I Introduction
Network coding outperforms routing in directed acyclic networks. However, for undirected unicast networks, it was conjectured in 2004 [1, 2] that network coding has no rate benefit over routing. This conjecture has found significance in theoretical computer science, see, e.g., [3, 4]. Despite its importance, the conjecture has been verified for only a handful of network instances and families of networks.
Three upper bounds on the symmetric information rate (i.e., the required information rate for each session is the same) in undirected unicast networks are known: the sparsest cut, the LP bound and the partition bound. The notion of the weighted sparsest cut was given in [5] for graphs, where weights are assigned to edges and can be seen as edge capacity. It was discussed in the context of multicommodity flow networks in [6] and in the context of information networks in [7]. Sparsest cut is an important upper bound for multisession undirected unicast commodity capacity but it also holds for information capacity. In [8], it has been shown that the decision version of the unweighted sparsest cut problem is NPcomplete where the term unweighted refers to restricting the edge capacity to be unit for every edge. This is shown by considering the problem for a special class of multisession undirected networks called uniform sparsest cut problem. The LP bound [9, Chapter 15], [7] has an exponential number of variables as well as an exponential number of constraints and hence it is not computable in practice for even small network instances, see, e.g., [10, Figure 6]. In [11], we presented a new informationtheoretic bound called the partition bound. A parameter is defined as an optimal partition which delivers the partition bound. A recursive formula to compute this parameter and an algorithm are given. The algorithm has exponential complexity.
This paper presents three results on the partition bound. In Section II, we present the network model and review the undirected unicast network coding conjecture, the partition bound and the associated parameter , and known results on the proof of the conjecture for a family of networks. In Section III, we present the main contributions of the paper summarized as follows:

We show that, as is the case with the unweighted sparsest cut problem, the decision version problem of computing the partition bound is NPcomplete.

In [12], TypeI and TypeII networks were defined and the conjecture was proved for these networks. In [11], these networks were generalized and the conjecture was proved. However, the proof of optimal routing schemes was omitted in both works. We establish a recurrence relation for number of edges in theses networks and give complete proofs of optimal routing schemes.

In [10], the conjecture was proved for a new class of networks and it was shown that all the network instances for which the conjecture is proved previously are elements of this class. We show the existence of a network for which the partition bound is tight and achievable by routing and is not an element of this class.
Finally, a conclusion is presented in Section IV.
Ii Background
An undirected information network is denoted where is the set of nodes, is the set of edges of the form and is the set of sessions with . Assume that each edge has 1 bit (unit) capacity. The source and sink node of a session are described by the mappings and respectively, e.g., session is available at (the source node) and demanded at (the sink node). We consider unicast networks in which a session located at the source node is demanded by exactly one sink node and .
Iia The partition bound
Theorem 1 (Partition bound, [11])
For an undirected network , the symmetric rate of information flow is upper bounded as
(1) 
where is a partition of into independent sets and .
Let be an optimal partition, i.e., a partition such that the total number of sourcesink pairs in a same partition set is maximum and opt be a biggest subset of such that for all we have for some . Since for some if then cannot be in opt, it is sufficient to restrict the search for opt in
(2) 
Now, let the set of neighboring nodes of be . There are two possibilities for any

for some then , where conf (abbreviated conflicting subset) is

for any then .
These two possibilities render a recursive formula
(3) 
Hence, alternatively, the partition bound can be described as
(4) 
The value for a given network can be computed using Algorithm 1 in [11]. The complexity of the algorithm is exponential.
IiB Proof of the conjecture for a class of networks
In this subsection, we review the main results in [10]. A cut is a partition of , e.g., . The cutset associated with a partition is
There exists a path of length from node to if there exists nodes end edges where . Here, a path from to , denoted , is defined as the set of edges involved in it. The distance between nodes and , denoted , is the number of edges in a shortest path connecting the nodes. Let be the set of simple paths from node to .
Definition 1
A set of edges is orthogonal to session if every shortest path from to crosses at most once, i.e., if then
(5) 
Definition 2
A set of edges is called compatible with session if every shortest path from to intersects the minimum number of times among all the paths, i.e., for all such that
(6) 
Theorem 2 (Theorem 3, [10])
Let be a family of networks that is closed under edge contractions. If contains a network with a coding advantage, there is a network satisfying the following properties:

every cutset is not orthogonal to some session;

for any two disjoint cut sets , , if is compatible with all sessions, then there are at least two sourcereceiver pairs with a shortest path intersecting more than twice.
In other words, if does not contain a network satisfying (P1) and (P2), network coding is unnecessary in .
Let be the class of network as defined in Theorem 2 such that it does not contain a network satisfying (P1) and (P2).
Iii Main Results
Iiia On computing
It is well known that the decision version of the sparsest cut problem is NPcomplete [8]
. The computation of linear programming bound
[7] too requires exponential time since it has exponential input constraint size for a given network. In this subsection, we show that the decision version problem of finding , OptimalPairs Decision, is NPcomplete by showing reduction of Independent Set problem which is a well known NPcomplete problem.OptimalPairs Decision problem: For , is at least ?
Independent Set problem: For , does there exists an independent set of size ?
Theorem 3
OptimalPairs Decision is NPcomplete.
Proof:
First note that OptimalPairs Decision is an NP problem: Consider a certificate that is the tuple where and is the corresponding partition. It can be checked in polynomial time whether for each , for some .
We prove NPcompleteness by reducing an instance of the Independent Set problem to an OptimalPairs Decision problem which is known to be NPcomplete. The reduction gadget is as follows: For a given graph , fix some node and let
Now for each , construct with such that each node in is a sink for a distinct and is the source node for all . It remains to show that there exists an independent set of size in if and only if has at least elements in for some .
Assume that there exists an independent set of size in . Then observe that there exists an independent set of size such that it has at least one element of . Hence, assume without loss of generality that there exists some (if does not contain any node from then we can obtain another independent set of size from by removing any one element of it and adding into it). Now, note that since the sessions associated with sourcesink pairs such that is the source and nodes in are sinks are elements of and there are such sessions.
Now assume that there exists such that it has elements in for some . Given this with we can obtain the set
Note that is an independent set in and .
IiiB Optimal routing schemes for TypeI and TypeII networks
In this subsection, we first present two expressions, (7) and (9), for the cardinality of the edge set for TypeI and TypeII networks and then present proof of routing schemes achieving the partition bound for these networks.
Definition 3
TypeI partite network is a complete partite network with partition sets for all such that for every unordered pairs of nodes in a partition set, there is a sourcesink pair and there are no other sourcesink pairs.
Definition 4
TypeII partite network is a complete partite network with partition sets for all such that for every unordered pairs of nodes in the network, there is a sourcesink pair and there are no other sourcesink pairs.
First, note that for a complete partite graph with partition sets , the cardinality of its edge set can be expressed via the recurrence relation
(7) 
and hence,
(8) 
Alternatively, note that for any two partitions , there are number of edges between them and hence,
(9) 
We remark that there is an error in the expression of the cardinality of the edge set of TypeI and TypeII networks in [11].
Theorem 4
For TypeI partite networks, the partition bound is attainable by a routing scheme and hence the conjecture holds for all TypeI partite networks.
Proof:
The partition bound for a TypeI partite network is
(10) 
We prove the statement by induction.
Base case (): Consider a bipartite TypeI network with partition sets and . Assume that . The partition bound is
(11)  
(12) 
Assume that each edge has the capacity
(13) 
Now, we show a routing scheme achieving the symmetric rate . First, note that each sourcesink pair in has disjoint 2hops paths via nodes in . For each sourcesink pair in , transmit 1 bit through each 2hops path, i.e., path of length 2. In this manner, each edge is involved in transmitting bits (since, for a TypeI network, each node in has total number of sources or sinks). In a similar manner the remaining capacity of each edge can be used to transmit 1 bit for each sourcesink pair in through each 2hops path via nodes in . Thus, we can obtain the routing capacity of (refer to (8) or (9)).
Induction step: Assume that the statement is true for partite TypeI networks. Now consider a partite TypeI network with partition sets . Then in its partite TypeI subnetwork with the partition sets , the rate
(14) 
is achievable by routing. Now, assume that each edge in the partite TypeI network has the capacity
(15) 
There are disjoint 2hops paths vie the nodes in for each sourcesink pair in . All sourcesink pairs in each have attained (in the partitie network by induction hypothesis) the symmetric rate of
(16) 
But towards proving the statement we need to show that each sourcesink pair in the partite network attains the symmetric rate bits in total. Hence, for each , the remaining bits need to be sent for each sourcesink in via the 2hops paths through nodes in are
where we have used the recurrence relation for , see (7). Assume that bits are sent in this manner: bits via each 2hops path through nodes in . In a TypeI network, each node in has total number of sources or sinks and thus, each edge carries bits. The remaining capacity of each edge between nodes in and is
(17) 
Hence,
(18) 
bits can be sent for each sourcesink pair in via the 2hops paths through the nodes in . Thus, the total number of bits which can be sent for each sourcesink pair in via the 2hops paths though the nodes in the partition sets is
(19) 
It remains to show that the above expression (19) is indeed .
Thus, the routing scheme attains the partition bound and in doing so it fully utilizes the capacity of each edge.
Theorem 5
For TypeII partite networks, the partition bound is attainable by a routing scheme and hence the conjecture holds for all TypeII partite networks.
Proof:
The partition bound for a TypeI partite network is
(20)  
(21)  
(22) 
Now, assume that each edge in the partite TypeI network has the capacity
(23) 
By Theorem 4, we can attain the rate for each session such that for some and this will utilize bits capacity for each edge in the network. Hence, each edge can carry further bits. This fraction of the capacity can be used to transmit bits for each session such that ; source and sink node for each of such sessions are directly connected by an edge. Thus the partition bound is achieve by considering the routing scheme for TypeI networks described in Theorem 4 and then superpositioning the flow for each session such that the respective sourcesink pair is one hop away.
IiiC A result on sets of networks: TypeI and
In this section, we show the existence of a TypeI partite network that satisfies the properties (P1) and (P2) in Theorem 2. Hence, there exists a network that is not in and for which the partition bound is tight and achievable by a routing scheme.
The following proposition shows that not all TypeI partite networks where satisfy both (P1) and (P2).
Proposition 1
There exist TypeI partite networks, , for which (P1) is violated.
Proof:
The smallest example is the following: Consider TypeI partite network with and hence . Consider the cutset where . Then it is straightforward to verify that is orthogonal to all sessions and hence the property (P1) is violated.
In fact, using the similar cutset construction, it can be shown that all TypeI partite networks with and violate (P1).
Theorem 6
There exists a TypeI partite network, , that satisfies (P1) and (P2).
Proof:
Consider the TypeI partite network in Figure 1. It has 7 nodes , 16 edges and 5 sessions, i.e., . The mappings and are depicted in the figure, e.g., and . Note that, in a typeI network, every shortest path from to is of length 2 for each session .
Note that there are possible distinct cutsets of the form in the network (similar bipartitions of a set are also considered in [13] but in a different context). Without loss of generality, assume that contains the node . Thus we need to consider all cutsets such that contains the node and it is a proper subset of (since form a partition, they cannot be the empty set by definition). Also, there are inherent symmetries in the network: permutations on the elements of the sets , , , results in essentially the same network since these permutations only changes the mappings and but do not affect the network topology.
In columns 13 of Table I, we list the sets for all such cutsets , corresponding nonorthogonal sessions, and symmetric cases of . We note that for each cutset there exists at least one nonorthogonal session and hence (P1) is satisfied.
Now, for each possible subset in column 1, column 4 shows all possible subsets of such that , i.e., the cutsets are disjoint. For each tuple such that , we verify that is not compatible with all sessions; details are omitted (refer to Definition 2 and note that every shortest path from a source to the respective sink is of length 2 and covers some shortest paths but not all shortest paths from to for at least one . This means that is not compatible with all sessions. For example, consider and . Then, is the set of all edges between the nodes in and and for session , a shortest path via has nonempty intersection with whereas a shortest path via has nonempty intersection with . Thus, is not compatible with session 3). Hence, (P2) is vacuously true for the TypeI partite network in Figure 1.
Nonorth.  Symmetric cases of  

, , ,    
, , , ,      
, ,    
,    
, , ,  ,    
, ,  ,    
, , , ,    
,    
,  , , , ,     
,  , , ,    
,  , ,    
, ,    
, , ,      
, ,  ,  
,  , , , , , , ,    
, , , ,      
, ,    
,  ,  , ,  
, , ,  , 
Also, note that the network in Figure 1 does not fall into the family of networks for which the conjecture is proved in Corollary 1. Thus, we have established the existence of a network not in such that the partition bound on the symmetric rate is tight and achievable by a routing scheme and hence the conjecture holds for the network. However, Proposition 1 shows that not all TypeI networks satisfy properties (P1) and (P2).
Iv Conclusion
We showed that, as is the case with the sparsest cut problem, the decision version of computing the partition bound is NPcomplete and gave a complete proof of the optimal routing schemes for TypeI and TypeII networks. Also, We showed that the partition bound is tight and achievable by routing for networks for which the conjecture has not been proved previously. One interesting future direction is to characterize the class of all networks for which the partition bound is tight and achievable by routing.
Acknowledgment
This work is supported by SERB, DST, Government of India, under Extra Mural Scheme SB/S3/EECE/265/2016.
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