Bipartite Index Coding

1. Bipartite Index Coding Arash Saber Tehrani Alexandros G. Dimakis Michael J. Neely Department of Electrical Engineering University of Southern California (USC)

2. Outline • Index Coding Problem • Introduction • Bipartite model • Our Scheme: Partition Multicast • Formulation • Partition Multicast is NP-hard • Connection to clique cover

3. Index Coding Problem • Introduced in [Birk and Kol 98], and further developed in [Bar-Yossef, Birk, Jayram, and Kol06 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.

4. 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 uito packet xjindicates that user ui knows packet xjas side info.

5. Index Coding Problem • A solution of the problem • A finite alphabet WX • an encoding function E: XmWX • each user uiis able to decode its designated packet from the broadcast message w and its side information. • Optimal solution is HARD to compute.

6. 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'sare 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)

7. X1 U1 U4 X1 U1 Our Scheme: Partition Multicast X2 U2 U5 X2 U2 X3 U3 U3 X4 • Induced subgraph by a subset of packets S

8. U4 X3 X1 X1 U1 Our Scheme: Partition Multicast X4 X2 X2 U2 U5 X3 U3 X4 • We are looking for a partition (valid packet decomposition) |{X1,X2}| = 2, d1 = 1 |{X3,X4}| = 2, d1 = 1 X1+X2 X3+X4

9. Our Scheme: Partition Multicast • Partition Multicast:

10. 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.

11. U1 U3 U1, X1 U2, X2 X3 U2 U5, X5 X1 X2 Partition Multicast is NP-hard X4 U4 X5 U5 U3, X3 U4, X4 • Undirected case: • We want to find a partition for which the sum of minimum knowledge is maximized • We call this problem “sum-degree cover”

12. 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

13. 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 GS1with 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.

14. 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

15. 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.

16. 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

17. Thanks, Questions?

18. T(GS1)=|S1|-d1 GS1 T(GS2)=|S2|-d2 GS2 T(GSk)=|Sk|-dk GSk Partition Multicast • Partition or Cover: • Let x∈S1, x∈S2 • Delete x from S1 to get set S1’ • New minimum knowledge for GS1, namely, d1’. • |S1’| =|S1|-1 and d1-1 ≤ d1’ ≤ d1.

19. 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

20. Index Coding Problem • A solution of the problem • A finite alphabet WX • an encoding function E: XmWX • each user uiis 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