On the Partition Bound for Undirected Unicast Network Information Capacity Mohammad Ishtiyaq Qureshi and Satyajit Thakor School of Computing and Electrical Engineering Indian Institute of Technology Mandi Himachal Pradesh, India email: D15063@students.iitmandi.ac.in, satyajit@iitmandi.ac.in 21-26 June, 2020
Outline 1 Introduction Network coding vs routing The conjecture Three bounds Three new results 2 Background Network model Partition bound Computing the bound Proof of the conjecture for a class of networks 3 Main results On computing opt( I ) Optimal routing for Type-I and Type-II networks A result on sets of networks 4 Conclusion 5 References Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 2 / 26
Introduction Network coding vs routing A network is unicast if each sink demands exactly one source. The tuple ( γ, δ ) = (capacity on the edge , one bit flow in the edge) in the figure. Figure 1 shows clear advantage of network coding over routing alone. No such advantage is seen in undirected unicast networks till date. (a) An optimal routing scheme can (b) Network coding can achieve a achieve a sum rate of 1 bit/network sum rate of 2 bit/network use. use. Figure 1: A two source-sink pairs directed unicast network. Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 3 / 26
Background A challenging conjecture Conjecture (Muliple unicast conjecture [1]) For k independent unicast sessions with a desired rate vector r = ( r 1 . . . , r k ) , if the network is undirected and fractional routing is allowed, then r is achievable with network coding if and only if it is achievable with routing alone. 1 Z. Li and B. Li, “Network coding in undirected networks,” in Proc. 38th Annu. Conf. Inf. Sci. Syst. (CISS) , 2004. Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 4 / 26
Introduction Three bounds on symmetric rate of information flow Sparsity bound is a trivial upper bound for commodity [6] and for information capacity [7]. Linear programming bound [9] is not computable in practice for even small network instances. Partition bound and its computation was given in [10]. 10 S. Thakor and M. I. Qureshi, “Undirected Unicast Network Capacity: A Partition Bound,” in IEEE International Symposium on Information Theory , pp. 1-5, July 2019. Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 5 / 26
Introduction Three new results 1 The decision version problem of computing the partition bound is NP-complete. 2 A proof of optimal routing schemes obtaining symmetric rate of information flow for two classes of networks are given. 3 The existence of a network for which the partition bound is tight and achievable by routing and is not an element of the netwoks in [12] is shown. 12 X. Yin, Z. Li, Y. Liu, and X. Wang, “A reduction approach to the multiple-unicast conjecture in network coding,” IEEE Trans. Inform. Theory , vol. 64, pp. 4530-4539, June 2018. Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 6 / 26
Background Network model An undirected network is denoted as G = ( V, E, I, s, t ). V is the set of nodes. E is the set of edges of the form e = { u, v } . I is the set of source indices. s : I �→ V and t : I �→ V are mappings. I ( α, β ) = { i : s ( i ) ∈ α, t ( i ) ∈ β } , α, β ⊆ V . Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 7 / 26
Background A bound for general networks Theorem (Partition bound [10]) For an undirected network G = ( V, E, I, s, t ) , the symmetric rate of information flow is upper bounded as | E | r ≤ � n | I | + max P i =1 | I ( P i , P i ) | where P is a partition of V into independent sets P 1 , . . . , P n . Definition (Symmetric rate of information flow) For undirected network G = ( V, E, I, s, t ), the symmetric rate of information flow is a scalar r such that ( r i = r | i ∈ I ) is achievable. 10 S. Thakor and M. I. Qureshi, “Undirected Unicast Network Capacity: A Partition Bound,” in IEEE International Symposium on Information Theory , pp. 1-5, July 2019. Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 8 / 26
Background Computing the bound To compute the bound, it is sufficient to consider a particular partition. A partition, such that total no. of source-sink pairs in a same partition set is maximized. Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 9 / 26
Background Let P ∗ be an optimal partition. opt( I ) be a biggest subset of I such that for all i ∈ opt( I ) we have { s ( k ) , t ( k ) } ⊆ P i for some P i ∈ P ∗ . Let the set of neighbouring nodes of u be ne( u ) = { v ∈ V : { v, u } ∈ E } . ˆ I = { i ∈ I : { s ( i ) , t ( i ) } / ∈ E } . For k ∈ ˆ I , conf( k ) = { l ∈ ˆ I : [ t ( l ) ∈ { s ( k ) , t ( k ) } , s ( l ) ∈ ne( s ( k ))] or [ s ( l ) ∈ { s ( k ) , t ( k ) } , t ( l ) ∈ ne( s ( k ))] } . Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 10 / 26
Background Figure 2: An optimal partition P ∗ for a network with 3 partition sets: P 1 = { v 1 , v 2 } , P 2 = { v 3 , v 4 } and P 3 = { v 5 , v 6 } . For example, for the network shown in Figure 2, ˆ I = { 1 , 2 , 3 , 5 } and conf( k = 3) = { 5 } . Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 11 / 26
Background There can be two possibilities for any k ∈ ˆ I . 1 { s ( k ) , t ( k ) } ⊆ P i for some P i then opt(ˆ I ) = { k } ∪ opt(ˆ I \ conf( k )). 2 { s ( k ) , t ( k ) } � P i for any P i then opt(ˆ I ) = opt(ˆ I \ { k } ). Hence the recursive formula | opt( I ) | = | opt(ˆ I ) | = max {| opt(ˆ I \ { k }| , 1 + | opt(ˆ I \ conf( k )) |} . Corollary For an undirected network G = ( V, E, I, s, t ) , the symmetric rate of information flow is upper bounded as | E | r ≤ . | I | + | opt(ˆ I ) | Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 12 / 26
Background The cut-set associated with a partition α, α c of V is: cs( α, α c ) = {{ u, v } ∈ E : u ∈ α, v ∈ α c } . There exists a path of length n from node u to v if there exists nodes u = v 1 , . . . , v n = v , and edges e 1 , . . . , e n − 1 where e i = { v i , v i +1 } . A path from u to v , denoted as P u,v , is defined as the set of edges involved in it. The distance between nodes u and v , denoted as d u,v , is the number of edges in a shortest path connecting the nodes. Let P u,v be the set of simple paths from node u to v . Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 13 / 26
Background Definition (Orthogonal set F ) A set of edges F is orthogonal to session i if every shortest path from s ( i ) to t ( i ) crosses F at most once, i.e., if | P s ( i ) ,t ( i ) | = d s ( i ) ,t ( i ) then | P s ( i ) ,t ( i ) ∩ F | ≤ 1 . Definition (Compatible set F ) A set of edges F is called compatible with session i if every shortest path from s ( i ) to t ( i ) intersects F the minimum number of times among all the ( s ( i ) , t ( i ))-paths, i.e., for all P s ( i ) ,t ( i ) , P ′ s ( i ) ,t ( i ) ∈ P s ( i ) ,t ( i ) such that | P s ( i ) ,t ( i ) | = d s ( i ) ,t ( i ) | P s ( i ) ,t ( i ) ∩ F | ≤ | P ′ s ( i ) ,t ( i ) ∩ F | . 12 X. Yin, Z. Li, Y. Liu, and X. Wang, “A reduction approach to the multiple-unicast conjecture in network coding,” IEEE Trans. Inform. Theory , vol. 64, pp. 4530-4539, June 2018. Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 14 / 26
Background Theorem (Theorem 3, [12]) Let G be a family of networks that is closed under edge contractions. If G contains a network with a coding advantage, there is a network G ∈ G satisfying the following properties: 1 every cut-set is not orthogonal to some session; 2 for any two disjoint cut sets cs ( α, α c ) and cs ( β, β c ) if F = cs ( α, α c ) ∪ cs ( β, β c ) is compatible with all sessions, then there are at least two source-sink pairs with a shortest path intersecting F more than twice. Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 15 / 26
Background Theorem (Contrapositive statement of Theorem 3, [12]) If G does not contain a network satisfying 1 (non-orthogonality) and 2 (compatibility), network coding is unnecessary in G . Corollary (Corollary 3, [12]) Network coding is unnecessary in networks with at most 7 nodes, 3 sessions and uniform link lengths, except the network shown in Figure 3 for which the network coding capacity is yet unknown. Figure 3: A network appeared in [12]. Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 16 / 26
Main results Main result-I 1 INDEPENDENT SET problem: For G = ( V, E ), does there exists an independent set of size k ? 2 OPTIMAL-PAIRS DECISION problem: For G = ( V, E, I, s, t ), is opt( I ) at least k ? Theorem OPTIMAL-PAIRS DECISION is NP-complete. NP-completeness is proved by reducing an instance of the problem 1 to a problem 2. Qureshi & Thakor (IIT Mandi) On the Partition Bound 21-26 June, 2020 17 / 26
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