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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

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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 achieve a sum rate of 1 bit/network use. (b) Network coding can achieve a sum rate of 2 bit/network use.

Figure 1: A two source-sink pairs directed unicast network.

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Background

A challenging conjecture Conjecture (Muliple unicast conjecture [1]) For k independent unicast sessions with a desired rate vector r = (r1 . . . , rk), 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.

• 1Z. Li and B. Li, “Network coding in undirected networks,” in Proc. 38th Annu.
• Conf. Inf. Sci. Syst. (CISS), 2004.

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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].

• 10S. Thakor and M. I. Qureshi, “Undirected Unicast Network Capacity: A

Partition Bound,” in IEEE International Symposium on Information Theory, pp. 1-5, July 2019.

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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.

• 12X. 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.

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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 .

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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 r ≤ |E| |I| + maxP n

i=1 |I(Pi, Pi)|

where P is a partition of V into independent sets P1, . . . , Pn. 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 (ri = r | i ∈ I) is achievable.

• 10S. Thakor and M. I. Qureshi, “Undirected Unicast Network Capacity: A

Partition Bound,” in IEEE International Symposium on Information Theory, pp. 1-5, July 2019.

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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.

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Background

Let P ∗ be an optimal partition.

• pt(I) be a biggest subset of I such that for all i ∈ opt(I) we have

{s(k), t(k)} ⊆ Pi for some Pi ∈ 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))]}.

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Background

Figure 2: An optimal partition P ∗ for a network with 3 partition sets: P1 = {v1, v2}, P2 = {v3, v4} and P3 = {v5, v6}.

For example, for the network shown in Figure 2, ˆ I = {1, 2, 3, 5} and conf(k = 3) = {5}.

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Background

There can be two possibilities for any k ∈ ˆ I.

1 {s(k), t(k)} ⊆ Pi for some Pi then opt(ˆ

I) = {k} ∪ opt(ˆ I \ conf(k)).

2 {s(k), t(k)} Pi for any Pi 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 r ≤ |E| |I| + |opt(ˆ I)| .

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Background

The cut-set associated with a partition α, αc of V is: cs(α, αc) = {{u, v} ∈ E : u ∈ α, v ∈ αc}. There exists a path of length n from node u to v if there exists nodes u = v1, . . . , vn = v, and edges e1, . . . , en−1 where ei = {vi, vi+1}. A path from u to v, denoted as Pu,v , is defined as the set of edges involved in it. The distance between nodes u and v, denoted as du,v, is the number of edges in a shortest path connecting the nodes. Let Pu,v be the set of simple paths from node u to v.

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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 |Ps(i),t(i)| = ds(i),t(i) then |Ps(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 Ps(i),t(i), P′

s(i),t(i) ∈ Ps(i),t(i)

such that |Ps(i),t(i)| = ds(i),t(i) |Ps(i),t(i) ∩ F| ≤ |P′

s(i),t(i) ∩ F|.

• 12X. 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.

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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, αc) and cs(β, βc) if

F = cs(α, αc) ∪ cs(β, βc) is compatible with all sessions, then there are at least two source-sink pairs with a shortest path intersecting F more than twice.

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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].

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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

• pt(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.

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Main results

Main result-II Definition (Type-I n-partite networks, [11], [10]) Type-I n-partite network G = (V, E(n), I, s, t) is a complete n-partite network with partition sets Pi, |Pi| ≥ 2 for all i ∈ {1, . . . , n} such that for every unordered pairs of nodes in a partition set, there is a source-sink pair and there are no other source-sink pairs. Definition (Type-II n-partite networks, [11], [10]) Type-II n-partite network G = (V, E(n), I, s, t) is a complete n-partite network with partition sets Pi, |Pi| ≥ 2 for all i ∈ {1, . . . , n} such that for every unordered pairs of nodes in the network, there is a source-sink pair and there are no other source-sink pairs.

• 11A. Al-Bashabsheh and A. Yongacoglu, “On the k-pairs problem,” in IEEE Int.
• Symp. Inform. Theory, pp. 1828-1832, July 2008.

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Main results

Main result-II continued . . . Theorem For Type-I n-partite networks, the partition bound is attainable by a routing scheme and hence the conjecture holds for all Type-I n-partite networks. The routing scheme is given by using mathematical induction. Theorem For Type-II n-partite networks, the partition bound is attainable by a routing scheme and hence the conjecture holds for all Type-II n-partite networks. The partition bound is achieved by considering the routing scheme for Type-I networks and then superpositioning a flow for each remaining session.

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Main results

Main result-III Theorem There exists a Type-I n-partite network, n ≥ 3, that satisfies the conditions 1 (non-orthogonality) and 2 (compatibility).

Figure 4: A Type-I 3-partite network with three partition sets: P1 = {v1, v2}, P2 = {v3, v4} and P3 = {v5, v6, v7}.

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Main results

Main result-III continued . . . To check the satisfiability of property 1. Consider α = {v1}. It can be seen, for a path Ps(3),t(3), |Ps(3),t(3) ∩ cs(α, αc)| = 2. In this way, it can be checked that all the cut-sets are non-orthogonal to some session.

Figure 5: The Figure 4 is repeated here.

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Main results

Main result-III continued . . . To check the satisfiability of property 2. Consider α = {v1}. For a given α, β is the all possible subsets of V such that cs(α, αc) ∩ cs(β, βc) = φ. Hence β = {v2}. F1 = cs(α, αc) ∪ cs(β, βc) is the set of all edges between the nodes in P1 and P c

1.

One can see that, F1 is not compatible with session 3, and this can be checked for all possible tuple (α, β).

Figure 6: The Figure 4 is again repeated here.

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Main results

Main result-III continued . . . Proposition There exist Type-I n-partite networks, n ≥ 3, for which condition 1 (non-orthogonality) is violated. We give a cut-set which is orthogonal to all the sessions in Type-I 3-partite network with Pi = {s(i), t(i)}, i ∈ {1, 2, 3}.

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Main results

Conclusion

1 The decision version of computing the partition bound is proved to

be NP-complete.

2 A complete proof of the optimal routing schemes for Type-I and

Type-II networks are given.

3 The partition bound is shown to be tight and achievable by

routing for networks for which the conjecture has not been proved previously.

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References I

[1]

• Z. Li and B. Li, “Network coding in undirected networks,” in Proc. 38th Annu. Conf. Inf. Sci. Syst. (CISS), 2004.

[2]

• N. J. A. Harvey, R. D. Kleinberg, and A. R. Lehman, “Comparing network coding with multicommodity

flow for the k-pairs communication problem,” Tech. Rep. MIT-LCS-TR-964, 2004. [3]

• M. Adler, N. J. A. Harvey, K. Jain, R. Kleinberg, and A. R. Lehman, “On the capacity of information

networks,” in ACM-SIAM Symp. on Disc. Algo., pp. 241-250, 2006. [4]

• A. Farhadi, M. Hajiaghayi, K. G. Larsen, and E. Shi, “Lower bounds for external memory integer sorting

via network coding,” CoRR, vol. abs/1811.01313, 2018. [5]

• D. W. Matula, “Concurrent Flow and Concurrent Connectivity on Graphs”, p. 543-559. USA: John Wiley &

Sons, Inc., 1985. [6]

• T. Leighton and S. Rao, “Multicommodity max-flow min-cut theorems and their use in designing

approximation algorithms,” J. ACM, vol. 46, pp. 787-832, Nov. 1999. [7]

• N. Harvey, R. Kleinberg, and A. Lehman, “On the capacity of informa- tion networks,” IEEE Trans. Inform.

Theory, vol. 52, pp. 2345-2364, Jun. 2006. [8]

• P. Bonsma, H. Broersma, V. Patel, and A. Pyatkin, “The complexity of finding uniform sparsest cuts in

various graph classes,” Journal of Disc. Algo., vol. 14, pp. 136-149, 2012. [9]

• R. W. Yeung, Information Theory and Network Coding. Springer, 2008.

[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. [11]

• A. Al-Bashabsheh and A. Yongacoglu, “On the k-pairs problem,” in IEEE Int. Symp. Inform. Theory, pp.

1828-1832, July 2008. [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.

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Thank you

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