BIPARTITE INDEX CODING

Arash Saber TehraniAlexandros G. DimakisMichael J. Neely

Department of Electrical Engineering University of Southern California (USC)

Outline

• Index Coding Problem– Introduction– Bipartite model

• Our Scheme: Partition Multicast– Formulation

• Partition Multicast is NP-hard– Connection to clique cover

Index Coding Problem• Introduced in [Birk and Kol 98], and further developed in

[Bar-Yossef, Birk, Jayram, and Kol 06 and 11].• Broadcast station • Set of m packets P ={x1, x2, … , xm} from a finite alphabet X • Set of n users U ={u1, u2, … , un} • Each user demands exactly one packet• Each user i knows a subset of packets denoted by Nout(ui) as

side info• Objective: Minimize the amount of broadcast data so that all

users decode their designated packets.

Bipartite model for IC

• The system can be represented by a bipartite graph

• A directed edge from packet xj to user ui indicates that user ui demands packet xj.

• A directed edge from user ui to packet xj indicates that user ui knows packet xj as side info.

Index Coding Problem

• A solution of the problem– A finite alphabet WX – an encoding function E: Xm WX

– each user ui is able to decode its designated packet from the broadcast message w and its side information.

• Optimal solution is HARD to compute.

Our Scheme: Partition Multicast

• When each user knows at least d packets as side information– We call d “minimum out-degree” or “minimum knowledge”

• Then there are at most m – d unknowns for each user. • With transmission of m - d independent equations in

the form a1x1 + a2x2 + … + amxm where ai's are taken from some finite field F, each user can decode the packet it demands as shown in Ho et al. (Given that |F| is large enough)

Our Scheme: Partition Multicast

• Induced subgraph by a subset of packets S

X1

X2

X3

X4

U1

U2

U3

U4

U5

X1

X2

U1

U2

U3

Our Scheme: Partition Multicast

• We are looking for a partition (valid packet decomposition)

X1

X2

X3

X4

U1

U2

U3

U4

U5

X1

X2

X3

X4

|{X1,X2}| = 2, d1 = 1 |{X3,X4}| = 2, d1 = 1X1+X2 X3+X4

Our Scheme: Partition Multicast

• Partition Multicast:

Our Scheme: Partition Multicast

• The scheme is optimal for known cases such as – Cliques– trees– Directed cycles

• It has cycle cover schemes proposed by Chaudhry et al. and Neely et al. as a special case and outperforms them.

Partition Multicast is NP-hard

• Undirected case:

– We want to find a partition for which the sum of minimum knowledge is maximized

– We call this problem “sum-degree cover”

U1, X1

U2, X2

U3, X3 U4, X4

U5, X5

X1U1

U2 X2

X3

X4

X5

U3

U4

U5

Partition Multicast is NP-hard

• Sum-degree cover and clique cover are equivalent– Partitioning a clique is strictly suboptimal

• For any graph T(GS) ≥1.• If GS is a clique, then T(GS) = 1, i.e., the minimum knowledge d

= |S| - 1.– We need to show that

• Solution of sum-degree cover gives the solution of clique cover• Solution of the clique cover gives the solution of sum-degree

cover

SD cover Clique cover• Let the solution of SD cover be GS1, … , GSK induced by

subsets S1, S2, …, Sk.• Clique cover is also a graph partition where each subgraph

requires exactly one transmission, so

• Consider subgraph GS1 with minimum knowledge d1. The complement of GS1 has maximum degree |S1| - d1 - 1.

• As is well known, any graph of maximum degree d has a vertex coloring of size d + 1.

SD cover Clique cover

• The complement of GS1 has a vertex coloring with |S1| - d1 color.

• Thus, GS1 has a clique cover of size |S1| - d1.• That is• Repeating the same procedure over all k

subgraphs, gives

• Jointly with the previous inequality we get

Partition Multicast is NP-hard

• Maps an undirected graph G to a bipartite graph.

• Solve the partition multicast.• Find the clique cover of all partitions through

coloring of complements of the subgraphs.• Find the clique cover.

Conclusion

• We introduced the bipartite graph model for the index coding problem

• We presented a new scheme “partition multicast” for index coding problem.

• We introduced the sum-degree cover problem.• We showed that finding the optimal partition is

NP-hard. • Future work: finding a ‘good’ partition

Thanks, Questions?

Partition Multicast

• Partition or Cover:

– Let x S∈ 1, x S∈ 2

– Delete x from S1 to get set S1’– New minimum knowledge for GS1, namely, d1’.– |S1’| =|S1|-1 and d1-1 ≤ d1’ ≤ d1.

GS1 GS2 GSk

T(GS1)=|S1|-d1 T(GS2)=|S2|-d2 T(GSk)=|Sk|-dk

Our Scheme: Partition Multicast• Bipartite case (Painful stuff)

– For set S P,⊆ define GS = (US,S,ES) to be the subgraph induced by S:

– A valid packet decomposition is set of k disjoint subgraphs such that

– It can be checked that for a valid packet decomposition

Index Coding Problem

• A solution of the problem– A finite alphabet WX – an encoding function E: Xm WX

– each user ui is able to decode its designated packet from the broadcast message w and its side information.

• The minimum coding length of the solution per input symbol:

where the minimum is over all encoding functions E.• Optimal broadcast rate