The Optimality of Partial Clique Covering for Index Coding

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by  Xinping Yi

1
The Optimality of Partial Clique Covering
for Index Coding
Xinping Yi and Giuseppe Caire
Technical University of Berlin, 10587 Berlin, Germany
{xinping.yi, caire}@tu-berlin.de
Abstract—Partial clique covering is one of the most basic to five messages were fully characterized, and by one-to-
coding schemes for index coding problems, generalizing clique one/subspace interference alignment, the broadcast rate of a
7 and cycle covering on the side information digraph and further family of index coding problems was identified under the
1 reducing the achievable broadcast rate. In this paper, we start
equivalent topological interference management (TIM) setting
0 with partition multicast, a special case of partial clique covering
2 withcovernumber1,andshowthatpartitionmulticastachieves [6]. Other attempts using linear programming formulation
the optimal broadcast rate of the multiple-unicast index coding were also made to improve the basic achievability schemes
n if and only if the side information digraph is partially acyclic. A through fractionalization, localization, and generalization, such
a digraphissaidtobepartiallyacyclicifitssub-digraphinducedby
J as fractional clique covering [16], (fractional) local graph
the vertex with maximum in-degree and its incoming neighbors
coloring [11], [25] as well as the recent development of
2 inthecomplementarydigraphisacyclic.Wefurtherextendtothe
2 generalpartialcliquecovering,offeringsufficientconditionsofits fractional local partial clique covering [26] and interlinked
optimalityandsub-optimalitywiththeaidofstrongconnectivity cyclecovering[27],amongmanyothers.Whiletheboundaryof

] decomposition. In addition, for some digraph classes, we also our knowledge towards the best possible achievability schemes
T
prove that the optimal broadcast rate can be approximated by
is pushed ahead, it is perhaps hopeless to expect that a single
I partialcliquecovering(aswellasbyotherbasicschemes)within
. schemeachievestheoptimalbroadcastrateforallindexcoding
s eitheraconstantfactor,oramultiplicativefactorofO( n ),or
c O(n(cid:15)) for some (cid:15)∈(0,1). logn instances.
[ Anotherlineofworkconsistsofevaluatingtheoptimalityof

IndexTerms—IndexCoding,PartialCliqueCovering,Partition
1 themostbasicachievabilityschemesathand,byidentifyingthe
Multicast, Confusion Graph
v fundamental structure for which such schemes are sufficient
1 and/or necessary to achieve the optimal broadcast rate. In
8 I. INTRODUCTION doing so, we could understand better the pros and cons
1
The index coding problem is one of the most intriguing of the available tools, and at the same time reduce the
6
0 open problems in network information theory and theoretic unsolvedproblemspace.Inparticular,[28]showedthatchordal
. computerscience,becauseofitsrichconnectionstodistributed network topology is the fundamental topological structure that
1
storage [1], [2], coded caching [3], network coding [4], [5], determines the optimality of fractional clique covering on the
0
7 topological interference management [6], and hat guessing side information digraph (or fractional vertex coloring on the
1 games[7].Themultiple-unicastindexcodingproblemconsiders underlying undirected graph of its complement). Specifically,
v: a noiseless broadcast transmission, where each receiver wants fractional clique covering achieves the all-unicast capacity
i one unique message from the transmitter and holds some other region of index coding problems if and only if the bipartite
X
receivers’ desired messages as side information. The goal is to network topology (see Definition 1) is chordal. Such an all-
r characterizetheminimumnumberoftransmissions,namelythe unicast setting includes all possible unicast messages. The
a
broadcast rate, such that all receivers are able to decode their characterizationoftheall-unicastcapacityregionautomatically
desired messages synchronously. Index coding has attracted yields the characterization of the capacity region for any
extensive research efforts from various fields using tools such arbitrarysubsetofunicastmessages,aswellasothertraditional
asgraphtheory[8]–[14],interferencealignment[6],[15],linear metricssuchassumorsymmetriccapacity(c.f.broadcastrate).
programming [16], [17], matrix completion [18], [19], random So, when fractional clique covering achieves the all-unicast
coding [20] as well as source coding [21], to name just a few. capacity region, it achieves the capacity region of any arbitrary
Although the index coding problem is very simple to subset of messages as well. With respect to side information
describe, it is very hard to solve in general. The difficulty digraphs, the chordal network topology implies the absence of
lies in the necessity of not only nonlinear coding schemes [22] directed cycle of length no less than 3, which turns out to be
but also non-Shannon information inequalities [23], neither one of the fundamental topological structures that determine
of which is well understood so far. The majority of the the optimality of clique covering.
progress made in the past decade consists of improving Beyond this, however, the fundamental topological structure
over the basic coding schemes (e.g., clique covering, cycle that determines the optimality of off-the-shelf achievability
covering,andpartialcliquecovering).Amongvariousadvanced schemes is still far less understood. On one hand, while the
achievability schemes, by composite coding [20] as well as sufficiency of the optimality of some sophisticated schemes,
linear coding schemes [24], all index coding instances up suchasinterferencealignment[6],[15],compositecoding[20],
2
and interlinked cycle covering [27], has been demonstrated for cliquecoveringcanorcannotachievetheoptimalbroadcastrate,
a family of network topologies, the necessity is still not known withtheaidofstrongconnectivitydecomposition.Wealsoshow
in general. On the other hand, while approximate capacity in Section III-C that (partial) clique covering approximates the
resultswerealsoreportedrecentlyin[29]forafamilyofindex broadcast rate for certain classes of directed graphs within
codingproblems,wherethebroadcastratecanbeapproximated either a constant factor, or a multiplicative factor of O( n ),
logn
by clique covering within a multiplicative factor of n23, the or O(n(cid:15)) for some (cid:15)∈(0,1), with the aid of the connection
main focus was merely on the undirected side information between the broadcast rate of index coding and the acyclic set
graphs, such as planar and line graphs, so that very little is coloring of side information digraphs.
known for the directed counterpart.
Partial cliques (see Definition 3) are a generalized version
of cliques and cycles [8]. Partial clique covering in general II. BACKGROUND
offers a smaller broadcast rate than clique or cycle covering,
In what follows, some preliminaries pertaining to index
becoming one of the most important and basic achievability
coding [8], confusion graph [9], [12], and partial clique
schemes. Partition multicast is a special case of partial clique
covering [8] are briefly recalled. The relevant graph theoretic
covering with cover number 1. As partial clique covering
definitions can be found in Appendix.
involves a complex combinatorial optimization problem, the
characterization of the optimality of partition multicast is
expected to lay down the fundamental building block of
the understanding of the general partial clique covering. So, A. Index Coding
a natural question is then under what condition partition
Consider a message set {0,1}t, from which a set of n
multicastachievestheoptimalbroadcastrate.Inthispaper,we
messagesisuniformlyandrandomlychosen.Let{1,...,n}be
answer such a question – partial acyclicity is the fundamental
themessageindexset.Letx(S)bethesetofmessagesindexed
topologicalstructurethatdeterminestheoptimalityofpartition by the subset S, where S ⊆[1:n](cid:44){1,...,n}. We use the
multicast. A digraph, as a single partial clique, is said to be
shorthand term x to represent the individual message x({j}),
j
partiallyacyclic(seeDefinition5)ifthesub-digraphinducedby
and the n-tuple xn to represent the whole set of message
thevertexwithmaximumin-degreeanditsincomingneighbors
x([1 : n]). The network topology of index coding is formed
in the complementary digraph is acyclic. In other words,
by a transmitter and n receivers, connected by a unit-capacity
partial acyclicity in digraphs indicates the absence of directed
sharednoiselesslink.Eachreceiverj hasasideinformationset
cycles in the dominant partial clique (see Definition 4). This
S , i.e., it already knows the messages x(S ). For the multiple-
j j
characterization leads to sufficient conditions for the optimality
unicastindexcodingproblem,eachreceiverj wishestoreceive
and the sub-optimality of partial clique covering. On one hand,
message x , aiming at the minimum number of transmission
j
partial clique covering is optimal, if every strongly-connected
(i.e., the minimum number of channel use of the unit-capacity
component (see Appendix) of the side information digraph is
noiseless link). Obviously, j ∈/ S , otherwise the demand is
j
partially acyclic. On the other hand, partial clique covering
trivial. The goal consists of the construction of a (t,r) index
is suboptimal if neither every strongly-connected component
codeC basedonthesideinformationindexsets{S ,...,S },
t 1 n
is partially acyclic, nor the strongly-connected components
including the following encoding/decoding functions:
that are not partially acyclic are divisible (see Definition 6).
A strongly-connected component is a partial clique, such that • An encoding function, φ:{0,1}tn×{2[1:n]}n (cid:55)→{0,1}r
every vertex has a directed path to any other vertices in this at the transmitter that encodes n-tuple of messages xn to
partial clique. Note that strong connectivity of a digraph is a length-r index coding codeword, i.e.,
an equivalence relation of the set of its vertices. As such, the
φ(xn,S ,...,S )(cid:55)→{0,1}r; (1)
partitioning of a digraph into strongly-connected components 1 n
(i.e., equivalent classes) is unique, in the sense that any vertex
belongs uniquely to one component. A digraph is said to be • A decoding function at each receiver j, ψj : {0,1}r ×
divisible if there exists a further partition such that the sum of {0,1}t|Sj| (cid:55)→{0,1}t, ∀j that decodes the received index
codingcodewordbacktox basedonthesideinformation
the minimum in-degree of each portion is strictly increased. j
x(S ) held at the receiver j. That is, an index code is
The rest of the paper is organized as follows. In the next j
feasible if
section, we introduce the multiple-unicast index coding prob-
lem, followed by the definition of confusion graph, and some
ψ (φ(xn,S ,...,S ),x(S ))=x , (2)
conceptsrelatedtopartialcliquecovering.InSectionIII-A,we j 1 n j j
startwithpartitionmulticast,andcharacterizethesufficientand
for all j ={1,...,n}.
necessary condition under which partition multicast achieves
the optimal broadcast rate. That is, the dominant partial clique The achievable broadcast rate of the index code C is
t
(see Definition 4) in the side information digraph is acyclic. β(C )= r. A broadcast rate β(C) is achievable for a family
t t
We show that partial acyclicity is the fundamental topological of index codes C given a configuration of the side information
structure that yields the optimality of partition multicast. sets {S ,...,S }, if there exists a sequence of feasible codes
1 n
Further,weextendinSectionIII-Btothegeneralpartialclique C = {C ,t ∈ N}, such that inf β(C ) ≤ β(C). The optimal
t t t
covering, identifying sufficient conditions for which partial broadcast rate (for given configuration of side information) is
3
the infimum of all achievable rates, defined as 1 information graphs G¯ are considered, using respectively the
acyclic set number α (D¯) and the independent set number
β =intfiCntfβ(Ct) (3) α(G¯). The definition oAf these two parameters can be found in
Appendix.
where the second infimum is over all (t,r) index codes C .
t
Sometimes, 1 is also referred to as the symmetric capacity, or Lemma 1 (Lower Bounds [8]). Consider an instance of the
β
the optimal symmetric degrees-of-freedom in the TIM setting. index coding problem with the undirected side information
graph G¯ and the side information digraph D¯. Then, the
Definition 1 (Graphic Representations). The index coding
broadcast rate satisfies the Maximum Independent Set (MIS)
problem with message index set V and side information index
bound and the Maximal Acyclic Induced Subgraph (MAIS)
sets {S ,∀j} can be fully represented by any of the following
j
bound, given respectively by
three graphs:
• Conflict digraph D = (V,A): A directed graph with β ≥α(G¯), β ≥αA(D¯). (4)
message index set V being vertices and an arc (i,j) ∈
A(D) if and only if i∈/ S ;
j B. Confusion Graph
• Side information digraph D¯ =(V,A¯): The complement
of the conflict digraph, with an arc (i,j)∈A¯(D¯) if and The confusion graph, first introduced in [9], is an undirected
graphwithverticesbeingthemessagetuples,andwithanedge
only if i∈S ; 2
j
between two vertices if the two corresponding message tuples
• Network topology H=(U,V,E): An undirected bipartite
are confusable. Given the n-tuple messages xn,zn ∈{0,1}tn,
graph with |U| = |V|, and an edge (u ,v ) ∈ E(H) if
i j
theyaresaidtobeconfusableatnodej ifx (cid:54)=z andx =z
and only if i=j or (i,j)∈A(D). The network topology j j i i
for all i∈S . Intuitively, two message tuples are confusable at
can also be represented instead by the sets {T ,v ∈V}, j
j j
where T (cid:44){i:(u ,v )∈E}=N−(D). node j if the receiver j cannot distinguish them only based on
j i j vj its side information S . This means that in order to determine
j
If the arcs in digraphs D and D¯ are bi-directed (see the message, some additional information (a codeword) must
Appendix), we also use respectively the undirected graphs be sent by the transmitter for disambiguation. Two message
G and G¯with the arc direction ignored to represent the conflict tuples xn,zn are confusable if they are confusable at some
graph and the side information graph. node j.
Here we present an alternative construction w.r.t. conflict
Example 1. In this example, we introduce a typical structure,
digraphs, which will be very useful for our proof. Given
referred to as the triangular network, which will be useful
two n-tuples xn and zn, let Q(xn,zn) (cid:44) {i : x (cid:54)= z , ∀i}
later. The triangular network topology H = Tri(K) refers i i
be the set of indices of the difference, and Q be the set of
to a bipartite graph with |U| = |V| = K and (i,j) ∈ E(H)
vertices indexed by Q(xn,zn). Here, N−(D) is the closed
for all i ≤ j, corresponding to the conflict digraph where v
in-neighborhood (see Appendix) of the vertex v in the digraph
(i,j)∈A(D) if and only if i<j. A triangular network Tri(4)
D.
with above three graphic representations is depicted in Fig. 1.
Definition 2 (Confusion Graph). For the conflict digraph D =
(V,A) and an integer t, the confusion graph Γ (D) is an
t
undirected graph with 2tn vertices (i.e., the n-tuple messages),
two of which are connected with an edge if and only if the
corresponding messages are confusable.
Given two message tuples xn and zn, they are confusable if
and only if there exists j ∈Q(xn,zn), such that Q(xn,zn)⊆
S¯ where S¯ = N−(D), or more specifically, if and only if
j j j
one of the following two conditions is satisfied:
Fig. 1: Three graphic representations of an index coding instance. (a)
The conflict digraph D, (b) the side information digraph D¯, and (c) (1) |Q(xn,zn)|=1;
the network topology H=Tri(4). (2) |Q(xn,zn)| ≥ 2, and there exists j ∈ Q(xn,zn) such
that ∀i∈Q(xn,zn)\{j}, (i,j)∈A(D).
Alternatively, two message tuples xn and zn are non-
In what follows, we introduce two lower bounds to the
confusableifandonlyif∀j ∈Q(xn,zn),Q(xn,zn)∩S (cid:54)=∅,
broadcast rate of the general index coding problems in the j
or specifically if and only if all vertices in the induced sub-
literature, represented by two graph theoretic parameters. Both digraph D¯[Q] have positive in-degree.
the side information digraphs D¯ and the undirected side
Remark 1. The above alternative construction of confusion
1Throughoutthepaper,anupperboundofthebroadcastrateindicatesan graphs is equivalent to the classical definition in [9]. When
achievablerateandisusuallyassociatedwithanachievabilitycodingscheme,
Q(xn,zn) = {i}, xn and zn are confusable at i regardless
alowerboundindicatesthatanyvaluebelowitcannotbeachievedbyany
codingscheme,andthentheequalitydenotestheoptimalbroadcastratein of the side information. When |Q(xn,zn)| ≥ 2, there exists
theinformationtheoreticsense. j ∈Q(xn,zn) such that ∀i∈Q(xn,zn)\{j}, (i,j)∈A(D),
2Noteherethatweadoptthemoreintuitiverepresentationdrivenbythe meaning that for all i∈Q(xn,zn)\{j}, i∈/ S and then for
interferencemanagementperspective,suchthattheconflictandsideinformation j
digraphshavearcswithdirectionsoppositetowhatwasoriginallyusedin[8]. all k ∈Sj, xk =zk. Thus, xn and zn are confusable at j. ♦
4
According to Definition 2, we have the following three above condition is equivalent to that if and only if ∀v ∈V[Q],
q
important observations for any two message tuples xn and |T | ≤ k+1, and ∃v∗ ∈ V[Q], |T | = k+1. The directed
vq q vq∗
zn, which are very useful for the proof of our main theorems. acyclic graph is a (n−1)-partial clique, the directed cycle is
Here, ∆−(D) denotes the maximum in-degree of the vertex a (n−2)-partial clique, and the clique is a 0-partial clique.
in D (See Appendix).
Lemma 3 (Achievable Broadcast Rate via Partial Clique
• Observation 1: If |Q(xn,zn)| ≥ ∆−(D)+2, then xn Covering[30]). Thebroadcastrateoftheindexcodingproblem
and zn are non-confusable, because there does not exist with side information digraph D¯ =(V,A¯) satisfies
any j such that Q(xn,zn) ⊆ N−(D) with |N−(D)| ≤
j j
∆−(D)+1. (cid:88)s
β ≤ min (k +1), (6)
• Oznbsaerrevacotinofnus2a:bIlfe|Qif(xannd,zonn)l|y=if∆Q−((xDn),+zn1),t=henNx−n(aDn)d, {V1,...,Vs}i=1 i
v∗
where v∗ = argmax |N−(D,v)|, because otherwise where the upper bound is achievable by partial clique
v∈V
theredoesnotexistanyj suchthatQ(xn,zn)⊆N−(D). covering, the minimum is over all possible partitions of
• Observation 3: If D¯[Q] is exactly a directed cyclej, then V = {V1,...,Vs}, and D¯[Vi] is a ki-partial clique for
xn and zn are non-confusable, because directed cycles i=1,...,s.
have positive in-degree for all nodes.
For each k -partial clique, a k + 1 Maximum Distance
i i
As an example, a conflict digraph D of an index coding Separable (MDS) code can make all the desired messages in
problem and its corresponding confusion graph Γt(D) with this partial clique recoverable. In this paper, we only consider
t=1 are given in Fig. 2(a) and Fig. 2(b), respectively. non-fractional covering, where partial clique covering boils
downtographpartitionintopartialcliques.Still,characterizing
the achievable broadcast rate via partial clique covering is
challenging, as the best partial clique partition to minimize the
sum is a complex combinatorial optimization problem.
The partition multicast [31] is a special case of partial
clique covering when s = 1. Roughly speaking, partition
multicastconsistsofsendinglinearlyindependentcombinations
of messages such that the receiver which collects enough
number of linear combinations can recover its desired message.
From a topological interference management perspective [6],
Fig. 2: (a) The conflict digraph D of an index coding instance partitionmulticastisalsoreferredtoasCodeDivisionMultiple
with ∆−(D)=1, and (b) its confusion graph corresponding Γ(D). Access (CDMA).
According to Observation 1, two codewords with Hamming distance
no less than 3 are non-confusable, and according to Observation 3, if Lemma 4 (Achievable Broadcast Rate via Partition Multicast).
the positions of difference of two codewords are exactly a directed The broadcast rate of the index coding problem with conflict
cycle in side information graphs D¯, then they are non-confusable. digraph D satisfies
β ≤∆−(D)+1, (7)
Lemma 2 (Broadcast Rate via Confusion Graph Construction
[9], [12]). The broadcast rate of the index coding problem where the upper bound is achievable by partition multicast,
with confusion graph Γ (D) is and ∆−(D) is the maximum number of incoming arcs of any
t
vertex in D (see Appendix).
log(χ (Γ (D)))
β = f t , (5)
t From Definition 3 and Lemma 3, we find that the vertex
with the minimum incoming arc in the side information
for some t.
digraph D¯ (i.e., the vertex with maximum incoming arcs in
the conflict digraph D) dominates the achievable broadcast
C. Partial Clique Covering
rate. We introduce the following definition to represent this
Let us first introduce the notion of partial cliques defined in dominant vertex and its neighbors.
[30] dedicated to side information digraphs. Here, d−(D,v) is
Definition 4 (Dominant Partial Clique). The dominant partial
the in-degree of the vertex v in the digraph D (see Appendix).
clique in a digraph D¯ is the induced sub-digraph D¯[N−(D)],
v∗
Definition 3 (Partial Clique). A side information sub-digraph where N−(D) is the largest closed in-neighborhood in its
v∗
D¯[Q] induced by the vertex set Q is a k-partial clique if and complement digraph D, i.e., v∗ =argmax |N−(D,v)|.
v∈V
only if ∀v ∈Q, d−(D¯[Q],v )≥|Q|−k−1, and ∃v∗ ∈Q,
q q q In words, the dominant partial clique of a digraph is the
d−(D¯[Q],v∗)=|Q|−k−1.
q sub-digraph induced by the vertex with maximum in-degree
With respect to the conflict sub-digraphs D[Q], the above and its incoming neighbors in the complementary digraph. The
condition is equivalent to that if and only if ∀v ∈ Q, dominant partial clique in D¯ is a ∆−(D)-partial clique with
q
d−(D[Q],v ) ≤ k, and ∃v∗ ∈ Q, d−(D[Q],v∗) = k. With ∆−(D)+1 vertices. The side information digraph in Fig. 1(b)
q q q
respect to the network topology H[Q]=(U[Q],V[Q],E), the is a dominant partial clique. A side information digraph can
5
havemultipledominantpartialcliques,whentherearemultiple contraposition, we need to prove that, if the side information
vertices v∗ with the same maximum in-degree in D. digraph D¯ is not partially acyclic, then partition multicast
cannot achieve the optimal broadcast rate. To prove this, we
Definition 5 (Partial Acyclicity). A partial clique is said to
show that, if every dominant partial cliques contains a directed
be partially acyclic in a digraph D¯ if there exists at least one
cycle, there exist coding schemes that achieve strictly less
dominant partial clique D¯[N−(D)] that is acyclic.
v∗ broadcast rate. Without knowing the structure of other parts
Remark2. Forthecasewithmultipledominantpartialcliques of the side information digraph, none of the coding schemes
inadigraph,aslongasoneofthemisacyclic,thenthedigraph that we know so far can guarantee this. So, we resort to the
is partially acyclic. The property of partial acyclicity is also constructionoftheconfusiongraph,showingthatsuchacoding
referred to as the absence of directed cycles in at least one of scheme does exist, although it is not explicitly constructed. By
the dominant partial cliques. ♦ the construction of the confusion graph Γ (D), we show that,
t
if every dominant partial clique contains a directed cycle, the
Definition 6 (Divisibility). Given a digraph D¯ = (V,A¯), it
fractional chromatic number of the confusion graph satisfies
is said to be divisible if there exists a further partition V =
χ (Γ (D)) < 2t(∆−(D)+1). By Lemma 2, we conclude that
{V ,...,V } with V ∩V =∅, ∀i(cid:54)=j such that f t
1 s i j there exists a coding scheme with β < ∆−(D)+1, making
δ−(D¯[V])<(cid:88)s δ−(D¯[V ]), (8) partitionmulticaststrictlysuboptimal.Thiscompletestheproof
i of the necessity.
i=1
Remark 3. The condition that a dominant partial clique
where δ−(D) is the minimum number of incoming arcs over
D¯[N−(D)]isacyclicisequivalenttothatthenetworktopology
all vertices in D. v∗
containsatriangularnetworkTri(∆−(D)+1)asasubnetwork.
According to the fact that As long as the above condition is satisfied, more conflict for
D, less side information for D¯, or more interference for H
s s
(cid:88) (cid:88)
min (k +1)= min (∆−(D[V ])+1) doesnotincreasethebroadcastrate.Theunderlyingundirected
i i
{V1,...,Vs}i=1 {V1,...,Vs}i=1 conflict graph is a clique with size ∆−(D)+1, which is not
s changed by these operations. ♦
=|V|− max (cid:88)δ−(D¯[V ]),
i
{V1,...,Vs}i=1 Through the proof of Theorem 1 in Section IV, we obtain
immediately the following corollary.
we conclude that a partial clique is divisible indicates that
there exists a further partition by which partial clique covering Corollary 1. If the dominant partial clique D¯[Nv−∗(D)] con-
further decreases the broadcast rate. It is easy to verify that, tainsexactlyonedirectedcycle,thenthebroadcastratesatisfies
if D¯ is partially acyclic, then D¯ is not divisible. So, directed ∆−(D)≤β <∆−(D)+1.
acyclic graphs, directed cycles, and cliques are not divisible.
Remark 4. The necessity proof of Theorem 1 mainly focus
ontheexistenceproofofanindexcodethatachievesbroadcast
III. MAINRESULTS rate strictly smaller than that achieved by partition multicast,
A. The Optimality of Partition Multicast if D¯[N−(D)] contains exactly one directed cycle. In this case,
v∗
it also shows that the best improvement (reduction) of the
The main result is a sufficient and necessary condition when
achievable broadcast rate of such an index code over partition
partition multicast achieves the optimal broadcast rate.
multicast is no better than ∆−(D) , although the upper bound
∆−(D)+1
Theorem 1 (Optimality of Partition Multicast). Partition might be not tight for some special cases. ♦
multicastachievestheoptimalbroadcastrate,ifandonlyif,the
side information digraph D¯ is partially acyclic. The optimal Example 2. Consider the example in [30, Proposition 4] with
broadcast rate of such a family of index coding problems is side information digraph in Fig. 3. The partial clique covering
yieldsachievablebroadcastrateof3,whilelocalgraphcoloring
β =∆−(D)+1. (9) achieves broadcast rate of 2. This is confirmed by the above
corollary, where ∆−(D) = 2, v∗ = {1,2,4} and the sub-
Proof: The sufficiency is straightforward, while the neces-
digraphinducedbypartialcliquesN− ={3,4},N− ={3,4}
sity is non-trivial. Here we present a sketch of the proof, and 1 2
and N− ={1,5} contain directed cycles in D¯. ♦
relegate the detailed proof to Section IV. 4
Sufficiency: For the “if” part, we need to show that, if The(K,L)regularnetworkreferstoafamilyofindexcoding
there exists one dominant partial clique in the side information problems with network topology {T ={j,j+1,...,j+L−
j
digraph D¯ that is acyclic, then partition multicast achieves the 1} mod K},∀j}. As the above partial acylicity condition is
optimal broadcast rate. As the size of the acyclic dominant satisfied, we characterize the broadcast rate as follows.
partialcliqueis∆−(D)+1,byLemmas1and4,theachievable
Corollary 2. For the (K,L) regular networks, partition
broadcast rate by partition multicast matches the MAIS lower
multicast achieves the optimal broadcast rate β =L.
bound. This completes the proof of the sufficiency.
Necessity: For the “only if” part, we need to show that, Theaboveresultconfirmsthosein[15],revealingthatpartial
partition multicast achieves the optimal broadcast rate, only acyclicity makes interference alignment unnecessary. The side
if the side information digraph D¯ is partially acyclic. By information digraph of a (|Q|,L) regular network topology is
6
where D¯[V ] is a k -partial clique.
i i
Proof: Before the proof, we show that the strong de-
composition V ={V ,...,V } is unique and any two SCCs
1 s
are disjoint. As the strong connectivity of a digraph is an
equivalence relation of the set of its vertices, the partitioning
of a digraph into SCCs (i.e., equivalent classes) is unique.3
This unique strong decomposition into disjoint SCCs provides
us a partition that is most suited for partial clique covering.
Fig. 3: (a) The side information digraph D¯ for the example in [30,
As such, we need to show that, if there always exists a
Proposition4],and(b)itsconflictdigraphD with∆−(D)=2anda
dominantpartialcliqueineachSCCthatisacyclic,thenpartial
proper coloring with {A,B,C} indicating distinct colors where the
number of local (in-neighborhood) colors for each vertex is 2. clique covering achieves the optimal broadcast rate.
Given the unique strong decomposition V ={V ,...,V },
1 s
foranyi,D¯[V ]isstrongly-connected,andallSCCsaredisjoint.
i
a (L−1)-partial clique (1≤L≤|Q|). When L=1, D¯[Q] is As such, V ={V ,...,V } is a valid partial clique partition.
1 s
a clique; when L=|Q|−1, D¯[Q] is a directed cycle. All the As D¯[V ] is a k -partial clique, the broadcast rate satisfies
i i
dominant partial cliques of the regular (L−1)-partial cliques β ≤ (cid:80)s (k +1), where the upper bound is achieved by
i=1 i
D¯[Q] are acyclic. partial clique covering with partition V ={V ,...,V }.
1 s
For the converse, as V = {V ,...,V } is a strong decom-
Example 3. Consider an index coding problem with regular 1 s
position, the strong component digraph S(D) (see Appendix)
network topology as shown in Fig. 4. For every node, given is acyclic [32]. For every portion D¯[V ], there exists a directed
the conflict sub-digraph induced by its closed in-neighborhood i
acyclicsetwithsizek +1,becausethedominantpartialclique
(in purple) in Fig. 4(b), its complement yields the dominant in the k -partial cliqueiD¯[V ] is acyclic. The strong component
partial clique, which is acyclic. The side information digraph i i
digraph S(D) with expansion of directed acyclic sets in all the
is a 2-partial clique. The network topology in Fig. 4 contains
SCCs will not result in directed cycles. Otherwise, there exist
a triangular bipartite graph Tri(3) in purple. The broadcast
directed cycles between two SCCs, and thus they should be
rate of such an index coding instance is β =3, which can be
contained in the same SCC, which contradicts with the fact
achieved by partition multicast (or CDMA). ♦
thattwoSCCsaredisjoint.Thus,thereexistsadirectedacyclic
induced sub-digraph with size at least (cid:80)s (k + 1), and
i=1 i
hence we have the lower bound β ≥(cid:80)s (k +1), according
i=1 i
to Lemma 1. The upper bound of broadcast rate achieved by
partialcliquecoveringcoincideswiththelowerbound,yielding
the optimality. This completes the proof.
Remark 5. Each strongly-connected component is a partial
clique in the side information digraph. The strong decom-
Fig.4:(a)ThenetworktopologygraphHofa(5,3)regularnetwork, position is unique and is a feasible partial clique covering
and (b) its conflict digraph D with ∆−(D)=2. scheme. Clique and cycle covering are special cases of strong
decomposition, where the strongly-connected components are
restricted to be cliques and cycles respectively. The arcs
between any two different strongly-connected components
B. The Optimality of Partial Clique Covering
belong to a directed cut and can be removed without affecting
In what follows, we present sufficient conditions of the the overall broadcast rate. ♦
optimality and the sub-optimality of partial clique covering,
Remark 6. The recent progress on the coding schemes for the
which can be seen as a partial generalization of Theorem 1.
index coding mainly focused on the exploitation of directed
Thedefinitionsofstrongdecompositionandstrongly-connected
cycles in the side information digraphs. Thanks to Theorem 2,
components can be found in Appendix.
ourfocuscanbeshiftedfromthewholedigraphtothedominant
Theorem 2 (Optimality of Partial Clique Covering). Partial partial cliques of the strongly-connected components. This
clique covering achieves the optimal broadcast rate, if every
strongly-connected component (SCC) of the side information 3Thispropertycanbeeasilyprovedasfollows.Assumebycontradiction
digraphD¯ =(V,A¯)ispartiallyacyclic.Thatis,fortheunique thatthereexisttwodifferentstrongdecompositionsV ={V1,...,Vs}and
sotnreondgomdiencaonmtppoasrittiiaolnclViq=ue{inV1e,v.e.r.y,SVCsC},Dt¯h[eVrie]ethxaisttsisaatclyecalisct. Vafonrd=aD¯n{y[VV1uj(cid:48)(cid:48),].∈w.h.Ve,riVe\t(cid:48)vV},iwt(cid:54)=hheerVerej(cid:48)e.axAivsstebrdtoeitrxhecvD¯te[bdVeilpo]anatghnssdtuDo¯[→tVwj(cid:48)o]vadraiefnfdsetrrevonnt→gSlyC-ucC,osnannD¯edc[Vtfeoidr],
The optimal broadcast rate of such a family of index coding any u(cid:48) ∈ Vj(cid:48)\v, there exist directed paths u(cid:48) → v and v → u(cid:48). So, there
must exist directed paths u → u(cid:48) and u(cid:48) → u through v, indicating that
problems is given by D¯[Vi∪Vj(cid:48)]arestrongly-connectedandthusneitherV ={V1,...,Vs}nor
s V ={V1(cid:48),...,Vt(cid:48)}isastrongdecomposition.Bycontradiction,itfollowsthat
(cid:88) thereexistsatmostoneuniquestrongdecomposition.Asstrongdecomposition
β = (k +1) (10)
i ofadigraphalwaysexists,weconcludethatthereexistsastrongdecomposition
i=1 V ={V1,...,Vs},whichisuniqueandmutuallydisjoint.
7
reduces the index coding problems with large size to a number achieved by the time division of two separate parts according
of smaller ones. ♦ to [12, Theorem 2], and thus the union does not decrease
broadcast rate.
Example 4. Let us take two examples shown in Fig. 5(a) [27,
Given the optimal partial clique partition for the third group,
Fig. 13(b)] and Fig. 5(b). In Fig. 5(a), there exists a strong
we need to show that if the SCCs are not partially acyclic,
decomposition V ={1,2} and V ={3,4,5}, and two strong
connectedcompo1nentsD¯[V ]and2D¯[V ]aredirectedcyclesand partial clique covering is strictly suboptimal. To this end, we
1 2 show that, similarly to Theorem 1, if for any SCC D¯[V ]
hence are partial cliques respectively with k =0 and k =1. i
1 2 of the strong decomposition, each dominant partial clique
InFig.5(b),thereexistsastrongdecompositionV ={1,2,8},
1 has a directed cycle, then there exists a coding scheme with
V2 = {3,4,5} and V3 = {6,7} with partial clique number achievablebroadcastratestrictlylessthan(cid:80)s (k +1).Given
k = 1, k = 1, and k = 0. The dominant partial clique i=1 i
1 2 3 the unique strong decomposition V = {V ,...,V }, for any
of every strongly-connected component is acyclic. As such, 1 s
two disjoint SCCs D¯[V ] and D¯[V ], there are only directed
partial clique covering achieves the optimal broadcast rate for i j
cuts (see Appendix) between them, i.e., only uni-directed arcs
Fig. 5(a) and Fig. 5(b) respectively, with total covering number
β =(cid:80)2 (k +1)=3 and β =(cid:80)3 (k +1)=5. ♦ alwaysfromonetotheother.Accordingto[12,Theorem2],we
i=1 i i=1 i have the confusion graph associated with the side information
sub-digraphs 4
Γ (D¯[V ∪V ])=Γ (D¯[V ])·Γ (D¯[V ]), (11)
t i j ti i tj j
for some integer t and t , where (·) is the lexicographic
i j
product (see Appendix) of two undirected graphs. By Lemma
8 in Appendix, we have
χ (Γ (D¯[V ∪V ]))=χ (Γ (D¯[V ]))χ (Γ (D¯[V ])).
f t i j f ti i f tj j
Fig. 5: (a) A side information digraph with strong decomposition
According to the necessity proof of Theorem 1, if there exists
{{1,2},{3,4,5}}, and (b) a side information digraph with strong
decomposition {{1,2,8},{3,4,5},{6,7}}. The dominant partial a directed cycle in every dominant partial clique of the strong
cliques of all strongly-connected components are acyclic, so that component D¯[V ], then
i
partial clique covering with such a partition achieves the optimal
broadcast rate (a) β =3 and (b) β =5. χf(Γti(D¯[Vi]))<2ti(ki+1). (12)
As such, by Lemma 2, we conclude that, there exists a coding
Theorem 3 (Sub-Optimality of Partial Clique Covering). scheme with strictly less broadcast rate than k +k +2 for
i j
Partial clique covering cannot achieve the optimal broadcast D¯[V ∪V ]. This argument can be straightforwardly extended
rate,ifthesideinformationdigraphD¯ =(V,A¯)fallsinneither i j
to the union of any number of SCCs.
of the following two conditions:
Thus,ifanyoneoftheSCCsisnotdivisibleandnotpartially
• Every SCC is partially acyclic. acyclic, the overall achievable broadcast rate of three groups
• Those SCCs that are not partially acyclic are divisible. will be strictly less than (cid:80)s (k +1), which is promised
i=1 i
by partial clique covering. This yields partial clique covering
Proof: Toprovethesub-optimality,weneedtoshowthat,
strictly suboptimal and completes the proof.
if there exist some SCCs that are not partially acyclic and not
divisible, then partial clique covering is strictly suboptimal. Remark 7. The sufficient condition of the sub-optimality of
Let us divide SCCs in the side information graph into three partial clique covering identified in Theorem 3 also gives a
groups: (1) SCCs that are partially acyclic, (2) SCCs that are necessary condition of its optimality. That is, if partial clique
not partially acyclic but divisible, and (3) SCCs that are not coveringisoptimalforacertainindexcodingproblem,thenthe
partially acyclic and not divisible. We follow the following corresponding side information digraph must fall in one of the
strategy. For the SCCs in the first group, they are also not two conditions of Theorem 3. We suspect that the conditions
divisible, and we use partition multicast (a special case of in Theorem 3 are both necessary and sufficient, but we do not
partial clique covering), which is optimal (see Theorem 1). have a proof for the sufficiency of the second condition nor a
For those SCCs in the second group, we use partial clique counterexample.
covering, which may be suboptimal but it does not matter. For
Example 5. Let us take partial clique covering as a two-step
the SCCs in the third group, we will show that, there exists a
procedure, where the first step using strong decomposition
codingschemethatisstrictlybetterthanpartialcliquecovering.
is always globally optimal, and the second step of further
Overall,ifthethirdgroupisnotempty,weshowthattheabove
partition on each SCC depends if it is partially acyclic or
strategy yields partial clique covering strictly suboptimal.
Let us focus on the third group. If every SCC in the third divisible. This categorizes the side information digraphs into
groupisnotdivisible,thenstrongdecompositionistheoptimal three types: (1) each SCC is partially acyclic, for which partial
partial clique partition, because (1) further partition of each cliquecoveringwithstrongdecompositionisoptimal,(2)some
SCC does not further decrease broadcast rate by partial clique
covering according to Definition 6, and (2) the capacity region 4Here, with a bit abuse of the notation, we take Γt(D¯) and Γt(D)
interchangeablytorepresentthesameconfusiongraphcorrespondingtothe
of the union of two sub-digraphs from two different SCCs is sideinformationdigraphD¯ andtheconflictdigraphD.
8
SCCs are not partially acyclic but divisible, for which partial Theorem 4 (Approximate Broadcast Rate with Multiplicative
clique covering may be still optimal and further partition of Gap). IfthesideinformationdigraphD¯ fallsintothefollowing
these SCCs is needed, and (3) some SCCs are not partially digraph classes, we have broadcast rate β lower bounds and
acyclic and not divisible, for which partial clique covering (partial) clique covering is within a factor Θ from the optimal.
is strictly suboptimal. Examples corresponding to these three • D¯ is oriented planar [34], β ≥ n, and Θ=3;
categories are presented in Fig. 6. ♦ • D¯ is planar with digirth 4 [35],3β ≥ 5n, and Θ= 12;
12 5
• D¯ is planar with digirth at least 5 [36], β ≥ n, and
2
Θ=2;
• D¯ is a tournament [37], β ≥ (cid:98)logn(cid:99)+1, and Θ =
(cid:16) (cid:17)
O n ;
logn
(cid:16) (cid:17)
• D¯ is oriented [37], β ≥logn, and Θ=O n ;
logn
• D¯ issparsewithm=|A(D¯)|≤n1+(cid:15)−n[38],β ≥n1−(cid:15),
Fig. 6: (a) A side information digraph with strong decomposition and Θ=O(n(cid:15)), for some (cid:15)∈(0,1),
{{1,2},{3,4,5}}, for which each SCC is partial acyclic and partial
clique covering is optimal (β =3), (b) an SCC that is not partially where n=|V(D¯)|.
acyclic but divisible with two partial cliques {{1,2},{3,4,5}}, for
Proof: The lower bounds of broadcast rate and the
which partial clique covering is still optimal (β = 3), and (c) an
SCCthatisnotpartiallyacyclicandnotdivisible,andfractionallocal multiplicative gap results are due to the upper bounds of
coloring achieves the optimal broadcast rate β = 5 strictly less than dichromatic number and the Sandwich bound in Lemma
2
partial clique covering with any partition. 6. If D¯ is oriented planar, oriented, and tournament, we
have respectively χ (D¯) ≤ 3 [39], α (D¯) ≥ (cid:98)logn(cid:99) +1,
A A
α (D¯) ≥ logn [37]. If D¯ is planar with digirth g = 4,
A
C. Capacity Approximation we have χ (D¯) ≤ 5n [35] and further if g ≥ 5, a recent
A 12
Before proceeding to the capacity approximation results, result showed that χ (D¯) ≤ 2 [36]. If D¯ is sparse with
A
we summarize two sandwich bounds for the side information m = |A(D¯)| ≤ n1+(cid:15) − n, it was shown in [38] that
graphsG¯anddigraphsD¯,someofwhichwerealreadyappeared α (D¯) ≥ n2 ≥ n1−(cid:15). By the Sandwich bound and the
A m+n
in the literature. fact that β ≤n, we obtain the results.
For the undirected side information graphs G¯(or bi-directed
digraphs) and their conflict graphs G, we have the following
sandwich bound. IV. PROOFOFTHEOREM1
There may be multiple dominant partial cliques. In the
Lemma 5 (Sandwich Bound for Graphs).
sufficiency proof, we only need to show that, if there exist at
|V|
≤α(G¯)≤β ≤χ(G)≤∆(G)+1. (13) least one dominant partial clique that is acyclic, then partition
χ(G¯)
multicast is optimal. In the necessity proof, we only need to
The first inequality is due to Lemma 7 in Appendix, the show that, if every dominant partial clique contains a directed
second one is the MIS bound of index coding problem in cycle, then partition multicast is strictly suboptimal.
Lemma 1, the third one is due to the achievability via vertex
coloring on conflict graphs, and the last inequality is due to
A. Sufficiency
Brooks’ theorem [33].
Similarly, for the general side information digraphs D¯ We focus on one of the dominant partial cliques. Let v∗ =
and their corresponding conflict digraphs D, we have an argmax |N−(D,v)| be the vertex with the maximum in-
v∈V
analogous sandwich bound. The definitions of the graph degree in the conflict digraph D, and let N− denote its closed
v∗
theoretic parameters can be found in Appendix. in-neighborhood in D, for notational convenience. It follows
that |N−|=∆−(D)+1, and D¯[N−] is the dominant partial
Lemma 6 (Sandwich Bound for Digraphs). v∗ v∗
cliqueinD¯.Inwhatfollows,weprovethatifD¯[N−]isacyclic,
v∗
|V|
≤α (D¯)≤β ≤χ (D)≤∆−(D)+1. (14) thenpartitionmulticastachievesthebroadcastrateoftheindex
χA(D¯) A L coding problem.
The first inequality is due to Lemma 7 in Appendix, the By the Sandwich bound in Lemma 6, we have
second one is the MAIS bound in Lemma 1, the third one is
α (D¯)≤β ≤∆−(D)+1. (15)
duetotheachievabilitybylocalgraphcoloring[11]onconflict A
digraphs,andthelastoneisduetothefactthatlocalchromatic
As D¯[N−] is acyclic, it follows that
numberisupperboundedbythesizeofclosedin-neighborhood v∗
[11], which is also the achievable broadcast rate via partition α (D¯)≥|N−|=∆−(D)+1. (16)
A v∗
multicast.
BytheSandwichboundinLemmas5and6,wealsoidentify As such, when D¯[N−] is acyclic, the broadcast rate of such
v∗
a family of index coding problems with side information di- index coding problems is exactly β =∆−(D)+1, which is
graph D¯ where (partial) clique covering achieves the broadcast achieved by partition multicast according to Lemma 4. This
rate within a multiplicative gap from the optimal. completes the sufficiency proof.
9
B. Necessity Second, we show that ω(Γt(D(cid:48)))≥ ω(Γt2(tDr(cid:48))), where D(cid:48) is
To prove the necessity, we show that, if each dominant obtained by adding/removing arcs to/from Dδ in Dr(cid:48). Adding
partial clique contains a directed cycle, then there exists an arcstoDδ doesnotdecreasethecliquenumberoftheconfusion
indexcodethatstrictlyoutperformspartitionmulticast.Inwhat graph,whileremovinganyarc,e.g.,(vi,vj),fromDδ decreases
follows, we focus on one of the dominant partial cliques, and the clique number to at most 21t. We divide the message tuples
the argument applies similarly to the other dominant partial considered above with difference only at M into 2t subgroups,
cliques. in each of which the messages have the same value in F2t at
To avoid confusion in notations, in this section, we let vj. Clearly, any two tuples xn and zn in the same subgroup
D¯(cid:48) denote the side information digraph whose dominant are still confusable, because (1) forcing xvj =zvj reduces the
partial clique contains a directed cycle, and D(cid:48) denote its differencefromQ(xn,zn)toQ(xn,zn)\{j},and(2)removing
ccolorsreesdpoinn-dnineigghcboonrflhiocotddiogfrapthhe. LveertteMx w=ithNtv−h∗e(Dm(cid:48))axbiemuthme (Tvhiu,vs,j)thferroemarDe adtoleesasntotωd(Γetc(rDera(cid:48))s)etNupv−lesfoirnaellacvh∈su[b1g:ronu]\p{,vajn}y.
2t
in-degree in D(cid:48) and δ = ∆−(D(cid:48)) be the maximum in- two of which with the same value at vj are still confusable. It
degree in D(cid:48). Assume that the dominant partial clique D¯(cid:48)[M] follows that χf(Γt(D(cid:48)))≥ω(Γt(D(cid:48)))≥ ω(Γt2(tDr(cid:48))) ≥2δt.
wcohnetraein2s≤all≤enδg+th1-l.RceyccalleltihnavtoQlv(ixnng,zvne)rt(cid:44)ice{si:{wxi1,(cid:54)=..z.i,,w∀il}},, χ C(Γlai(mD(cid:48))2)-≤χχf(Γ(Γt(D(D(cid:48))(cid:48))))≤fo(cid:80)ruδjn=d0i2rejctt:edGgirvaepnhst[h4e0,fTahcetotrhemat
and let d (xn,zn) (cid:44) |Q(xn,zn)| be the Hamming distance f t L t
H 3], we need to show that there exists a proper coloring for
of two message tuples. Γ (D(cid:48)) such that there are at most (cid:80)δ 2jt colors in the
To prove the existence of a coding scheme that satisfies t j=0
closed neighborhood of any vertex, such that any two adjacent
β <δ+1, we have to rely on confusion graphs, as we have
vertices in such a closed neighborhood receive distinct colors.
noknowledgeaboutthestructureofotherpartinD(cid:48).LetΓt(D(cid:48)) First, we show that any vertex is locally (cid:16)(cid:80)δ 2jt(cid:17)-
denote the corresponding confusion graph of D(cid:48). In what j=0
follows, we will show that 2δt ≤ χ (Γ (D(cid:48))) ≤ (cid:80)δ 2jt, colorable, that is, the closed neighborhood of any vertex can
f t j=0 be properly assigned with at most (cid:80)δ 2jt colors. Before
for some t. The proof consists of two parts. j=0
proceeding further, we introduce a general coloring function
Claim 1 - χ (Γ (D(cid:48)))≥2δt: To show this claim, we only
f t
thatwillberepeatedlyusedlater.Givenanintegerk,wedefine
need to show ω(Γ (D(cid:48)))≥2δt. Given an arbitrary digraph D(cid:48),
t a proper coloring function
it is not easy to count the clique number of its confusion graph
Γ (D(cid:48)) directly.5 So, we do this in a different way. c:(F )n (cid:55)→(F )k, (17)
t 2t 2t
Let M = {v ,v ,...,v } denote the vertices in N−(D(cid:48)).
0 1 δ v∗ which assigns a color associated with a k-tuple vector to a
We introduce anther digraph D(cid:48), which is a copy of D(cid:48) with
r vertex associated with an n-tuple message in confusion graphs.
the dominant part D(cid:48)[M] replaced by the conflict digraph of
Specifically, let {a ,i={1,...,n}} be a set of distinct values
thetriangularnetworkTri(δ+1)(SeeExample1),denotedby i
in the finite field F . Let us treat each message x as an
D , as shown in Fig. 7, in which ∀ i < j, (v ,v ) ∈ A(D ). 2t i
δ i j δ element of the finite filed F such that the products with
We assume that D(cid:48) is obtained by removing some arcs from 2t
field elements a(cid:96), and the corresponding linear combinations
and adding some other arcs to D . i
δ are well-defined. For a tuple xn =(x ,...,x ), the coloring
1 n
function is given by
(cid:32) n n n (cid:33)
(cid:88) (cid:88) (cid:88)
c(xn)(cid:44) x , a x ,..., ak−1x (18)
i i i i i
i=1 i=1 i=1
whichmapsamessagexn toak×1vector(c ,...,c ).Taking
1 k
Fig. 7: The reference sub-digraph Dδ. a closer look at this coloring function, we can find that it is
a syndrome computation of (n,k,n−k+1) Reed-Solomon
First, we show that ω(Γ (D(cid:48)))≥2(δ+1)t, given the digraph
t r (RS) codes, which can detect up to n−k errors. In other
Dr(cid:48) anditsconfusiongraphΓt(Dr(cid:48)).Consideranytwomessage words,thesyndromesoftwodistinctmessagesxn andzn with
tuples xn and zn where xj = zj, ∀j ∈/ M, and xj,zj ∈ d (xn,zn) ≤ n−k should not be all the same, i.e., there
F2t, ∀j.If|Q(xn,zn)|=1,xnandznareconfusableandthus exHistsomej’ssuchthat(cid:80)n aj−1x (cid:54)=(cid:80)n aj−1z .Thus,it
adjacent in Γt(Dr(cid:48)). If |Q(xn,zn)| ≥ 2, as (vi,vj) ∈ A(Dr(cid:48)) followsthatanytwoadjacein=t1veirticesiwithdi=1(xni,zni)≤n−k
for all i < j, it follows by Definition 2 that xn and zn are receivedistinctcolors,i.e.,c(xn)(cid:54)=c(zn)foHralld (xn,zn)≤
confusable and thus adjacent in Γt(Dr(cid:48)). As such, any two n−k. For each k, there are 2kt such RS codewHords, which
tuples xn and zn with xj = zj, ∀j ∈/ M are adjacent in correspond to 2kt distinct colors.
Γ (D(cid:48)). As there are 2(δ+1)t such distinct tuples, we conclude
t r Because any two message tuples with Hamming distance
that ω(Γ (D(cid:48)))≥2(δ+1)t.
t r no less than δ +2 are non-confusable (see Observation 1)
and thus non-adjacent, we only have to consider the adjacent
5Averyrecentresultin[14]hasestablishedthebroadcastrateintermsof
thecliquenumberoftheconfusiongraph.Onemaywonderifthebroadcast vertices whose corresponding messages have distance no more
rate can be directly calculated by counting clique numbers in the conflict than δ+1. Given any message xn, without loss of generality
graph.Nevertheless,itseemscountingcliquenumberintheconfusiongraph
assuming the all-zero message tuple, its confusable messages
islessconvenientthancountingcoloringintheneighborhoodasshowninthe
proofofClaim2. (i.e., adjacent vertices) lie in the following categories:
10
Category A: This category consists of all messages zn with because d (xn,zˆn)≤δ. In short, each subgroup of messages
H
d (xn,zn) ≤ δ. According to Hamming distance between in Category B can be colored with the same color assigned to
H
xn and zn, we divide the messages in Category A into δ a message zˆn in Category A.
sub-categories, and assign colors for each sub-category. For Second,giventhefactthatΓ (D(cid:48))islocallyr-colorablewith
t
(cid:16) (cid:17)
the sub-category j (1 ≤ j ≤ δ), we consider the messages r = (cid:80)δ 2jt , there exists a homomorphism of Γ (D(cid:48)) to
j=0 t
zn where d (xn,zn) = j and assign such messages with a
H theuniversalgraphU(m,r)definedin[40,Definition6]witha
color c(zn) given by the syndrome of the (n,k,n−k+1) RS
sufficiently large m. For instance, U(m,r) can be constructed
codewords, 6 where k is specified by as follows associated to Γ (D(cid:48)). Each vertex of U(m,r) rep-
t
(cid:26) j, if 1≤j ≤n−δ−1 resents a pair (xn,A) where A⊆N(xn) is the set of vertices
k = n−δ−1, if n−δ−1<j ≤δ (19) withdistinctcolorsintheneighborhoodofxn.So,|A|=r−1.
Two vertices (xn,A) and (yn,B) in U(m,r) are connected
wherek ≤j forbothcases.Thus,thenumberofcodewordsfor withanedgeifxn ∈Bandyn ∈A.Thishappensifxn andyn
the(n,k,n−k+1)RScodeisnomorethan2jt,corresponding areconfusableinΓ (D(cid:48)).By[40,Lemma1],weconcludethat,
to no more than 2jt distinct colors. there exists a propetr coloring for Γ (D(cid:48)) such that the number
For any two adjacent messages zn and z(cid:48)n in such a sub- t
of colors in the closed neighborhood of any vertex is at most
category, we have, by Observation 1, (cid:80)δ 2jt, and thus χ (Γ (D(cid:48))) ≤ χ (Γ (D(cid:48))) ≤ (cid:80)δ 2jt.
j=0 f t L t j=0
dH(zn,z(cid:48)n)≤δ+1≤n−k (20) T2δotge≤theχr (wΓit(hDχ(cid:48))f)(Γ≤t((cid:80)D(cid:48)δ)) ≥2jtω<(Γ2t((δD+(cid:48)1)))t,≥whe2rδet,thweeuhpapveer
f t j=0
for both 1 ≤ j ≤ n−δ −1 and n−δ −1 < j ≤ δ. As bound is strict.
such, any two messages in this sub-category have Hamming In conclusion, if every dominant partial clique D¯(cid:48)[M]
distance no more than the error detection capability n−k of contains a directed cycle, then there exists a coding scheme
the (n,k,n−k+1) RS code. As detailed before, any two such that the broadcast rate satisfies
adjacent vertices with dH(xn,zn)=j receive distinct colors, logχ (Γ(cid:48)(D(cid:48)))
i.e., c(xn) (cid:54)= c(zn) for all d (xn,zn) = j. For each sub- β ≤ f t <δ+1, (23)
H t
category j, we assign no more than 2jt colors to zn such that
which is strictly less than that achieved by partition multicast.
any two adjacent zn’s receive distinct colors. It follows that
This completes the necessity proof.
the total required number of colors in Category A is no more
than
V. CONCLUDINGREMARK
1+(cid:88)δ 2jt = 2t(δ+1)−1 <2t(δ+1). (21) For the multiple-unicast index coding problems, we have
2t−1 characterized the sufficient and necessary condition for which
j=1
partition multicast, a special case of partial clique covering
Category B: This category consists of all messages z¯n with with cover number 1, achieves the optimal broadcast rate.
dH(xn,z¯n)=δ+1.AccordingtoObservation2,thepositions It has been proven that partial acyclicity (i.e., the absence
of difference between xn and z¯n correspond exactly to the of directed cycles in the dominant partial clique) is the
verticesofpartialclique,becauseotherwisexn andz¯n arenon- fundamentaltopologicalstructurethatdeterminestheoptimality
confusable. The indices of the length-l directed cycle belong of partition multicast. Such a result indicates that the benefits
to a subset of these positions, i.e., of sophisticated achievable schemes (such as local coloring,
{w ,...,w }⊆Q(xn,z¯n), (22) random coding, subspace interference alignment) should focus
1 l
on the exploitation of directed cycles in the dominant partial
for sure. For these vertices, we will color them with the same cliquesratherthanthoseinthewholesideinformationdigraphs.
colors used for Category A messages. We divide messages in On the other hand, it is also shown that maximum acyclic
Category B into subgroups, each of which contains messages induced sub-digraph bound is tight for the optimality of
with different value only at {w ,...,w }. According to Ob- partition multicast, in addition to the tightness for clique
1 l
servation 3, each subgroup of messages forms an independent covering shown in [28].
set, and can be assigned with the same color, because the We have further extended to the general partial clique
side information sub-digraph induced by {w ,...,w } is a covering, providing a sufficient condition for which partial
1 l
directedcycleandinturnanytwomessagesarenon-confusable. clique covering is optimal, and a sufficient condition for
For each subgroup, there exists a message say zˆn, where which it is strictly suboptimal. Nevertheless, the complete
Q(z¯n,zˆn) = {w ,...,w } for some z¯n’s in this subgroup, characterization of the fundamental structural property of the
1 l
such that there exists at least one position w in {w ,...,w } optimality of partial clique covering is not fully understood.
j 1 l
with zero value, i.e., x = zˆ . Together with the fact that Thesufficientandnecessityconditioninthemostexplicitform
wj wj
d (xn,z¯n) = δ+1, it follows that zˆn is non-confusable to is still unknown due to the difficulty of the characterization
H
all messages in such a subgroup, and belongs to Category A of optimal partial clique partition, which is an NP-hard
combinatorial optimization problem.
6Whenn−δ−1<j≤δ,theresultingRScodeshavethesamelength Furthermore, by the connection of the lower bound of
ofinformationsymbols.Todistinguishonefromanother,wechoosedifferent
broadcast rate and acyclic set coloring of digraphs, we have
ai’sormakearotation/permutationtogeneratedifferentcoloringfunctions
c(zn)fordifferentsub-categoryj. shown that, for a family of index coding instances, (partial)