1. B IPARTITE I NDEX C ODING 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 Kol 06 and 11]. Broadcast station Set of m packets P ={x 1, x 2, …, x m } from a finite alphabet X Set of n users U ={u 1, u 2, …, u n } Each user demands exactly one packet Each user i knows a subset of packets denoted by N out (u i ) 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 x j to user u i indicates that user u i demands packet x j. A directed edge from user u i to packet x j indicates that user u i knows packet x j as side info.

5. Index Coding Problem A solution of the problem – A finite alphabet W X – an encoding function E: X m W X – each user u i is 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 a 1 x 1 + a 2 x 2 + … + a m x m where a i '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)

7. Our Scheme: Partition Multicast Induced subgraph by a subset of packets S X1X1 X2X2 X3X3 X4X4 U1U1 U2U2 U3U3 U4U4 U5U5 X1X1 X2X2 U1U1 U2U2 U3U3

8. Our Scheme: Partition Multicast We are looking for a partition (valid packet decomposition) X1X1 X2X2 X3X3 X4X4 U1U1 U2U2 U3U3 U4U4 U5U5 X1X1 X2X2 X3X3 X4X4 |{X 1,X 2 }| = 2, d 1 = 1|{X 3,X 4 }| = 2, d 1 = 1 X 1 +X 2 X 3 +X 4

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. 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” U 1, X 1 U 2, X 2 U 3, X 3 U 4, X 4 U 5, X 5 X1X1 U1U1 U2U2 X2X2 X3X3 X4X4 X5X5 U3U3 U4U4 U5U5

12. Partition Multicast is NP-hard Sum-degree cover and clique cover are equivalent – Partitioning a clique is strictly suboptimal For any graph T(G S ) ≥1. If G S is a clique, then T(G S ) = 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 G S1, …, G SK induced by subsets S 1, S 2, …, S k. Clique cover is also a graph partition where each subgraph requires exactly one transmission, so Consider subgraph G S1 with minimum knowledge d 1. The complement of G S1 has maximum degree |S 1 | - d 1 - 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 G S1 has a vertex coloring with |S 1 | - d 1 color. Thus, G S1 has a clique cover of size |S 1 | - d 1. 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. Partition Multicast Partition or Cover: – Let x ∈ S 1, x ∈ S 2 – Delete x from S 1 to get set S 1 ’ – New minimum knowledge for G S1, namely, d 1 ’. – |S 1 ’| =|S 1 |-1 and d 1 -1 ≤ d 1 ’ ≤ d 1. G S1 G S2 G Sk T(G S1 )=|S 1 |-d 1 T(G S2 )=|S 2 |-d 2 T(G Sk )=|S k |-d k

19. Our Scheme: Partition Multicast Bipartite case (Painful stuff) – For set S ⊆ P, define G S = (U S,S,E S ) 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 W X – an encoding function E: X m W X – each user u i 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