≤),r ≤ λ ·w = (1,1,1)}. We
rC∗1 ⊕r∗ C2is theCl1ex-opCt2imal 3bas3e3w.r.t. w = (1,1,1,1) in cannot increasVe λ any4more aloVng the1direVction λ·χ since
BC(1fˆ#,≤C)2, which is also the Jain’s fairest mVinimum sum-rate otherwise we will be out of P(fˆ#,≤) so that the reVsulting
6 4
strategy. vector does not belong to B(fˆ#,≤). For the set {2,3} that
4
is associated with the boundary, we know that we can at
V. PATH TOWARDSr∗ least allocate fˆ4#({2,3}) = 2 rates evenly to users 2 and
V 3. Therefore, by letting λ = 1 and S = {2,3}, we assign
1 1
TheproblemunsolvedinTheoremIV.3ishowtodetermine rates r∗ =λ w =1 and r∗ =λ w =1 so that the rate r∗
2 1 2 3 1 3 V
thelex-optimalbaseineachusersubset.Inthissection,weuse is updated to (0,1,1). By knowing that we cannot increase
theexistingresultsin[15]toshowthatthereexistsapiecewise any further along any dimension in S ,5 we set the direction
1
linear path towards the lex-optimal base, which can be found λ·χ =(λ,0,0) and continue to increase the value of λ
by a decomposition algorithm in the existing literature. fromVr\∗S1=(0,1,1).The purposeis to see ifwe candistribute
Let the distinct values of wri∗i of the lex-optimal base r∗V the remVaining rates fˆ4#(V)−fˆ4#(S1) = 2 evenly among the
Sbe0 =λ1,∅.a.n.,dλSNns=uc{hit|hwrai∗ti 0≤=λnλ,0i∈<Vλ}1.<We..h.a<veλalNl.SDnefnoormte iussesrestibnyVth\eSc1o.nAsttrλai=nt2r,(w{1e,r2e,a3c}h)th≤efsˆe4#co({n1d,b2o,3u}n)da=ry4thiant
a chain [15] P(fˆ#,≤). We set λ =2 and S = {1,2,3}. For S \S =
4 2 2 2 1
{3}, we assign r∗ = λ w = 2. r∗ is updated to (2,1,1).
C :∅=S ⊂S ⊂...⊂S =V 3 2 3 V
S 0 1 N Here, the second boundary point (2,1,1) is the maximizer of
suchthatri∗ =λnwi,foralli∈Sn\Sn−1.Sincedep(r∗V,i)⊆ max{r(V) | rV ∈ P(fˆ4#,≤),rV ≤ λ2 · wV = (2,2,2)}.
Sn,∀i ∈ Sn, r∗(Sn) = fˆα#(Sn). It means that we cannot Now, we have V \ S2 = ∅, which means all fˆ4#(V) = 4
increase the value of r∗ for all i∈S because otherwise the rates have been allocated. We get r∗ = (2,1,1) as the lex-
i n V
condition r∗(S ) ≤ fˆ#(S ) in the polyhedron P(fˆ#,≤) is optimal rate vector w.r.t. w = (1,1,1). In the same way, we
n α n α
breached.Alternativelyspeaking,r∗ ∧λ isthemaximizerof canshowthepath(0,0,0)→(0,1,0)→(2.4,1,0.6)towards
V n
max{r(V)|rV ∈P(fˆα#,≤),rV ≤λn·wV}[15],wherer∗V∧ r∗V =(2.4,1,0.6) w.r.t. wV =(4,2,1) in Fig. 2(b). Based on
λn·wV = (min{ri∗,λn·wi}: i ∈ V) is the componentwise this path, we have λ1 = 0.5 corresponds to S1 = {2} and
minimization of r∗V and λn·wV. It is clear that a piecewise λ2 =0.6 corresponds to S2 ={1,2,3}.
linear path in P(fˆ#,≤) towards the lex-optimal base r∗ can
α V Due to the min-max theorem [7]
be determined by considering the problem
max{r(V)|rV ∈P(fˆα#,≤),rV ≤λ·wV}, (5) max{r(V)|rV ∈P(fˆα#,≤),rV ≤λ·wV}=
min{fˆ#(X)+λw(V \X)|X ⊆V}, (6)
where λ ∈ [0,+∞). This path also determines the values of α
all S and λ .
n n
5ItmeansthatS1 isr∗V-tight, i.e.,r∗(S1)=fˆ4#(S1).
5
Algorithm 1: decomposition algorithm (DA) [16] user4inC ={4}todeterminethelex-optimalbaser∗ and
r∗ , respec2tively. C1
input : S,S in CS C2
output: the set that contains S, S and all Sn in CS such that
S ⊂Sn ⊂S VII. CONCLUSION
1 begin We considered the problem of minimizing a weighted
2 λ= fˆα#(S)−fˆα#(S); quadratic function over the set of all achievable rate vectors
w(S\S) with the same sum-rate for CO. The objective function was
3 get S as the maximal minimizer of
a generalization of the Jain’s fairness index. We showed that
min{fˆα#(X)−λw(X):X ⊆V}; if the constraint set was nonempty the optimizer was a lex-
optimal base that could be searched by a DA algorithm in
if S =S then return (S,S) else return
DA(S,S)∪DA(S,S) stronglypolynomialtime.Wealsoshowedthatthelex-optimal
4 end minimum sum-rate strategy for CO could be determined by
directly merging the lex-optimal bases for all user subset in
the fundamental partition, which could be completed in a
problem (5) reduces to min{fˆ#(X) − λw(X) | X ⊆ V}, decentralized manner with lower complexity.
α
which is an SFM problem and can be efficiently solved by
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1