1. Multicut Lower Bounds via Network Coding Anna Blasiak Cornell University

2. Multicut Given: Directed graph G = (V,E) Capacities on edges k source-sink pairs Find: A min-cost subset of E such that on removal all source-sink pairs are disconnected s1s1 s2s2 s3s3 t1t1 t2t2 t3t3

3. Directed Multicut The State of the Art Õ( n 11/23 ) - approx algorithm [Agarwal, Alon, Charikar ’07] 2 Ω( log 1-ε n) hardness [Chuzhoy, Khanna ‘09] Nothing non-trivial known in terms of k

4. Dual Problem: Maximum Multicommodity Flow Given: A directed graph G = (V,E) Capacities on nodes k source-sink pairs Find: A maximum total weight set of fractional s i -t i paths. s1s1 s2s2 s3s3 t1t1 t2t2 t3t3 All approximation algorithms for multicut are based on the LP and compare to the maximum multicommodity flow. Limited by large integrality gap: Ω(min(( k, n δ )) [Saks, Samorodnitsky,Zosin ’04, Chuzhoy, Khanna ’09 ]

5. Better Lower Bound? Network Coding Good News: Coding Rate ≥ Flow Rate, can be a factor k larger Bad News: Multicut ≱ Coding Rate, can be a factor k smaller

6. Results Identify a property P of a network code such that any code satisfying P is a lower bound on the multicut. Show P is preserved under a graph product. Main Corollary: – Improved and tight lower bound on the multicut in the construction of Saks et al. and give a network code with rate = min multicut

7. Simplifying Assumptions A network is: Undirected graph G = (V,E ) Capacity one for each node Subsets of V : {S i } i ∈[k], {T i } i ∈[k] s i - t i pairs, i ∈[k], connect to G: s i connects to v ∈S i v ∈T i connect to t i Type of networks giving: Ω(min(( k, n δ )) [Saks, Samorodnitsky,Zosin ’04, Chuzhoy, Khanna ’09 ]

8. Saks et al. Construction Begin with The n-path network P n : v 1 v 2 v 3 v 4 v 5 v n-1 v n s1s1 t1t1 Hypergrid( n, k ) is the k -fold strong product of P n. It has n k nodes and k s-t pairs and flow rate n k-1 Saks et al. show it has multicut at least has kn k-1 THEOREM Hypergrid( n,k ) has a code with rate n k - (n-1) k that is a lower bound on the multicut. S1S1 T1T1

9. Hypergrid(3,2) = P 3 ☒ P 3 s 2 t2t2 s1s1 ( v 2, v 1 ’ ) ( v 1, v 2 ’ )( v 3, v 2 ’ )( v 2, v 2 ’ ) ( v 2, v 3 ’ ) t1t1 ( v 1, v 1 ’ )( v 3, v 1 ’ ) ( v 1, v 3 ’ ) ( v 3, v 3 ’ )

10. Hypergrid(3,3) = P 3 ☒ P 3 ☒ P 3 S 1 T 1 S 2 T 2 S 3 T 3

11. Coding Matrix s 2 t2t2 s1s1 ( v 2, v 1 ’ ) ( v 1, v 2 ’ )( v 3, v 2 ’ )( v 2, v 2 ’ ) ( v 2, v 3 ’ ) t1t1 ( v 1, v 1 ’ )( v 3, v 1 ’ ) ( v 1, v 3 ’ ) ( v 3, v 3 ’ ) ( v 1,v 1 ’ ) ( v 2,v 1 ’ ) ( v 3,v 1 ’ ) ( v 1,v 2 ’ ) ( v 2,v 2 ’ ) ( v 3,v 2 ’ ) ( v 1,v 3 ’ ) ( v 2,v 3 ’ ) ( v 3,v 3 ’ ) a1a1 111000000 b1b1 000111000 c1c1 000000111 a2a2 100100100 b2b2 010010010 Column v describes the linear combination of messages sent by v. Column v is a linear combination of columns of predecessors of v. t i can decode messages from s i. Column v describes the linear combination of messages sent by v. Column v is a linear combination of columns of predecessors of v. t i can decode messages from s i. Columns labeled with v in V Rows labeled with messages a i, b i, c i originate at s i rate of code = # of messages Rows labeled with messages a i, b i, c i originate at s i rate of code = # of messages a 1 +a 2 b1b1 a 1 +b 2 a1a1 c1c1 b 1 +b 2 c 1 +a 2 c 1 +b 2 a 2, b 2, c 2 a 1, b 1, c 1 b 1 +a 2 Entries in finite field

12. Coding Matrix as a Lower Bound L gives a lower bound on the multicut: For M a minimum multicut, |M | = rank ( I M ) ≥ rank (L I M ) ≥ p. L gives a lower bound on the multicut: For M a minimum multicut, |M | = rank ( I M ) ≥ rank (L I M ) ≥ p. DEFINITION A coding matrix L is p -certifiable if 1.For any multicut M, rank (L I M ) ≥ p. 2.Column v of L is a linear combination of columns of incoming sources and predecessors of v that form a clique. |V| x |M| matrix, column v ∈ M is an indicator vector for v.

13. Main Theorem* Given networks N 1 and N 2 with coding matrices L 1 and L 2, there is a coding matrix for N 1 ☒ N 2 : L = such that: 1.If L 1 and L 2 have rates p 1 and p 2 then L has rate p := n 1 p 2 + n 2 p 1 - p 1 p 2. 2.If L 1 and L 2 are p 1 and p 2 certifiable then L is p- certifiable. In1⊗ L2In1⊗ L2 L1⊗ In2L1⊗ In2

14. Saks et al. Construction Begin with The n-path network P n : v 1 v 2 v 3 v 4 v 5 v n-1 v n s1s1 t1t1 Hypergrid( n, k ) is the k -fold strong product of P n. It has n k nodes and k s-t pairs and flow rate n k-1 Saks et al. show it has multicut at least has kn k-1 THEOREM Hypergrid( n,k ) has a code with rate n k -(n-1) k that is a lower bound on the multicut. P n is 1-certifiable.

15. Conclusions and Open Questions Our result: a certain type of network coding solution is a lower bound on directed multicut – Is there a more general class of network coding solutions that is a lower bound? Multicommodity flow can be far from the multicut, what about the network coding rate? – Does there exist an α = o(k) s.t. multicut ≤ α network coding rate?

16. LP Relaxation All approximation algorithms use LP relaxation with dual = maximum multicommodity flow problem Problem: Large integrality gap – Ω(min(( k, n δ )) [Saks, Samorodnitsky,Zosin ’04, Chuzhoy, Khanna ’09 ] min Σ v in V x v s.t. Σ v in p x v ≥ 1 for all 1 ≤ i ≤ k, for all p s i -t i path x v ≥ 0 for all v in V min Σ v in V x v s.t. Σ v in p x v ≥ 1 for all 1 ≤ i ≤ k, for all p s i -t i path x v ≥ 0 for all v in V

17. Butterfly network + infinite capacity edges between s i and t j for all i ≠ j x 1 ⊕ x 2 ⊕ ∙∙∙ ⊕ x k x1x1 x2x2 xkxk x4x4 x5x5 x6x6 x k-1 Coding Rate = k, Flow Rate = Multicut = 1

18. Guiding Questions When is network coding a lower bound on multicut in directed graphs? When it is an lower bound, is it a better lower bound than maximum multicommodity flow?

19. Strong Graph Product Kronecker Product – A: m ✕ n matrix – B: p ✕ q matrix – A ⊗ B = a 11 B … a 1n B … … a 1m B … a nm B Adjacency matrix of G 1 ☒ G 2 : (G 1 + I n 1 ) ⊗ (G 2 + I n 2 ) – I n 1 n 2 G i here denotes adjacency matrix of G i

20. IMIM Cuts as matrices v 11 00000 v 2 1 10000 v 31 00000 v 12 01000 v 2 2 00100 v 32 00010 v 13 00000 v 23 00001 v 33 00000 Matrix I M represents a cut M. s 2 t2t2 s1s1 ( v 2, v 1 ’ ) ( v 1, v 2 ’ ) ( v 3, v 2 ’ ) ( v 2, v 2 ’ ) ( v 2, v 3 ’ ) t1t1 ( v 1, v 1 ’ ) ( v 3, v 1 ’ ) ( v 1, v 3 ’ ) ( v 3, v 3 ’ ) L gives a lower bound on the multicut: For M a minimum multicut, |M | = rank ( I M ) ≥ rank (L I M ) ≥ p. L gives a lower bound on the multicut: For M a minimum multicut, |M | = rank ( I M ) ≥ rank (L I M ) ≥ p. DEFINITION A coding matrix L is p -certifiable if 1.For any multicut M, rank (L I M ) ≥ p. 2.Column v of L is a linear combination of columns of incoming sources and predecessors of v that form a clique.

21. Flows, Cuts, and Codes Multicast Flow Rate Coding Rate Cut Bound Directed Ω((log n/ log log n) 2 ) Undirected 8/7 Multicommodity Flow RateCut Bound Directed Ω(min(( k, n δ )) Undirected Ω(min(( log n, log k )) Coding Rate?

22. Multicut Lower Bound Certificates Multicut M gives Cut Matrix B : – n x |M| matrix, each column is an indicator vector for one node in the cut If there is a matrix A s.t. rank (AB) ≥ p for all cut matrices B, then this shows lower bound p – Proof: |M| = rank (B) ≥ rank (AB) Definition A coding matrix A is p -certifiable for multicut instance G if rank (AB) ≥ p for any cut matrix B of G Definition A coding matrix A is p -certifiable for multicut instance G if rank (AB) ≥ p for any cut matrix B of G

23. Multicut instance G 1 : Coding matrix A 1, rate p 1, p 1 – certifiable Multicut instance G 2 : Coding matrix A 2, rate p 2, p 2 – certifiable Then is a r – certifiable coding matrix with rate r for G 1 × G 2, r := n 1 p 2 + n 2 p 1 - p 1 p 2. Result* A 1 × I n 2, I n 1 × A 2 Definition A coding matrix A is p -certifiable for multicut instance G if rank (AB) ≥ p for any cut matrix B of G and for each v in V there is a clique K(v) with A1 K(v) = 0. Definition A coding matrix A is p -certifiable for multicut instance G if rank (AB) ≥ p for any cut matrix B of G and for each v in V there is a clique K(v) with A1 K(v) = 0.

24. Multicut instance G 1 : Coding matrix A 1, rate p 1, p 1 – certifiable Multicut instance G 2 : Coding matrix A 2, rate p 2, p 2 – certifiable Then is a r – certifiable coding matrix with rate r for G 1 × G 2, r := n 1 p 2 + n 2 p 1 - p 1 p 2. Result* A 1 × I n 2, I n 1 × A 2 But how do we get the original certifiable code??

25. Certifiable Coding Matrices Easy Observation: – If A is the “coding” matrix for a solution that routes messages along p node disjoint paths (each row is an indicator vector for one path) – Then A is p - certifiable. 11000000 00011010 00100101 000 010 000 000 000 100 000 001 Disjoint Paths: at most one non-zero in each column of AB Multicut: at least one non-zero in each row of AB

26. Saks et al Construction [Saks, Samorodnitsky,Zosin ’04] Ω(k) flow-cut gap Seed Network N : – Path of length n : x 1, …, x n Final network: k -fold strong product of N – n k nodes – k s-t set pairs – max flow = n k-1 S t S1S1 t1t1 S2S2 t2t2 Corollary The minimum multicut and the network coding rate in the Saks et al. construction is n k - (n-1) k. ( Saks et al. showed multicut is at least k(n-1) k-1 ) Corollary The minimum multicut and the network coding rate in the Saks et al. construction is n k - (n-1) k. ( Saks et al. showed multicut is at least k(n-1) k-1 )

27. Result Theorem* Suppose multicut instances G 1 and G 2 have linear network coding solutions instances A 1 and A 2 with rates p 1 and p 2 respectively, and are p 1 and p 2 certifiable respectively. Let r := |V 1 |p 2 + |V 2 |p 1 - p 1 p 2 Then multicut instance G 1 × G 2 has a linear network coding solution of rate r given by a matrix A that is r – certifiable. Theorem* Suppose multicut instances G 1 and G 2 have linear network coding solutions instances A 1 and A 2 with rates p 1 and p 2 respectively, and are p 1 and p 2 certifiable respectively. Let r := |V 1 |p 2 + |V 2 |p 1 - p 1 p 2 Then multicut instance G 1 × G 2 has a linear network coding solution of rate r given by a matrix A that is r – certifiable. Proof: A 1 × I p, I n × A 2

28. Graph Constructions Strong Graph Product G 1 ✕ G 2 – Vertex Set: V 1 ✕ V 2 – Edge Set: (u,v) ~ (u’,v’) if (u = u’ OR u ~ u’) AND (v = v’ OR v ~ v’) ✕

29. Network Products An undirected graph G=(V,E) k source sets S 1, …, S k paired with k sink sets T 1 …T k. An undirected graph G=(V,E) k source sets S 1, …, S k paired with k sink sets T 1 …T k. An undirected graph G ✕ G ’ 2k source sets S 1 ✕ V ’, …, S k ✕ V ’, V ✕ S ’ 1, …, V ✕ S ’ k paired with 2k sink sets T 1 ✕ V ’, …, T k ✕ V ’, V ✕ T ’ 1, …, V ✕ T ’ k. An undirected graph G ✕ G ’ 2k source sets S 1 ✕ V ’, …, S k ✕ V ’, V ✕ S ’ 1, …, V ✕ S ’ k paired with 2k sink sets T 1 ✕ V ’, …, T k ✕ V ’, V ✕ T ’ 1, …, V ✕ T ’ k. An undirected graph G ’ =(V ’,E ’ ) k source sets S ’ 1, …, S ’ k paired with k sink sets T ’ 1 …T ’ k. An undirected graph G ’ =(V ’,E ’ ) k source sets S ’ 1, …, S ’ k paired with k sink sets T ’ 1 …T ’ k.

30. Network Products S1S1 T1T1 S’1S’1 T’1T’1

31. Multicut Lower Bound Certificates Maximum Multicommodity Flow = n k-1 – Route between 1 s-t pair Network Coding? – Network Coding Rate ≥

32. Vertex Multicut Given: A graph G = (V,E) k source-sink pairs All nodes capacity 1 Find: A min-cost subset of V such that on removal all source-sink pairs are disconnected S S S t t t

33. Hypergrid(3,3)

34. A 1 × I p, A 2 × I n t S t s ( v 1, v 1 ’ ) ( v 5, v 1 ’ ) ( v 4, v 1 ’ ) ( v 3, v 1 ’ ) ( v 2, v 1 ’ ) ( v 1, v 2 ’ ) ( v 5, v 2 ’ ) ( v 4, v 2 ’ ) ( v 3, v 2 ’ ) ( v 2, v 2 ’ ) ( v 1, v 3 ’ ) ( v 5, v 3 ’ ) ( v 4, v 3 ’ ) ( v 3, v 3 ’ ) ( v 2, v 3 ’ ) ( v 1, v 4 ’ ) ( v 5, v 4 ’ ) ( v 4, v 4 ’ ) ( v 3, v 4 ’ ) ( v 2, v 4 ’ ) ( v 1, v 5 ’ ) ( v 5, v 5 ’ ) ( v 4, v 5 ’ ) ( v 3, v 5 ’ ) ( v 2, v 5 ’ )