Hypergraph Modeling and Approximation Algorithms for the Minimum Length Link Scheduling in Multiuser MIMO Networks

This paper investigates the problem of the minimum length link scheduling (MLLS) in multiuser MIMO (MU-MIMO) networks. Generally, in the networks with MU-MIMO capability, the number of concurrent transmissions can be as large as that of antenna elements at the receiver. As a result, link interference is no longer binary but demonstrates a strong correlation among multiple links, which cannot be captured by the conventional conflict graph interference model. Hence, we propose a novel hypergraph interference model, which can accurately and efficiently characterize the relationship of multiple interferences induced by concurrent transmissions, and provide a tractable formalization of theminimum length link scheduling inMU-MIMOnetworks (MU-MIMO MLLS). Afterwards, we prove that the MU-MIMO MLLS problem is NP-hard and introduce two approximation algorithms to find the near-optimal feasible schedule. Finally, extensive simulation experiments are presented.


Introduction
Link scheduling in wireless networks is one of the key and classical research topics for the communication and network communities.In wireless networks, nodes communicate with each other through a shared common channel.On one hand, to shorten the communication latency, node pairs are required to communicate concurrently as much as possible; on the other hand, concurrent transmissions induce the cochannel interference, which results in severe deterioration or even interruption of an ongoing transmission.To control the cochannel interference, link scheduling is required to coordinate the transmission of different links.A good schedule not only avoids communication failures by silencing the interferers of every receive node in each time slot but also minimizes the number of time slots and hence the communication latency.To sum up, link scheduling in wireless networks usually requires the minimum length conflict-free assignment of time slots in which each communication link is activated at least once [1,2] and the link scheduling problem is difficult because many subsets of nonconflicting nodes are candidates for each time slot, and the subset selected for transmission in one slot affects the number of concurrent transmissions in the next time slot.
The interference model, which characterizes the interference relationship of communication links, has a major impact on the complexity of wireless link scheduling.The conflictgraph based interference model, as a simple and powerful modeling tool, has been widely employed [3,4].In [3], Hajek and Sasaki consider the MLLS problem and propose an optimal scheduling scheme with polynomial time complexity.However, they consider only the primary interference; that is, two links are interfered with each other when they are neighboring.In practice, the transmission of a link can interrupt that of the other even when they are not neighboring.At this time, link scheduling is shown as an NPhard problem.Just recently, constant factor approximation bounds have emerged for link scheduling under the conflictgraph interference model with -hop interference [4].
The conventional conflict graph interference model cannot capture the interference when multiple-input multipleoutput (MIMO) is deployed.As a revolutionary technology from an information theoretic perspective and physical layer communications, MIMO has been introduced as a de facto component of wireless standards [5,6].In the networks with multiuser MIMO capability (MU-MIMO networks), a node with multiple antenna elements can decode a composite signal from different transmitters.Therefore, more communication links are permitted to transmit concurrently.In general, the number of concurrent transmissions can be as large as that of antenna elements at the receiver.For example, given a network consisting of a receiver  1 with two antenna elements, and three senders  1 ,  2 , and  3 with single antenna, respectively, consider the interference of  1 on  2 : when  3 is silent, there is no interference; however, when  3 is active,  1 and  2 cannot transmit simultaneously.That is, the interference relationship between  1 and  2 has strong correlation with the activity of  3 .Such dependence has not been characterized in the binary conflict graph model.
Recently there is a few existing works in the literature trying to explore the optimal link scheduling problem in MU-MIMO networks.Chu and Wang [7] presented an integer linear programming (ILP) formulation of the optimal link scheduling problem in MU-MIMO networks, and proposed an opportunistic scheduling algorithm that can adaptively select different transmission strategies.Blough et al. [8] considered the same problem in MU-MIMO networks with interference suppression capabilities, and formulized the one-shot optimal link scheduling problem as an integer linear programming.However, all of them cannot be applied directly to the MLLS with MU-MIMO capability: on one hand, the goal of [7,8] is maximizing throughput while the objective of MU-MIMO MLLS is minimizing the schedule length; on the other hand, both work in [7,8] model the optimal link scheduling problem in MU-MIMO networks as an ILP formulation, which is well-defined in mathematics while difficult to find approximation algorithms providing a guaranteed performance.
In this paper, we address the issue of the minimum length link scheduling in MU-MIMO networks, aiming for providing at least one transmission time slot for every communication link with a minimum schedule length.Firstly, we propose a novel interference model based on hypergraph to characterize multiple interferences in MU-MIMO networks.The concept of hypergraph is extended from graph, and is a nature way for representing multiple interferences among multiple links.Afterwards, two algorithms for the problem of MU-MIMO MLLS are proposed: the first one is a performance guaranteed approximation algorithm whose approximation ratio is at most 2/log 2 (); the second one is a time efficient heuristic algorithm based on degrees of interference (DOI) greedy.Extensive simulations are presented to test how well the proposed algorithms based hypergraph model may be applied in practice.
The remainder of this paper is organized as follows.Section 2 presents the integer programming formalization and the proposed hypergraph interference model of MU-MIMO MLLS.Section 3 introduces two algorithms to approximate the optimal solution.Section 4 shows the extensive simulation results.Finally, in Section 5, conclusions are given.

System Model
2.1.Problem Statement.Consider a wireless network  of  communication links, and  stationary nodes, each of which is equipped with  antenna elements.A link with transmit node  and receive node  is denoted by   .We assume that: (i) nodes work in the half-duplex mode, that is a node can either transmit or receive, but not at the same time; (ii) every link is of a single data stream as to enhance as many concurrent transmissions as possible.
The use of a MIMO antenna array is typically modeled as degree of freedom (DOF) model [9].For simplicity, a receive node with  antenna elements has up to  DOFs, which can be used for multiplexing  desired data streams and suppressing (-) interferences from other neighboring concurrent transmission links.And if and only if the number of the concurrent transmission link in vicinity of receive node , including link   , does not exceed , then receive node  can successfully decode the data stream carried by link   .
Firstly, notations used in the IP formulation are specified as follows:  = { 1 ,  2 , . . .,   }: set of transmission time slots for link scheduling;  = { 1 ,  2 , . . .,   }: set of communication links to be scheduled;  = {V 1 , V 2 , . . ., V  }: set of wireless nodes; (V): set of links incident into node V;   ,   : transmit node and receive node of link   , respectively; (): set of links interfered by   ;    : an integer variable, equals to 1 iff link   is active at the th time slot, otherwise equals 0.
The objective function (1) states that we aim to minimize the length of scheduling time slots.Constrain (2) ensures that each of communication links is assigned at least one transmission time slot in a schedule period.The next constraint (3) provides that every active communication link does not suffer too much cochannel interference.And to make constrain (3) always feasible when    = 0, the constant parameter   needs to be set large enough.A sufficiently large value is   = −.From constraint (4), we know that the size of search space is 2 () , rising at an exponential rate with the network size .
The disadvantage of the IP formulation is that it is difficult to find approximation algorithms, but the IP formulation provides a way to calculate the optimal solution through exhaustive searching in cases of networks with a small size.

Hypergraph Modeling MU-MIMO MLLS.
Typically, a wireless network can be modeled as a directional graph  = (, ).The set  includes all the nodes in the network, and  is a link set, of which for a pair of nodes , V ∈ , a directional edge  = (, V) is a member of  if node 's signals can be decoded successfully at node V in the absence of interference.Further, the interference relationship between transmission links is modeled as a conflict graph   = (  ,   ), where   is the set of links (in fact,   is the set  in the graph ) and   is the set of conflicts; that is, (, ) ∈   means that the concurrent transmission of link   and   is invalid.However, the conflictgraph interference model is restrictive and cannot descript the interference relationship among three links or more.For example, even when (, ) ∉   , (, ) ∉   , and (, ) ∉   are given, we cannot conclude whether the concurrent transmission of link   ,   , and   is valid or not.While generally the interference relationship in MU-MIMO networks involves multiple communication links.Thus, the conflict graph interference model is not appropriate to handle the case of MU-MIMO networks.

Hypergraph Model
Construct.Here, we present a new graph model, hypergraph [10,11], to characterize the interference relationship in MU-MIMO networks.Let   = (  ,   ) denote a hypergraph, where   is the set of links, the same as   and , and   = { 1 ,  2 , . . .,   } is also the set of conflicts, similar to   .The difference between   and   is that the element of   is limited to 2-tuple of , while the element of   , named as hyperedge, is extended to a subset of .And for any link set  ⊆   ,  ∈   if and only if satisfies three rules as follows.

R1: (Invalidity Rule
).When all the links of  are assigned in a same transmission time slot, there is at least one link in  failing to transmit data successfully.

R2: (Minimality Rule).
If any link in  is removed, no failure will occur if only the remaining links in  are scheduled.

Hypergraph Model
Problem Formalization.Given a hypergraph model   = (  ,   ), the problem of MU-MIMO MLLS can be transformed to the problem of finding the optimal coloring of the hypergraph   : give all node in   a color, and for every hyperedge  ∈   , having at least two nodes as endpoints, the nodes of this hyperedge that connect  are not all of the same color.Denote the hypergraph coloring solution by  = { 1 ,  2 , . . .,   }, where   is the link set with the same color .Then, the coloring solution with the least number of colors is the optimal solution to the problem of MU-MIMO MLLS.
And the problem of MU-MIMO MLLS could be formulized as follows: minimize , where where colors() means the number of colors appeared in the hyperedge .
Example 1.Consider the network in Figure 1, where there are four communication links and seven nodes, all of which are equipped with two antenna elements.According to the proposed hypergraph model, we can model the network as hypergraph   = (  ,   ) showed in Figure 2, in which   = {  conflicts with the Minimality Rule R2.Considering (1) and (2) together, we can conclude that, for any  ∈   , card() =  + 1.

Lemma 3. The MU-MIMO MLLS problem is NP-hard.
Proof.We reduce the problem of finding the chromatic number of a  + 1-uniform hypergraph to the MU-MIMO MLLS problem.The chromatic number of a hypergraph   is the smallest number  such that   is -colorable and   is -colorable if its vertices can be colored using  different colors in such a way that for every hyperedge in   , the nodes of this hyperedge are not all of the same color.According to the hypergraph modeling above mentioned and Lemma 2, we know that every link scheduling solution corresponds to a coloring solution of hypergraph   .Meanwhile, due to the three construction rules, every hypergraph coloring is also a link scheduling solution.Thus, the minimum scheduling length is equal to the chromatic number of the +1-uniform hypergraph   .And several results [12,13] show that it is NP-complete to optimally color -uniform hypergraphs, for various values of  ( ≥ 3).Specific statements can be found in the corresponding papers.

Algorithms
In this section, two algorithms based on the uniform hypergraph coloring for the problem of MU-MIMO MLLS are proposed.Here, we firstly give some definitions: given a uniform hypergraph   = (  ,   ), a link set  ⊆   is called independent if  does not contain any hyperedge in   .An -coloring of   is a mapping :   → { 1 , . . .,   } such that no hyperedge of   with one single color.Equivalently, an -coloring of   is a partition of the vertex set   into  independent sets.The chromatic number of   , denoted by (  ), is the minimal , for which   admits an -coloring.[14], we find an independent set with the following algorithm, named as performance-guaranteed (-) algorithm, whose approximation ratio is 2/log 2 ().It is worth noting that the idea of partitioning the vertex set of a -colorable hypergraph   on  into groups of size log   and performing an exhaustive search for an independent set of size log   in each group is due to Berger and Rompel's paper [15].Define
The idea of our proposed Algorithm - is simple: as long as there are some uncolored vertices (their union is denoted by ), call Procedure 1 for finding an independent set  in the subhypergraph spanned by , denoted by    , give the output  a fresh color, and update .
) times, and according to Lemma 4, the time complexity of Procedure 1 is   (1) ; hence, the time complexity of Algorithm - is polynomial in .Lemma 6.The approximation ratio of Algorithm - is at most 2/log 2 () (see, Algorithm 1).
Proof.For a hypergraph   = (  ,   ), we denote the chromatic number of   by ; that is,   is -colorable.Furthermore, the optimal coloring of   is the solution of coloring   by  colors.In addition, for any subhypergraph   , which is spanned by the subset  and denoted by    = (   ,    ) as above mentioned, due to    ⊆   , it is obvious that    is also -colorable.Through Procedure 1, the size of the independent set  picked by Algorithm - is at least ℎ  (|   |).And we know that ℎ  () is a positive, nondecreasing function.Then, Algorithm - produces a coloring with at most   () = ∑  =1 (1/ℎ  ()) colors, since each link in   contributes at most 1/ℎ  (  ) colors, where   is the number of links remaining in the subhypergraph    at the time when the link was assigned a color.It is proofed that when ℎ  () grows no faster than  1/ ( > 1),   () is at most /( − 1) ⋅ /ℎ  () [16].Specifically in our proposed Algorithm -, ℎ  () = log / log  grows no faster than  3/4 .Thereby, Algorithm - colors the -colorable hypergraph   on  vertices in at most 4 ⋅ log / log , then the approximation ratio of Algorithm - is at most () = 4 ⋅ log /( ⋅ log ).Finally, () ( ≥ 2) is decreasing function that is max(()) = 2/log 2 .

DOI-Based Greedy (𝐷-𝐺) Heuristic Algorithm. Although
Algorithm - can provide a guaranteed performance, its time complexity is   (1) .The distinguishing time complexity promotes us to design another time efficient heuristic coloring algorithm.
Recall that the goal of link scheduling is to keep the cochannel interference at a proper level.Here, the concept of degree of interference, denoted by DOI (), is defined to measure the interferences induced by a link  and presents the impact of the link  brought to the other links.DOI () is equal to the amount of communication links affected by the link  when link  is in activation.The main idea of Algorithm - is to assign the link with maximum degrees-of-interference as early as possible; hence, the interferences in the whole network would reduce drastically and then a relative larger independent link set might be picked in next scheduling time slots.To improve the performance, we refresh the DOI information every time after picking an independent link set.

Lemma 7. The time complexity of Algorithm
Proof.The bottleneck of Algorithm - is the procedure of QuickSort (   ), which is called at most  times, and the expected running time of QuickSort (   ) is ( log ).

Simulation and Discussion
We evaluate the performance of our proposed Algorithms - and - via extensive simulations.The network settings include wireless nodes that are distributed in a square area of 1250 m × 1250 m and form an ad hoc network with random or grid topology; each node is equipped with  antennas and has a communication range of 250 m and an interference range of 400 m (Figure 3).A simulation result is obtained by averaging over 50 runs of simulations, and all simulations run on a 2.8 GHz Intel Core Duo machine with 4 GB of RAM.  with the network size.Although Algorithm - has ASL performance advantage, the performance difference between Algorithm - and Algorithm - is less than 10% when the network size is controlled below 115 communication links.The comparison of the ART performance of three algorithms is shown in Figure 4(b).To represent the performance difference more effectively, we adopt the denary logarithm value of average running times.The results verify that the running time of Algorithm - is rising at an exponential rate with the network size; thus, Algorithm - can be only executed in the case of very small networks; Algorithm -, Algorithm -, and Algorithm - are of polynomial time complexity, while Algorithm - is of orders-of-magnitudes advantage over Algorithm - and Algorithm - in terms of ART performance.

Impact of Network Topology.
We perform extensive simulations with two types of network topology: a randomly generated topology and a grid.In both two networks, the number of communication links to be scheduled is set to the same value,  = 112.And the compared algorithms  are Algorithm -, Algorithm -, and Algorithm -.From Figure 5, we see that the efficiency of all the three algorithms is higher with random topology: with random topology, fewer transmission time slots and little running time are required.The reason is that in a grid topology, each wireless node is equidistant from its neighbors; hence, the distribution of communication links is more homogeneous and the signals of neighbors arrive at a receiver node with similar powers; this limits the choices for the scheduling policies, especially for Algorithm -.Besides, we find that in both networks, Algorithm - is of a huge ART performance benefits, compared with Algorithm - and Algorithm -.

Impact of Degree of
Freedom.Finally, we turn to perform simulations in networks with varied degree of freedom.The networks are with a rand generated topology and 112 communication links to be scheduled.As shown in Figure 6, the value of degree of freedom impacts the ASL performance significantly while it has relative little impact to the ART performance.That is because as the number of antenna elements (degree of freedom) is increased, the receive node is more effective and, thus, more communication links can be grouped in a same schedule time slot.

Conclusions
In this paper, we study the issue of the minimum length link scheduling in a wireless network with the MU-MIMO capability.Firstly, we formulate the MU-MIMO MLLS problem as an integer programming problem.Secondly, we introduce a novel and straight hypergraph method to model the multiple interferences, which cannot be captured by the traditional conflict graph model, and then we reformulate the MU-MIMO MLLS problem based on the new hypergraph model.And, we show that, the scheduling problem is NP-hard.Thirdly, we propose two efficient approximation solutions.For the first algorithm, named as Algorithm -, we show that the approximation ratio is at most 2/log 2 ().To reduce the time complexity, we also present a time efficient heuristic algorithm based on degree of interference greedy (Algorithm -).Finally, extensive simulation results show that (1) both Algorithm - and Algorithm - can scale to quite a large number of links and produce satisfying suboptimal solutions; (2) generally, Algorithm - has a better ASL performance than Algorithm -, while the time complexity of Algorithm - is largely reduced compared with that of Algorithm -.

Figure 6 :
Figure 6: Impact of degree of freedom.
condition in the DOF model, all links in  could concurrently transmit together; that is, the link set  is a feasible link set, which evidently conflicts with the Invalidity Rule R1; (2) if there is any hyperedge  ∈   , satisfying card() ≥  + 2, then after removing one link  in , card( \ {}) equals  + 1 and  \ {} is still an infeasible link set, which directly Procedure B(G H , k):