Optimal and Approximation Algorithms for Joint Routing and Scheduling in Millimeter-Wave Cellular Networks
11 Optimal and Approximation Algorithms for JointRouting and Scheduling in Millimeter-WaveCellular Networks
Dingwen Yuan, Hsuan-Yin Lin, Jörg Widmer and Matthias Hollick
Abstract —Millimeter-wave (mmWave) communication is apromising technology to cope with the exponential increase in5G data traffic. Such networks typically require a very densedeployment of base stations. A subset of those, so-called macrobase stations, feature high-bandwidth connection to the corenetwork, while relay base stations are connected wirelessly.To reduce cost and increase flexibility, wireless backhauling isneeded to connect both macro to relay as well as relay to relaybase stations. The characteristics of mmWave communicationmandates new paradigms for routing and scheduling. The paperinvestigates scheduling algorithms under different interferencemodels. To showcase the scheduling methods, we study themaximum throughput fair scheduling problem. Yet the proposedalgorithms can be easily extended to other problems. For a full-duplex network under the no interference model, we proposean efficient polynomial-time scheduling method, the schedule-oriented optimization . Further, we prove that the problem is NP-hard if we assume pairwise link interference model or half-duplex radios. Fractional weighted coloring based approximationalgorithms are proposed for these NP-hard cases. Moreover,the approximation algorithm parallel data stream schedulingis proposed for the case of half-duplex network under the nointerference model. It has better approximation ratio than thefractional weighted coloring based algorithms and even attainsthe optimal solution for the special case of uniform orthogonalbackhaul networks.
Index Terms —millimeter-wave, 5G, backhaul, max-min fair-ness, full-duplex, half-duplex, matching, coloring, pairwise linkinterference, single user spatial multiplexing.
I. I
NTRODUCTION
5G and beyond cellular systems are embracing millimeterwave (mmWave) communication in the - GHz bandwhere abundant bandwidth is available to achieve Gbps datarate. One of the main challenges for mmWave systemsis the high propagation loss. Although it can be partiallycompensated by directional antennas [1], [2], the effectivecommunication range of a mmWave base station (BS) isaround meters for typical use cases. Thus, base stationdeployment density in 5G will be significantly higher thanin 4G [3], [4]. This leads to high infrastructure cost for theoperators. In fact, besides the cost of site lease, backhaul linkprovisioning is the main contributor to this expense because
D. Yuan and M. Hollick are with Secure Mobile Networking Lab(SEEMOO), Technische Universität Darmstadt, Darmstadt, Germany (e-mail:[email protected]; [email protected]).H. Lin is with Simula UiB Research Lab, Bergen, Norway (e-mail: [email protected])J. Widmer is with Institute IMDEA Networks, Madrid, Spain (e-mail:[email protected]) Fig. 1. mmWave self-backhauling setup. the mmWave access network may require multi-Gbps backhaullinks to the core network. Currently, such a high data rate canonly be accommodated by fiber-optic links which have highinstallation costs and are inflexible for reconfiguration.Recent studies show that mmWave self-backhauling is acost-effective alternative to wired backhauling [3], [5]. Thisapproach is particularly interesting in an NG-RAN (NextGeneration Radio Access Network) where one or more (gNBs) have fiber backhaul to the core networkand act as gateways for the other gNBs [6]. We refer to thegateway gNBs as macro BSs and the other gNBs as relayBSs . Fig. 1 illustrates such a setup in which each relay BScan be reached by at least one macro BS directly or viaother relay BSs. Moreover, the directionality of mmWavecommunication reduces or removes the wireless backhaulinterference and allows simultaneous scheduling of multiplelinks over the same channel as long as their antenna beamsdo not overlap. However, the number of simultaneous datastreams a base station can handle is limited by the number ofits radio frequency (RF) chains. Furthermore, a base stationmay only support half-duplex communication, i.e. it can not work as a transmitter and a receiver at the same time.As of now, much of the research on mmWave communica-tion has been dedicated to issues that the mobile users (UEs)face in the access networks. How to maximize performancesuch as throughput and energy efficiency in mmWave backhaulnetworks has received less attention. Transmission schedulingthat incorporates the possibility of multi-hop routing is a mostcrucial research question to be addressed.A naive scheduling which lets a macro BS serve all therelay BSs in its macrocell in a round robin fashion is neither a r X i v : . [ c s . N I] J u l practical nor efficient. If a relay BS’ link to a macro BS isweaker than its links to other nearby relay BSs (which inturn have high-capacity links to a macro BS), a scheduleallowing multi-hop routing can be more favorable since italleviates the bottleneck at the macro BSs. At the same time,the limited interference at mmWave frequencies makes itefficient to maximize spatial reuse and operate as many datastreams simultaneously as possible. The goal of this paperis to design a scheduler that exploits these characteristicsto optimize mmWave backhaul efficiency for full-duplex andhalf-duplex radios assuming pairwise link interference or not,and realistic or maximum single-user spatial multiplexing.The paper is organized as follows: related work is discussedin §II. The system model is described in §III. The maxi-mum throughput fair scheduling (MTFS) serves as a concreteproblem for presenting our methods for backhaul scheduling.In §IV, we present a polynomial time optimal algorithm forthe full-duplex MTFS problem assuming no interference (NI)between links. In §V, we show that the MTFS problem is NP-hard for both cases of pairwise link interference (PI) modeland half-duplex radios. Yet it is solvable in polynomial timefor a special case—the so-called uniform orthogonal back-haul network . In §VI, we propose two general approximationalgorithms based on fractional weighted coloring for the NP-hard cases. For half-duplex radios under the NI model, wepropose the approximation algorithm—parallel data streamscheduling (PDS) in §VII, which has better a performancebound than the fractional weighted coloring based algorithms.It provides the optimal scheduling for the case of uniformorthogonal backhaul networks. §VIII elaborates on the exten-sion of the scheduling algorithms to more general scenarios.§IX demonstrates the efficiency and effectiveness of the algo-rithms through numerical evaluations. Finally, §X concludesthe paper. II. R ELATED W ORK
Our previous paper [7] discussed the optimal full-duplexscheduling of mmWave backhaul networks and its approxima-tion algorithm assuming zero interference between any pair oflinks and the linearity of link capacity in terms of RF chains.This paper is a continuation of the work, which includes bothfull-duplex and half-duplex scheduling of mmWave backhaulnetworks for pairwise link interference and realistic single-userspatial-multiplexing model. A few works studying mmWavenetwork scheduling [8]–[12] share the assumption that thetraffic demand is measured in discrete units of timeslots orpackets. The resulting optimization problems are all formu-lated as mixed integer programs (MIP). As MIPs are in generalNP-hard, optimal solutions can only be computed for verysmall networks. For practical use, they all rely on heuristics,which are based for example on greedy edge coloring [8], [10]or finding the maximum independent set in a graph [9], [11],[12]. Furthermore, [8], [9], [11], [12] assume that routing ispre-determined, which does not fully exploit the freedom givenby a reconfigurable backhaul, and may thus limit performance.In contrast, we relax the constraint of in-order flow schedul-ing (i.e., if needed, packets are queued for a short time) which does not harm the long-term throughput, and allowsa timeslot to be of any length (a following step can be usedto discretize the length). Based on these assumptions, we pro-pose polynomial time optimal and approximation schedulingalgorithms. We show by simulation that they are practical formmWave cellular networks. Moreover, the scheduling takesQoS optimization goals or QoS requirements as input and findsan efficient routing automatically. The first attempt to solvethe problem of joint routing and scheduling in a network withEdmonds’ matching formulation goes back to [13]. Hajek’sscheduling algorithm differs from ours in that it minimizesschedule length. Furthermore, we use a one-step schedule-oriented approach while they first compute the optimal linktime and then compute the minimum length schedule giventhe link time. The matching-based approach is also usedin [14] to study the problem of throughput-optimal routing andscheduling in a wireless directed acyclic graph (DAG) witha time-varying connectivity, to which a throughput-optimaldynamic broadcast policy is proposed.Two of our approximation algorithms (F WC) are adaptedfrom [15] which proposed scheduling algorithms for multihopwireless networks based on fractional weighted vertex coloringof conflict graphs. We show the generality of F WC algorithmsby demonstrating that all constraints of half-duplex, RF chainnumber and pairwise link interference can be modeled in aconflict graph.While current mmWave radios only support half-duplexoperation, full-duplex is feasible through proper analog anddigital cancellation [16] and is likely to be used in the future.More details about the design of a full-duplex mmWave radioare discussed in [17]. In addition, we assume that multiple datastreams can be transmitted concurrently between a single pairor multiple pairs of BSs. This assumption is backed by thefeasibility of single-user and multi-user MIMO in mmWavecommunication [18]–[20], exploiting beamforming, multipath,spatial and polarization diversity.Existing research works on mmWave backhaul schedulingfavor centralized solutions for various optimization goals,including throughput [7], [10], [11], delay [21], energy con-sumption [12], makespan [8], [22], wireless bandwidth [23]and flows with satisfied QoS requirements [9]. To achievethese optimization goals, most of the works perform linkscheduling to maximize spatial reuse while some control rout-ing [7], [8], [10], [22], [23], transmission power [11], [12], [23]and bandwidth allocation [23]. 3GPP is currently performinga study on Integrated Access and Backhaul (IAB) [24] whichdescribes mechanisms for sharing radio resources betweenaccess and backhaul links. Yet designing a high-performanceIAB is still an open problem [25], [26]. Our algorithms can beequally applied to IAB networks because of our graph-basedscheduling method, thus contributing to this emerging field.III. S
YSTEM MODEL
The system model considers a backhaul network whichconsists of one or more macro BSs and multiple relay BSs.The macro BSs act as the backhaul gateways for the relayBSs. We assume that any two macro BSs are connected by fiber optics with infinite capacity (data rate). Thus, data canbe exchanged between two macro BSs with zero delay .A node (BS) is equipped with multiple RF chains. Twonodes may have a different number of RF chains. We assumeanalog or hybrid beamforming which allows up to the samenumber of simultaneous data streams incident to a node as itsnumber of RF chains. In addition, the maximum number ofsimultaneous data streams supported by a link is determinedby the spatial diversity of that link (see. §III-A). These datastreams may have different capacities. Finally, the networkis either full-duplex or half-duplex , meaning all nodes areequipped with full-duplex or half-duplex radios, respectively.We use the convention that arc and edge refer to directed and undirected edge of a graph. The notations ( u, v ) and { u, v } are used to represent an arc from u to v and an edge between u and v , respectively. To model the backhaul network, westart with a directed network D , which is an arc weighteddirected multigraph . In this paper, we use the notation V ( G ) and E ( G ) to refer to the vertex set and arc (edge) set of adirected (or undirected) graph G . Thus, V ( D ) and E ( D ) arethe vertex set and arc set of D , representing the set of allBSs and the set of all possible data streams. Let B ( D ) and M ( D ) be the set of macro BS vertices and the set of relay BSvertices, respectively. Let L be the link network of D . It is asubgraph of D which is constructed by replacing each set ofparallel arcs by a single arc. Let l = ( u, v ) ∈ E ( L ) denote thelink from node u to node v and d ( l ) the maximum number ofsimultaneous data streams allowed by l . Then there are d ( l ) parallel arcs from u to v in D , which are labeled as l . . . l d ( l ) .Let c : E ( D ) (cid:55)→ Q + be the capacity function defined for eacharc (data stream) where Q + is the set of non-negative rationalnumbers. The value c ( e ) is the capacity of a data stream e .Fig. 2 illustrates a toy example for a directed network and thecorresponding link network. Fig. 2. Example of directed network and link network . (cid:50) and (cid:35) represent amacro and a relay BS, respectively. In addition, we define an
RF chain number function r : V ( D ) (cid:55)→ N where N is the set of natural numbers { , , . . . } . r ( u ) is the number of RF chains of node u . Wedenote the total number of RF chains of D by r ( D ) , i.e., r ( D ) = (cid:80) v ∈ V ( D ) r ( v ) . Scheduling is made based on thedirected network D . A. Single User Spatial Multiplexing Model
Parallel data streams from a transmitter u to a receiver v (cid:54) = u may be scheduled simultaneously if both u and v use We can equally support non-zero communication delay between macroBSs. This can be done by adding wired links of fixed capacity between macroBSs. These links do not interfere with each other or with the wireless links.For presentation clarity, we assume that wired links introduce zero delay. A directed multigraph allows parallel arcs with the same head node andthe same tail node, but disallows loops. multiple RF chains. This is referred to as single user spatialmultiplexing (SU-SM) [18]. In this paper, we assume that everyRF chain of a node v is assigned the same transmission power p tx ( v ) . p tx ( u ) and p tx ( v ) may be different if u (cid:54) = v . This isa reasonable assumption for a multi-transceiver node that hasmultiple RF chains, each with their own amplifiers and phasedarrays (see [27] for an example MIMO system). We investigatetwo models for SU-SM:1) REAL-SU-SM: This is a general and realistic model.A link l = ( u, v ) ∈ E ( L ) supports at most d ( l ) data streams where d ( l ) ≤ min (cid:0) r ( u ) , r ( v ) (cid:1) and isdetermined by the spatial diversity of link l such as the channel matrix rank . Since only the total capacity ofthe data streams is relevant in our scheduling problem,we can equivalently model their capacities such that c ( l ) ≥ · · · ≥ c ( l d ( l ) ) > . This means that we havecapacity c ( l ) if only one data stream is active. Witheach additional data stream we have a more marginalincrease in total capacity. Using k ≤ d ( l ) RF chains atboth ends, we get a total capacity of (cid:80) ki =1 c ( l i ) .2) MAX-SU-SM: This model achieves the maximum pos-sible capacity of SU-SM, where each link providessufficient spatial diversity. It is a special case of theREAL-SU-SM model with d ( l ) = min (cid:0) r ( u ) , r ( v ) (cid:1) and c ( l ) = · · · = c ( l d ( l ) ) > . In this model, capacity isalso defined for a link such that c ( l ) = c ( l ) . B. RF Chain Number Constraint and Half-duplex Constraint
The constraint due to the limited number of RF chainsis that given a set of simultaneously scheduled data streams E ∈ E ( D ) , for each node v , the number of arcs in E thatare incident to v is at most r ( v ) . Furthermore, a schedulecan be either full-duplex (FD) or half-duplex (HD). FD allowsincoming and outgoing arcs of a vertex to be scheduledsimultaneously, whereas HD does not. We will prove in thepaper that the duplex state is a key parameter for scheduling.It determines whether the computational complexity is poly-nomial time or NP-hard. C. Interference Model
Compared to lower frequency omnidirectional communica-tion, mmWave frequencies significantly reduce the interferencebetween concurrent links due to the short communicationrange and the directionality brought by beamforming. We willstudy and compare two interference models in this paper:1) NI: no interference between concurrent links.2) PI: pairwise link interference.
Fig. 3. Two interfering links l (1) and l (2) . An angle between two dashedrays starting from a node is the beamwidth of that node. The NI model (no interference assumption) is reasonablefor the situation of narrow beams and sparse deployments.
The PI model specifies that given two links l (1) , l (2) ∈ E ( L ) sharing no common vertex, the function intf( l (1) , l (2) ) = 1 if there is interference between l (1) and l (2) , i.e., they cannotbe scheduled simultaneously; otherwise intf( l (1) , l (2) ) = 0 .As shown in Fig. 3, we say that l (1) interferes l (2) if (1) r and t are within the beamwidths of each other, and (2) SINR = p t r p t r + N < τ , where p tr is the received power atreceiver r due to transmitter t ; N is the noise power and τ is aSINR threshold. intf( l (1) , l (2) ) = 1 if l (1) interferes l (2) or l (2) interferes l (1) . Otherwise, intf( l (1) , l (2) ) = 0 . For the case thattwo links have vertices in common, we assume that advancedsignal processing (MIMO precoding and combining) can beperformed at the common vertices (transmitters or receivers)to remove the interference.According to [28], the protocol model is a sufficiently accu-rate interference model for directional mmwave communica-tion. Due to the high gain of directional antennas, interferencefrom a single node is typically either negligible or destructive.Obviously, the protocol model with an arbitrary interferencerange can be translated into pairwise link interference. Thisjustifies that the PI model is reasonably accurate and the NImodel is optimistic. D. Optimization Goal
Our paper focuses on downlink communication. With sim-ple adaptation, the proposed algorithms can also be appliedto an uplink or a joint uplink and downlink optimization. Wewill explain this in §VIII, after presenting the algorithms. Fordownlink scheduling, the only data sources are the macroBSs, and we can thus remove all incoming arcs to macroBSs in D . We can observe in Fig. 2 that there are manyways to schedule downlink communication among the macroand relay BSs. Our goal is to find a schedule that is ofunit length and is optimal with respect to a QoS metric,while satisfying given QoS requirements and the constraints onsimultaneous transmissions (number of RF chains at a node,radio duplexity, spatial multiplexing model and interferencemodel). In practice, the unit time length corresponds to theduration of a radio frame.IV. O PTIMAL M AXIMUM T HROUGHPUT F AIR S CHEDULING F OR A F
ULL -D UPLEX N ETWORK U NDERTHE
NI M
ODEL
This section assumes a full-duplex backhaul network underthe NI (No Interference) model. Given these assumptions, weprovide a polynomial-time optimal algorithm for the maximumthroughput fair scheduling problem (MTFS).The goal of the MTFS problem is to maximize the downlink network throughput under the condition that the max-minfairness [29], [30] in throughput is achieved at the relay BSs.
Definition 1 (Maximum Throughput Fair Schedule) . Given adirected network D and a unit time schedule S , let h S =[ h Sv | v ∈ M ( D )] be the throughput vector of S , where h Sv denotes the throughput of a relay BS v , that is the total amountof data entering v minus that leaving v when S is applied. (i) A feasible unit time schedule S f is said to satisfy the max-min fairness criterion if min v ∈ M ( D ) h S f v ≥ min v ∈ M ( D ) h Sv forany feasible unit time schedule S . The value min v ∈ M ( D ) h S f v is called the max-min throughput .(ii) A feasible unit time schedule S ∗ is an optimal solutionof the MTFS problem if S ∗ satisfies the max-min fairnesscriterion in (i) and the network throughput (cid:80) v ∈ M ( D ) h S ∗ v ismaximum. In the following, we present our generally applicable opti-mization method— schedule-oriented optimization . A. Schedule-Oriented Optimization
To characterize the scheduled data streams in each timeslot,we need the definition of simple b -matching of a graph [31]. Definition 2.
Let G be an undirected graph with numbers b : V ( G ) (cid:55)→ N and weights c : E ( G ) (cid:55)→ R , then a simple b -matching in G is a function f : E ( G ) (cid:55)→ { , } and (cid:80) e ∈ δ ( v ) f ( e ) ≤ b ( v ) for all v ∈ V ( G ) where δ ( v ) is the setof edges incident to v . A maximum weight simple b -matching f is a simple b -matching whose weight (cid:80) e ∈ E ( G ) c ( e ) f ( e ) ismaximum. Let b = [ r ( v ) | v ∈ V ( D )] , then it is obvious the arc setscheduled in a timeslot is a simple b -matching of D . On theother hand, a simple b -matching of D is an arc set that can bescheduled simultaneously in a timeslot. The schedule-orientedoptimization solves a linear optimization problem, the solutionto which is exactly the optimal schedule. For the mathematicalformulation, we construct the node-matching matrix . Definition 3 (Node-matching Matrix) . Given a directed net-work D , suppose the number of all possible simple b -matchings of D is K where b = [ r ( v ) | v ∈ V ( D )] . Then thenode-matching matrix A = [ a i,j ] is a | V ( D ) |× K matrix. Eachelement a i,j is equal to the sum capacity of all arcs in the j -thsimple b -matching that enter the i -th vertex of D minus thesum capacity of all arcs in the j -th simple b -matching thatleave the i -th vertex of D . A : α α β α , α α , β α , β (cid:34) (cid:35) v − − − − − v − v Fig. 4. Node- b -matching matrix A . Every node in the backhaul network has2 RF chains. Let two links α = ( v , v ) and β = ( v , v ) . The maximumnumber of data streams of each link is d ( α ) = 2 , d ( β ) = 1 . The arc capacitiesin the directed network D are: c ( α ) = 8 , c ( α ) = 6 , c ( β ) = 3 . As we will see, the node-matching matrix helps to formulatethe throughput constraints at each relay BS. Fig. 4 gives anexample of node-matching matrix for a directed network.Let A be the node-matching matrix of D . We define A M as a submatrix of A , which consists of the rows related to relay BSs. As the set of arcs scheduled in each timeslot ofa schedule must be a simple b -matching in D , we define t S as a K × length vector , each element of which is thelength of a potential timeslot corresponding to a simple b -matching. Let the minimum throughput among all relay BSsbe θ . Then we can solve the MTFS problem in two steps: (i)maximizing θ ; the solution θ ∗ is the max-min throughput, and(ii) computing the optimal schedule S ∗ that offers the highestnetwork throughput subject to the constraint θ ≥ θ ∗ . Linear programs for MTFS.
The linear program to max-imize θ in step (i) ismax θ (1a)s.t. A M t S ≥ θ (1b) T t S = 1 and t S ≥ , (1c)where and represent the all-one and all-zero columnvectors. The superscript ‘ T ’ denotes the vector transpose. (1b)is the constraint that the throughput at each relay BS shouldbe at least θ . (1c) is the constraint that the schedule shouldbe of unit length. The feasibility of the schedule is implicitlyguaranteed by the formulation in terms of simple b -matchings.After we have obtained the solution θ ∗ from (1), wecan formulate the linear program that maximizes the net-work throughput, under the condition that each relay BS hasthroughput at least θ ∗ :max c T t S (2a)s.t. A M t S ≥ θ ∗ (2b) T t S = 1 and t S ≥ , (2c)Here, c is the capacity vector whose element c j isthe cumulative capacity of all macro-BS-to-relay-BS datastreams in the j -th simple b -matching M j , i.e., c j = (cid:80) { e | e ∈ M j , tail ( e ) ∈ B ( D ) } c ( e ) , where B ( D ) is the set of allmacro BSs.The difficulty in solving (1) and (2) is due to the hugenumber of elements in t S (equal to the number of simple b -matchings of D , which is exponential in | V ( D ) | ). Yet, weshow that it is unnecessary to enumerate all of them, and both(1) and (2) can be solved in polynomial time. Theorem 1.
Under the assumption of a full-duplex backhaulnetwork and the NI model, the MTFS problem can be solvedin polynomial time with the ellipsoid algorithm.Proof.
See Appendix A for the proof.Although polynomial, in practice the ellipsoid algo-rithm [32] almost always runs much slower than the simplexalgorithm . Therefore, we propose algorithms based on the revised simplex algorithm [33] which does not require thegeneration of all columns of A M . Conceptually, the algorithmsfirst create a feasible schedule. In each iteration, to improvethe optimization goal, we replace a timeslot in the scheduleby another simple b -matching (a set of simultaneous datastreams) while keeping the schedule feasible, until the opti-mum is reached. The optimum is guaranteed to be reachabledue to the correctness of the simplex algorithm in solvinglinear programs. The maximum weight simple b -matching Algorithm 1:
Compute the max-min throughput θ ∗ Set the basis B according to the initial schedule S ; while True do Compute the dual variable p T = f T B B − ; Set weight w ( e ) to each arc e = ( v i , v j ) l ∈ E ( D ) where w ( e ) (cid:44) (cid:40) c ( e )( p j − p i ) if v i ∈ M ( D ) c ( e ) p j otherwise . Do max weight simple b -matching on D and let the maxweight be z . Compute η = − z − p | M ( D ) | +1 ; Compute η = − (cid:80) | M ( D ) | k =1 p k ; Compute η = min ≤ k ≤| M ( D ) | p k ; Compute η = min( η , η , η ) and let the correspondingcolumn be u ∈ U ; if η ≥ then return θ ∗ = θ and B θ ∗ = B ; else Update B by replacing a column of B with u according to the simplex algorithm; end end algorithm [34] is used to choose a better simple b -matching(column) to enter the schedule (basis). B. Solving the MTFS Problem
To optimize θ , we need an initial basic feasible solution to(1). Suppose that each relay BS is reachable from at least onemacro BS by following a sequence of arcs in D . Let D bea subgraph of D such that only the first data stream of eachlink (the first arc of each set of parallel arcs in D ) is keptin D . We add a root vertex v r to D and add an arc from v r to each macro BS vertex in D . We perform a breadth-first-search (BFS) in D starting from v r . The result is a tree T spanning v r and all BSs. Removing v r from T , we get aforest T (cid:48) that has exactly | M ( D ) | arcs. The initial schedule S is constructed as follows: S has | M ( D ) | timeslots, each ofwhich contains a different arc in T (cid:48) . Moreover, it is requiredthat the throughput of every relay BS is the same and theschedule takes unit time. This initial solution is unique. Weconvert the linear program (1) to the standard form (3) byintroducing | M ( D ) | surplus variables s i .min f T x (3a)s.t. U x = g and x ≥ , (3b)where U (cid:44) [ U | U | U ] (cid:44) (cid:20) A M − − I T T (cid:21) , f T = (cid:2) T | − | T (cid:3) , x T (cid:44) (cid:2) ( t S ) T | θ | s T (cid:3) , and g T (cid:44) (cid:2) T | (cid:3) . Alg. 1shows the computation of the max-min throughput θ .The basis B is a square matrix that consists of | M ( D ) | + 1 columns from U . f B are the elements of f correspondingto B . Lines 4, 5 and 6 compute the minimum reduced costof a column in the matrices U , U and U respectively. Todecrease − θ , we need to find a column of U , u k that hasnegative reduced cost f k − p T u k < to enter the basis,according to the simplex algorithm. In each iteration of thealgorithm, we find the column u in U that produces the Algorithm 2:
Solving the MTFS problem Set the basis B = B θ ∗ ; while True do Compute the dual variable p T = f T B B − ; Set weight w ( e ) to each arc e = ( v i , v j ) l ∈ E ( D ) where w ( e ) (cid:44) (cid:40) c ( e )( p j − p i ) if v i ∈ M ( D ) c ( e )( p j + 1) otherwise . Do max weight simple b -matching on D and let the maxweight be z . Compute η = − z − p | M ( D ) | +1 ; Compute η = (cid:80) | M ( D ) | k =1 p k ; Compute η = min ≤ k ≤| M ( D ) | p k ; Compute η = min( η , η , η ) and let the correspondingcolumn be u ∈ U ; if η ≥ then return the optimal schedule S ∗ corresponding to B ; else Update B by replacing a column of B with u ; end end minimum reduced cost η . If η ≥ , then no columns canbe used to decrease − θ , thus we have reached the optimum.Let the max-min throughput be θ ∗ and the related basis be B θ ∗ . To directly use B θ ∗ as the initial basis for the solution (2),we add an artificial scalar variable y ≥ to (2) and replacethe constraint A M t S ≥ θ ∗ with A M t S − y ≥ θ ∗ . Since θ ∗ is the max-min throughput, the feasible y must be 0. Hence,the optimal solution to (2) is unaffected. Again, we convert(2) into the standard form of (3), which is solvable with therevised simplex algorithm.In the standard form, U remains unchanged. We redefine f T (cid:44) (cid:2) − c T | | T (cid:3) , x T (cid:44) (cid:2) ( t S ) T | y | s T (cid:3) , and g T (cid:44) (cid:2) T θ ∗ | (cid:3) .The optimization algorithm is similar to Alg. 1 and is outlinedin Alg. 2. Since the basis B is a square matrix of dimension | M ( D ) | + 1 , it follows that the optimal schedule S ∗ containsno more than | M ( D ) | + 1 timeslots. Additionally, since thelinks of a flow from a macro BS to a destination relay BSmay not be scheduled in sequential order, some transmissionopportunities of the flow in the first few frames may be wasted.Therefore, maximum throughput is achieved in the long-term. C. Reducing Simple b -matching to Matching To do maximum weight simple b -matching, we can eitheruse dedicated algorithms such as [34], or reduce it to amatching (equivalent to simple -matching) problem [35],for which highly efficient algorithms and implementationsare available. In this work, we use the state-of-the-art C++implementation for maximum weight perfect matching on ageneral graph [36], where a perfect matching matches allvertices of a graph. Thus, we need to reduce a maximumweight simple b -matching problem to a maximum weightperfect matching problem.
1) Reduction for the case of MAX-SU-SM:
We use thereduction by Tutte [37]. Given a strict graph G whose edges A graph is strict if it has no loops or parallel edges between any pair ofvertices. have positive weights w and whose vertices have numbers b = r (cid:44) [ r ( v ) | v ∈ V ( G )] , we create a graph G (cid:48) asfollows. Each vertex v ∈ V ( G ) is mapped to r ( v ) vertices v (1) . . . v ( r ( v )) . Each edge e = { u, v } ∈ E ( G ) with weight w ( e ) , is mapped to r ( u ) · r ( v ) edges (cid:8) { u ( i ) , v ( j ) }|∀ i, ∀ j (cid:9) in G (cid:48) , each of which is assigned the weight w ( e ) . Then agraph G (cid:48)(cid:48) is created by duplicating G (cid:48) and connecting eachpair of symmetric vertices by an edge of zero weight. Anexample of the reduction is shown in Fig. 5. It is obvious thata maximum weight b -matching in G can be deduced from amaximum weight perfect matching in G (cid:48)(cid:48) . An edge { u ( i ) , v ( j ) } in a matching is mapped to the edge { u, v } in a b -matching. Fig. 5. Reduction for the case of MAX-SU-SM. u, v, x have 1, 2, 3 RF chains,respectively. The edges of the same style have the same weight. Dotted edgeshave zero weights.
2) Reduction for the case of REAL-SU-SM:
We adaptthe reduction in [38] for this case. Given a multigraph G whose edges have positive weights w and whose verticeshave numbers b = r (cid:44) [ r ( v ) | v ∈ V ( G )] , we assume that r ( v ) ≤ deg( v ) for any v ∈ V ( G ) where deg( v ) is the degreeof v ; otherwise we set r ( v ) = deg( v ) . We create a graph G (cid:48) as follows. For each vertex v in G , we create r ( v ) vertices in G (cid:48) , labeled as v (1) . . . v ( r ( v )) . These vertices are called outervertices . For each edge e = { u, v } in G , we add two innervertices e u , e v to G (cid:48) , and r ( u ) + r ( v ) + 1 edges of weight w ( e ) , which are { u ( i ) , e u } , ∀ i ; { e u , e v } and { e v , v ( j ) } , ∀ j .Then G (cid:48)(cid:48) is created by duplicating G (cid:48) and connecting eachpair of symmetric outer vertices by an edge of zero weight. Anexample of the reduction is shown in Fig. 6. It is obvious thata maximum weight simple b -matching in G can be deducedfrom a maximum weight perfect matching in G (cid:48)(cid:48) . An edge { u ( i ) , e u } in a matching is mapped to the edge e in a simple b -matching. Fig. 6. Reduction for the case of REAL-SU-SM. u, v, x have 1, 2, 3 RFchains, respectively. The edges of the same style have the same weight. Dottededges have zero weights.
V. F
URTHER C OMPLEXITY R ESULTS O N MTFSS
CHEDULING
In this section, we will develop further computational com-plexity results on MTFS scheduling, for different duplexitymodes, interference models and single-user spatial multiplex-ing (SU-SM) models.
A. MTFS Under The Pairwise Link Interference Model is NP-hard
For both full-duplex and half-duplex scheduling, the MTFSproblem is NP-hard if we assume an arbitrary pairwise linkinterference (PI) model.
Theorem 2.
The MTFS problem is NP-hard under the pair-wise link interference model for both half-duplex and full-duplex backhaul networks.Proof.
The proof can be done by relating the MTFS prob-lem to computing the fractional chromatic number. See Ap-pendix B for details.
B. Half-duplex MTFS is NP-hard
Different from the polynomial-time solvable problem offull-duplex MTFS under the NI model, half-duplex MTFS isNP-hard, which will be proved in the following. Then we willshow a special case that allows a polynomial-time optimalsolution to the half-duplex MTFS problem. Since it is provedin §V-A that half-duplex MTFS is NP-hard under the PI model,we assume the NI model in this subsection.
1) Linear Programs for Half-duplex MTFS Problem:
Com-pared to the full-duplex case, the half-duplex scheduling hasthe additional half-duplex constraint—a node cannot workas transmitter and receiver simultaneously. Therefore, thematching-based optimization method that works successfullyfor the full-duplex scheduling cannot be applied directly. Forthe half-duplex case, the data streams scheduled in eachtimeslot must be a half-duplex subgraph J ⊆ D which isdefined as follows. Definition 4 (Half-duplex Subgraph) . A half-duplex subgraphof a directed network D is a subgraph J ⊆ D such that (i) J is a simple b -matching of D with b = [ r ( v ) | v ∈ V ( D )] and(ii) J is a directed bipartite graph , i.e., V ( J ) can be dividedinto two disjoint sets V , V where each arc of E ( J ) has thehead in V and the tail in V . The constraint (i) is due to the number of RF chains. Theconstraint (ii) reflects the half-duplex property, because theactive nodes in a timeslot can be divided into the sender set V and the receiver set V , where a data stream only goes froma sender to a receiver. Analogous to the node-matching matrix,we define the node-hd-subgraph matrix for the formulation ofthe half-duplex MTFS problem. Definition 5 (Node-hd-subgraph Matrix) . Given a directednetwork D , suppose that the number of all half-duplex sub-graphs of D is K . Then the node-hd-subgraph matrix L = [ l i,j ] is a | V ( D ) | × K matrix. Denote the i -th vertex of D as v i ,which is related to the i -th row of L . Denote the j -th half-duplex subgraph of D as J j , which is related to the j -thcolumn of L . Each element l i,j is equal to the sum capacityof all arcs in J j that enter v i minus the sum capacity of allarcs in J j that leave v i . Fig. 7 gives an example of the node-hd-subgraph matrixfor a directed network. Similar to the definition of A M in§IV-A, L M is the submatrix of L that only consists of the rows L α β γ δ α, β α, δ β, δ (cid:34) (cid:35) v − − − − − v − v − − − Fig. 7. Node-hd-subgraph matrix L . v , v and v have 2, 2 and 1 RF chainrespectively. The arc capacities are: c ( α ) = c ( β ) = 8 , c ( γ ) = c ( δ ) = 3 . related to relay BSs. The linear program formulation of thehalf-duplex MTFS problem is the same as (1) and (2) exceptthat A M is replaced by L M . Yet, different from the full-duplexMTFS problem, the half-duplex MTFS problem is NP-hard.The intuitive reason is that the solution of these two problemsrequires computing a maximum weight simple b -matching anda maximum weight half-duplex subgraph, respectively. Thefirst can be done in polynomial time while the second is NP-hard. We will give a formal proof in the following.
2) The Half-duplex MTFS Problem is NP-hard:
Definition 6 (Maximum Weight Half-duplex Subgraph(MWHS) Problem) . Given a directed network D and a weightfunction w ( e ) defined for each arc e , find a half-duplexsubgraph J ⊆ D such that (cid:80) e ∈ E ( J ) w ( e ) is maximum. Since the maximum weight simple b -matching problemcan be solved in polynomial time, so can the full-duplexMTFS problem (see §IV). Analogously, if the MWHS problemcould be solved in polynomial time, so could be the half-duplex MTFS problem. Unfortunately, this is not the case. TheMWHS problem is NP-hard for a directed network even if it isa directed acyclic graph (DAG). Furthermore, by extending thetechnique for proving NP-hardness of MWHS, we can provethat the half-duplex MTFS problem is also NP-hard. Lemma 1.
The MWHS problem is NP-hard for a directednetwork that is a DAG.Proof.
The proof is done by reduction from the SAT problem.For further details, see Appendix C.
Theorem 3.
The half-duplex MTFS problem is NP-hard for ageneral directed network.Proof.
See Appendix D for the proof.In summary, assuming the NI model, the optimal schedulingproblem of a mmWave backhaul networks can be solved inpolynomial time when all BSs are full-duplex. In contrast, theproblem is NP-hard when all BSs are half-duplex, i.e., it isimpossible to obtain an optimal schedule in polynomial time.
C. Special Case: Half-duplex MTFS is Solvable in PolynomialTime
We now study a special case of the half-duplex MTFS thatis solvable in polynomial time. We refer to such backhaulnetworks as uniform orthogonal backhaul networks . Definition 7 (Uniform Orthogonal Backhaul Network) . Abackhaul network that is represented by the directed network D that satisfies the following conditions: There is no interference (NI model) between any pair oflinks. The MAX-SU-SM model is assumed for single-userspatial multiplexing. Each relay BS has the same number of RF chains, i.e., r ( v ) ≡ r M ∈ N , ∀ v ∈ M ( D ) . In addition, any macroBS has the RF chain number that is a multiple of r M ,i.e., for each i = 1 . . . | B ( D ) | and n i ∈ B ( D ) , r ( n i ) = k i · r M , for some k i ∈ N . Assuming the NI model, half-duplex MTFS is solvablein polynomial time if every node has a single RF chain ,because the half-duplex constraint is automatically satisfied ifevery node can serve only one data stream. Thus, the optimalschedule can be obtained with the optimal matching-basedalgorithm. Next we will prove that half-duplex MTFS is alsosolvable in polynomial time for uniform orthogonal backhaulnetworks and provide an optimal algorithm. First we look atthe case that each node has the same number of RF chains R .In this case, the directed network D is a multi-digraph, eacharc of which belongs to a set of R equivalent (same head, tailand capacity) parallel arcs. Theorem 4.
Given a uniform orthogonal backhaul network D , each node of which has the same number of RF chains: r ( v ) ≡ R , ∀ v , the half-duplex MTFS problem can be solvedin polynomial time as follows: Compute the optimal schedule S with the matching-based optimal MTFS algorithm on D ’s link network L where the capacity of an arc in L is the same as thatof an arc in D with the same head and tail, assumingeach node has one RF chain. The optimal schedule S ∗ consists of R copies of S running in parallel.Proof. See Appendix E for the proof.We can now relax the condition that all nodes in D havethe same number of RF chains. Corollary 1.
Given a uniform orthogonal backhaul network D , the optimal solution to the half-duplex MTFS problem canbe solved in polynomial time as follows. Assume that eachrelay BS has the same number of RF chains, r ( v ) ≡ r M ∈ N , ∀ v ∈ M ( D ) . In addition, any macro BS has an RF chainnumber that is a multiple of r M , i.e., for each i = 1 . . . | B ( D ) | and n i ∈ B ( D ) , r ( n i ) = k i · r M with k i ∈ N . Replace each macro BS vertex n i by k i vertices n (1) i . . . n ( k i ) i , each of which has r M RF chains. Theconnection of n ( j ) i to the relay BSs is the same as thatof n i (same number of arcs with the same head andcapacity). Let the resulting directed network be D (cid:48) . The optimal half-duplex MTFS schedule for D is equiva-lent to that for D (cid:48) , which is obtained with the algorithmin Theorem 4.Proof. The optimal half-duplex schedule for D is equivalentto the optimal one for D (cid:48) , since a half-duplex schedule for D can be translated into one for D (cid:48) and vice versa. Moreover D (cid:48) satisfies the condition of a uniform orthogonal backhaulnetwork.VI. A PPROXIMATION A LGORITHMS BASED ON F RACTIONAL W EIGHTED C OLORING
As explained in §V, the MTFS problem is NP-hard ifthere is pairwise link interference or the backhaul networkis half-duplex. For both cases, we must rely on approximationalgorithms. Two such algorithms F WC-FAO and F WC-LSLO are based on the method fractional weighted vertexcoloring of conflict graphs , as proposed by Wan [15]. TheMTFS problem can be transformed into a fractional weightedvertex coloring problem since we can embody all four types ofconstraints in a conflict graph: 1) pairwise link interference, 2)number of data streams for a link restricted by spatial diversity,3) number of data streams incident to a node restricted by thenumber of RF chains, and 4) half-duplex. A. Conflict Graph
Conflict graph is a powerful tool for modeling schedulingconstraints. It is an undirected simple graph (it has neitherloops nor parallel edges) denoted by C , in which each vertexrepresents a data stream and each edge represents that twodata streams cannot be scheduled simultaneously. The conflictgraph is derived from the expanded network H , a directedgraph that explicitly models the RF chains. H is the collectivenotation of four variants: H RFD , H
MFD , H
RHD and H MHD , dependingon the modeling. Each vertex of C is one-to-one mapped toeach arc of H , V ( C ) = E ( H ) .
1) Full-Duplex Scheduling:
Let us first consider the mostgeneral backhaul network which is subjected to the PI modeland the REAL-SU-SM model. Given a backhaul network D ,we assume that the RF chain number r ( v ) ≤ deg( v ) , ∀ v ∈ V ( D ) . Otherwise, we set r ( v ) = deg( v ) as the extra RFchains are redundant. The expanded network H RFD is created bymapping each vertex v ∈ V ( D ) into vertices v (1) . . . v ( r ( v )) .Each arc l i = ( u, v ) i ∈ E ( D ) (the i -th data stream from u to v ) is mapped to r ( u ) · r ( v ) arcs { ( u ( j ) , v ( k ) ) i |∀ j, k } ,each with capacity c ( l i ) . We define the expanded arcs ofa link ( u, v ) in L as X (cid:0) ( u, v ) (cid:1) = { ( u ( j ) , v ( k ) ) i |∀ i, j, k } .We define the expanded arcs of a data stream ( u, v ) i in D as X (cid:0) ( u, v ) i (cid:1) = { ( u ( j ) , v ( k ) ) i |∀ j, k } . An example for theexpanded network is shown in the middle of Fig. 8.The conflict graph C is constructed as follows. Let a complete graph K ( V ) be an undirected graph such thatthere is an edge between each pair of vertices in V . Theedges of C are constructed by first adding the union of theedge sets of a number of complete graphs. They are 1) theones formed by the arcs incident to each vertex in H RFD , K ( δ H RFD ( v )) , ∀ v , and 2) the ones formed by the expanded arcsof each data stream X ( e ) , ∀ e ∈ E ( D ) . Then we add theedges representing pairwise link interference. For each pairof interfering links in L , say l and l (cid:48) , we add to C the edges (cid:8) { e, e (cid:48) }| e ∈ X ( l ) , e (cid:48) ∈ X ( l (cid:48) ) (cid:9) .If the MAX-SU-SM model is assumed instead of the REAL-SU-SM model, then the expanded network H MFD contains fewerarcs than H RFD . Again each vertex v ∈ V ( D ) is mapped into Fig. 8. Transforming a directed network D into an expanded network H RFD and H RHD (full-duplex/half-duplex and REAL-SU-SM model). The number insidea node is its number of RF chains. The arcs of the same style have the same capacity. r ( v ) vertices. If there is a link ( u, v ) ∈ E ( L ) of capacity c ,then D contains min( r ( u ) , r ( v )) arcs (data streams) from u to v with the same capacity c . The link is mapped into r ( u ) · r ( v ) arcs { ( u ( j ) , v ( k ) ) |∀ j, k } in H MFD , all having capacity c . The expanded arcs of a link ( u, v ) in L are defined as X (cid:0) ( u, v ) (cid:1) = { ( u ( j ) , v ( k ) ) |∀ j, k } . An example for the expanded network isshown in the middle of Fig. 9.We first add the edges of the complete graphs formed bythe arcs incident to each vertex in H MFD . Then for each pairof interfering links in L , say l and l (cid:48) , we add to C the edges (cid:8) { e, e (cid:48) }| e ∈ X ( l ) , e (cid:48) ∈ X ( l (cid:48) ) (cid:9) .
2) Half-Duplex Scheduling:
Again let us first considerthe most general backhaul network subject to the PImodel and the REAL-SU-SM model. The expanded net-work H RHD is more sparse than the full-duplex counter-part H RFD . Given a backhaul network D , we set r ( v ) =min (cid:16) max (cid:0) deg − ( v ) , deg + ( v ) (cid:1) , r ( v ) (cid:17) where deg − ( v ) and deg + ( v ) are the number of incoming and outgoing arcs of v in D . The reason is that a higher number of RF chains isunnecessary. H RHD is created by first mapping the vertices in D the same way as before. Then each arc l i = ( u, v ) i ∈ E ( D ) ismapped as follows. If r ( u ) (cid:54) = r ( v ) , l i is mapped to r ( u ) · r ( v ) arcs { ( u ( j ) , v ( k ) ) i |∀ j, k } , each with capacity c ( l i ) the sameway as for H RFD . Otherwise, r ( u ) = r ( v ) , l i is mapped to r ( u ) arcs { ( u ( j ) , v ( j ) ) i |∀ j } , each with capacity c ( l i ) . An examplefor the expanded network is shown in the right side of Fig. 8.For a vertex v in D , we denote δ − H RHD ( v ) and δ + H RHD ( v ) asthe arcs in H RHD that enter or leave the vertices v ( j ) for all j ,respectively. The conflict graph C is first constructed with themethod for H RFD . Then we add to C the edges (cid:8) { e, e (cid:48) }| e ∈ δ − H RHD ( v ) , e (cid:48) ∈ δ + H RHD ( v ) , ∀ v ∈ V ( D ) (cid:9) . These edges model thehalf-duplex constraint.If the MAX-SU-SM model is assumed, then the expandednetwork H MHD is even more sparse than H RHD . Each link ( u, v ) ∈ E ( L ) with capacity c is mapped as follows. If r ( u ) (cid:54) = r ( v ) , ( u, v ) is mapped to r ( u ) · r ( v ) arcs { ( u ( j ) , v ( k ) ) |∀ j, k } ,each with capacity c . Otherwise, it is mapped to r ( u ) arcs { ( u ( j ) , v ( j ) ) |∀ j } , each with capacity c . An example for theexpanded network is shown in the right side of Fig. 9.The conflict graph C is first constructed with the method for H MFD . Then we add to C the edges (cid:8) { e, e (cid:48) }| e ∈ δ − H MHD ( v ) , e (cid:48) ∈ δ + H MHD ( v ) , ∀ v ∈ V ( D ) (cid:9) . The conflict graph for H MHD in Fig. 9is shown in Fig. 10.A sparse expanded network leads to a conflict graph withfewer vertices and hence shorter execution time for the al-gorithms. We will prove in the following why the sparseexpanded networks H RHD and H MHD can be used for half-duplexscheduling.
Algorithm 3:
First-fit fractional weighted coloring.
Input : C , t ∈ R V ( C )+ , and an ordering of V ( C ) . Output:
A fractional weighted coloring Π of ( C, t ) . Π ← ∅ ; U ← { v ∈ V ( C ) | t v > } ; while U (cid:54) = ∅ do I ← the first-fit MIS (maximal independent set) of U ; λ ← min v ∈ I t v ; add ( I, λ ) to Π ; for each v ∈ I do t v ← t v − λ ; if t v = 0 then remove v from U ; end end end output Π ; Theorem 5.
Given a directed network D , then any half-duplexsubgraph of D can be represented by a matching in H MHD .Proof.
See Appendix F for the proof.
B. General Procedure of Fractional Weighted Coloring BasedApproximation Algorithms
A fractional weighted coloring based approximation algo-rithm consists of three steps: (i) computing the data streamtime vector t = [ t e | e ∈ E ( H )] = [ t v | v ∈ V ( C )] , (ii)sorting the vertices V ( C ) and performing F WC with thegiven ordering, and (iii) scaling the schedule. We use theresults of [15] and adapt two approximation algorithms basedon fractional weighted coloring. The difference of the twoalgorithms lies in the linear programs for computing the linktime vector t and the ordering of V ( C ) for coloring. Thecoloring step uses the so-called first-fit fractional weightedcoloring (F WC) algorithm from [15], listed in Alg. 3.How to compute t depends on the specific algorithm.The minimum makespan scheduling for t is the same as the minimum fractional weighted coloring of ( C, t ) . The latter isdefined as a set of K ∈ N pairs ( I i , λ i ) where each I i isan independent set (a set of nonadjacent vertices) of C and λ i ∈ R + for ≤ i ≤ K satisfying that (cid:80) ≤ i ≤ K,v ∈ I i λ i = t v , ∀ v ∈ V ( C ) and the sum (cid:80) Ki =1 λ i is the minimum. But theproblem of finding a minimum fractional weighted coloring isNP-hard [39]. Let P be the independence polytope of C , i.e.,the convex hull of the incidence vectors of the independent setsof C . Then any point in P corresponds to a feasible unit timeschedule. The minimum fractional weighted coloring problemcan be expressed as a linear program with the help of P . Fig. 9. Transforming a directed network D into an expanded network H MFD and H MHD (full-duplex/half-duplex and the MAX-SU-SM model). The numberinside a node is its number of RF chains. The arcs of the same style have the same capacity.Fig. 10. The conflict graph for H MHD in Fig. 9
We assume that an algorithm provides a γ -approximate( γ > ) independent polytope Q ◦ , i.e., Q ◦ ⊆ P ⊆ γQ ◦ .Specifically, we have two options—F WC-FAO or F WC-LSLO with γ = α ∗ , Q ◦ = Q and γ = 2 β ∗ , Q ◦ = Q (cid:48) ,respectively (the definition of these variables will be clear inthe following). Step (i) is to solve the following two linearprograms.max θ (4a)s.t. (cid:88) e ∈ δ − H ( U ( v )) c ( e ) t e − (cid:88) e ∈ δ + H ( U ( v )) c ( e ) t e ≥ θ ∀ v ∈ M ( D ) (4b) t ∈ Q ◦ , (4c)where U ( v ) = { v ( i ) ∈ V ( H ) |∀ i } .With the max-min throughput solution θ , we go on tocompute t for the maximum network throughput.max (cid:88) e ∈{ δ + H ( U ( v )) | v ∈ B ( D ) } c ( e ) t e s.t. (4b) and (4c) . Step (ii) is to sort V ( C ) with the given method and then toperform the F WC algorithm (Alg. 3) with the computed t and vertex ordering. Since, it is guaranteed by step (i) and (ii)that the schedule length after performing the F WC algorithmis no more than one, we perform the last step to scale theschedule length to exactly unit time. The goal is to improveperformance by fully utilizing the available time resource.
C. Fixed and Arbitrary Ordering (F WC-FAO)
Assume that (cid:104) v . . . v n (cid:105) is an arbitrary but fixed orderingof V ( C ) where n = | V ( C ) | . We denote v i < v j if i < j .Let V i be the set of vertices of v i and all its smaller neighbors (neighbors in { v . . . v i − } ). Define the inductiveindependence polytope Q of C by the ordering (cid:104) v . . . v n (cid:105) as Q (cid:44) (cid:110) t ∈ R V ( C )+ (cid:12)(cid:12)(cid:12) max ≤ i ≤ n t ( V i ) ≤ (cid:111) , (6)where t ( V i ) = (cid:80) v ∈ V i t v . Q is an approximation of theindependence polytope P . D. Largest Surplus Last Ordering (F WC-LSLO)
The largest surplus last ordering of V ( C ) is done by firsttransforming the undirected graph C into a directed graph C d by imposing a certain orientation on each edge. We specifythe following orientation.Suppose that the vertices of the directed network D havean ordering. That is, given two different vertices w, w (cid:48) ∈ V ( D ) , if w comes before w (cid:48) in the ordering, we denote w < w (cid:48) . Given two different vertices u ( i ) and v ( j ) ofthe expanded network H , we denote u ( i ) < v ( j ) if andonly if u < v or ( u = v and i < j ). For the MAX-SU-SM model, given two different vertices ( u ( i ) , v ( j ) ) and ( s ( k ) , t ( l ) ) ∈ V ( C d ) , ( u ( i ) , v ( j ) ) < ( s ( k ) , t ( l ) ) , if and only if u ( i ) < s ( k ) or ( u ( i ) = s ( k ) and v ( j ) < t ( l ) ) . For the REAL-SU-SM model, given two different vertices ( u ( i ) , v ( j ) ) m and ( s ( k ) , t ( l ) ) n ∈ V ( C d ) , ( u ( i ) , v ( j ) ) m < ( s ( k ) , t ( l ) ) n if and onlyif ( u ( i ) , v ( j ) ) < ( s ( k ) , t ( l ) ) or (cid:0) ( u ( i ) , v ( j ) ) = ( s ( k ) , t ( l ) ) and m < n (cid:1) .The orientation is chosen according to the following rulesfor each edge in C . Note, the subscripts m, n are taken asempty for the MAX-SU-SM model.1) An edge between ( u ( i ) , v ( j ) ) m and a vertex of the form ( v ( k ) , x ( l ) ) n such that u (cid:54) = x has the orientation fromthe first to the second.2) Otherwise, an edge between two vertices has the orien-tation from the small one to the large one.Let D (cid:48) be a digraph. For a vertex u ∈ V ( D (cid:48) ) , let N in ( u ) denote the set of in-neighbors of u in D (cid:48) , and let N in [ u ] denote { u } ∪ N in ( u ) . N out ( u ) and N out [ u ] are definedcorrespondingly. For any t ∈ R V ( D (cid:48) )+ , the surplus of a vertex u is defined as t ( N in ( u )) − t ( N out ( u )) . The largest surplus lastordering is constructed as follows. Let t ∈ R V ( C d )+ . Initialize D (cid:48) to C d . For i = n down to 1, let v i be a vertex of the largestsurplus in ( D (cid:48) , t ) and then delete v i from D (cid:48) and the element t v i from t . The ordering of (cid:104) v . . . v n (cid:105) is the largest surpluslast ordering of ( C d , t ) . The independence polytope Q (cid:48) of C d is defined as Q (cid:48) = (cid:110) t ∈ R V ( C d )+ (cid:12)(cid:12)(cid:12) max u ∈ V ( C d ) t ( N in [ u ]) ≤ / (cid:111) , (7)which is another approximation of the independence polytope P . E. Approximation Ratios In Terms of Max-Min Throughput
The following theorem presents the worst-case approxi-mation ratios in terms of max-min throughput of the twoalgorithms F WC-FAO and F WC-LSLO. Theorem 6.
The algorithms F WC-FAO and F WC-LSLOsolve the MTFS problem by producing a unit-time schedule.They achieve a max-min throughput θ (cid:48) ≥ θ ∗ /α ∗ and θ (cid:48) ≥ θ ∗ / (2 β ∗ ) respectively, where θ ∗ is the optimum and • for the case of a full-duplex network, PI and REAL-SU-SM model: α ∗ ≤ max (cid:16) , max l ∈ E ( L ) (cid:0) (cid:88) l (cid:48) | intf( l (cid:48) ,l )=1 d ( l (cid:48) ) (cid:1)(cid:17) + 2 β ∗ ≤ max (cid:16) , max l ∈ E ( L ) (cid:0) (cid:88) l (cid:48) | l (cid:48) PPROXIMATION A LGORITHM OF P ARALLEL D ATA S TREAM S CHEDULING This section proposes an effective approximation algorithmfor half-duplex MTFS scheduling under the NI model. ThePDS (Parallel Data Stream Scheduling) approximation algo-rithm (listed in Alg. 4) extends the optimal half-duplex MTFSalgorithm in §V-C to cover the situation that an optimal MTFSschedule cannot be found in polynomial time. It is based on theidea that the parallel data streams between a pair of BSs arealways scheduled simultaneously. An example of the graphtransformation step (Line 1 to 3) of the PDS algorithm isshown in Fig. 11. Fig. 11. Example of the graph transformations in PDS. (cid:50) and (cid:35) representsmacro BS and relay BS, respectively. The number inside a node v is r G ( v ) and the number next to an arc e is c G ( e ) where G is the related graph. Theorem 7. Suppose that the optimal max-min throughput ofthe half-duplex MTFS problem on a directed network D is θ ∗ under the NI model. Let the max-min throughput obtainedwith the PDS algorithm be θ and r min = min u ∈ V ( D ) r D ( u ) be the minimum RF chain number of any BS. Let r Mmax =max u ∈ M ( D ) r D ( u ) be the maximum RF chain number of any Algorithm 4: PDS algorithm. Create a network D e based on the directed network D . D e copies the relay BS vertices and the arcs between them from D while keeping the values of RF chain number and capacityunchanged. Let the minimum data stream number of any linkin the link network L be d min = min l ∈ E ( L ) d ( l ) . Each macroBS vertex v in D is mapped into s ( v ) macro BS vertices v (1) . . . v ( s ( v )) in D e where s ( v ) (cid:44) (cid:98) r D ( v ) /d min (cid:99) . (8)The RF chain number of each macro BS vertex v ( i ) is definedas r D e ( v ( i ) ) (cid:44) (cid:40) d min if i < s ( v ) m ( v ) = r D ( v ) − [ s ( v ) − d min otherwise . (9)For each i , create d = min( r D e ( v ( i ) ) , r D e ( w ) , d ( v, w )) arcs ( v ( i ) , w ) j in D e such that c D e (( v ( i ) , w ) j ) = c D (( v, w ) j ) , ∀ j = 1 . . . d , for each neighbor w of v in D ; Make a copy D s of D e and replace each set of parallel arcwith a single arc. For each arc ( u, v ) in D s , define thecapacity function associated with D s as c D s (( u, v )) (cid:44) (cid:80) j c D e (( u, v ) j ) . Define the RF chain numberfunction associated with D s as r D s ( u ) (cid:44) , ∀ u ∈ V ( D s ) ; Compute the optimal MTFS schedule S for D s with thecapacity c D s and RF chain number r D s using the method in§IV-B; Create the final schedule S ∗ based on S , by mapping theactivation of an arc in D s into the simultaneous activation ofparallel arcs in D ; relay BS, and d min = min l ∈ E ( L ) d ( l ) be the minimum datastream number of any link. We have θ ≥ θ ∗ /γ ∗ , where • γ ∗ = max( r Mmax , max v ∈ B ( D ) m ( v )) ≤ max( r Mmax , d min − , where m ( v ) is defined in (9) , ifthe REAL-SU-SM model is assumed; • γ ∗ = max( r Mmax , max v ∈ B ( D ) m ( v )) r min ≤ max( r Mmax , r min − r min , ifthe MAX-SU-SM model is assumed.Proof. See Appendix H for the proof.Let us consider some special cases of the MAX-SU-SMmodel. If each relay BS in D has the same RF chain number r M and each macro BS has an RF chain number that is amultiple of r M , then the PDS algorithm attains the optimalMTFS schedule. On the other hand, if each relay BS has r M RF chains and any macro BS has at least r M RF chains, thenPDS has a worst-case performance ratio of / for the max-min throughput. Corollary 2. Assume the NI model and the MAX-SU-SMmodel. Given a directed network D , assume that each relayBS has the same number of RF chains r M and any macro BShas at least r M RF chains. The PDS algorithm achieves themax-min throughput θ > θ ∗ / where θ ∗ is the optimum forthe half-duplex MTFS problem.Proof. According to Theorem 7, θ ≥ r M r M − θ ∗ > θ ∗ / . In summary, under the NI model, the three algo-rithms: PDS, F WC-FAO and F WC-LSLO are respectively r Mmax , max v ∈ B ( D ) m ( v )) ≥ r max , / max l ∈ E ( L ) ( r ( l )) and / (2 r max + 2) -approximate algorithms for the half-duplexMTFS problem, where r max = max v ∈ V ( D ) r D ( v ) . A ρ -approximate ( ρ ≤ ) algorithm achieves a max-min throughput θ that is at least ρ times that of the optimal value θ ∗ , θ ≥ ρθ ∗ .Theoretically, PDS has the best performance and F WC-LSLOhas the worst.VIII. E XTENSION T O I NTEGRATED A CCESS AND B ACKHAUL To date, 3GPP is investigating the standardization of In-tegrated Access and Backhaul (IAB) for mmWave cellularnetworks [24]. Yet designing a high-performance IAB networkis still an open problem [25], [26]. This paper offers jointrouting and scheduling algorithms for IAB networks withoptimal or guaranteed QoS. Both the optimal algorithm andapproximation algorithms proposed in this paper can readilybe applied to the scenario of integrated backhaul and access(IAB) networks. Due to the graph-based network modeling,our approach is applicable to both IAB networks and backhaulnetworks. However, the runtime efficiency may be an issue, ifthe IAB network includes numerous user equipments (UE).The proposed algorithms in this paper solve a downlink opti-mization problem. Yet with slight modification, they can solvean uplink or a joint uplink and downlink optimization problem.By doing so, the optimal algorithms still retain their optimalitywhile the approximation algorithms keep their approximationratios. A joint uplink and downlink optimization may use theresources better than two separate optimizations. Conceptually,every algorithm in this paper has two parts, the routing part andthe data stream conflict resolving part, which are performedeither sequentially or intertwined. The routing part is a linearprogram that finds an efficient routing scheme for arbitrarythroughput requirements on sources and destinations. So itnaturally supports an uplink or a joint uplink and downlinkoptimization. The data stream conflict resolving part useseither the matching technique for the optimal algorithms orthe conflict graph technique for the approximation algorithms.In addition, our algorithms can be extended to solve otherproblems than MTFS. These include problems that can beformulated as a linear program whose variables are the activetime of data streams and QoS metrics. For example, we canoptimize for the constraint that each relay BS has a minimumthroughput requirement. Another example is to optimize theenergy consumption as it can be translated into the minimiza-tion of total transmission time in a schedule. We do not furtherelaborate on them as the extension is straightforward.IX. N UMERICAL E VALUATION In this section, we evaluate the proposed optimal andapproximation algorithms for the MTFS problem in terms ofmax-min throughput, network throughput and execution time. A. Evaluation Setting We simulate an mmWave backhaul network, which consistsof n × n relay BSs and j × k macro BSs. The relay BSs areplaced on the intersections of n horizontal and n vertical gridlines. The distance between two neighboring grid lines is d g . Fig. 12. An example backhaul network consisting of × relay BSs and × macro BSs. TABLE IS IMULATION PARAMETERS Parameter Value Distance between 2 grid lines, d g 80 mCarrier frequency, f 28 GHzPath loss parameters α, β, σ in P L ( d ) = α + 10 β log d + ξ LOS: α = 61 . , β = 2 , σ = 5 . NLOS: α = 72 , β = 2 . , σ = 8 . Transmission power, p tx 30 dBDirectivity gain, g x 30 dBBandwidth, b N = kT + F +10 log b kT = − dBm/Hz, F = 4 dBMin SINR threshold for recep-tion, τ dBNumber of data streams, K K ∼ max { Poisson ( λ ) , } , λ = 1 . Beamwidth, φ φ = 20 ◦ Correlation coefficient in the ex-ponential correlation matrix, r r = 0 . The grid plane is divided into j × k equal rectangles and amacro BS is placed at each rectangle center (see Fig. 12 foran example). We assume channel reciprocity in the simulation.The capacity of a link is computed with the formula ofShannon capacity. This is the value if one RF chain is usedto serve the link on both ends. The received power is givenby p rx = p tx + g x − P L where p tx is the transmission power, g x is the directivity gain and P L is the path loss. We assumea carrier frequency of 28 GHz. The channel state of a linkis simulated according to the statistical model derived fromthe real-world measurement [40]. There are three possiblechannel states—LOS (line-of sight), NLOS (non line-of-sight)or outage. We only keep the links that are in LOS or NLOSstate and have an SNR higher than dB. The simulationparameters are listed in Tab. I.For the PI model, we simulate the pairwise link interferenceaccording to the model in §III-C. As illustrated in Fig. 3,the 4 links ( t , r ) , ( t , r ) , ( t , r ) , ( t , r ) are assumed to beindependent.To simulate the REAL-SU-SM model, we assume that themaximum number of data streams supported by a link isPoisson distributed with the mean value . (Tab. I), followingthe empirical model of [40]. The total capacity of a linkincreases sublinearly to the number of data streams andis simulated according to the exponential correlation matrixmodel in [41] by choosing the correlation coefficient r = 0 . .A comparison of the total capacity of parallel data streams ξ represents the shadowing effect. It is a normally distributed randomvariable with zero mean and σ standard deviation. Num of data streams T o t a l c apa c i t y ( G bp s ) MAX-SU-SM (SNR = 5 dB)REAL-SU-SM (SNR = 5 dB)MAX-SU-SM (SNR = 30 dB)REAL-SU-SM (SNR = 30 dB) Fig. 13. Comparison of the total capacity of parallel data streams for theREAL-SU-SM and MAX-SU-SM models. for the REAL-SU-SM and MAX-SU-SM models is shown inFig. 13. algo OPT-FD-MTFS OPT-HD-MTFSslot v → v v → v v → v v → v v → v v → v v → v time 1 0.4286 0.5714Fig. 14. v is the macro BS, and v and v are relay BSs. Each node has 2RF chains. We assume NI and MAX-SU-SM models. The optimal max-minthroughput for full-duplex MTFS and half-duplex MTFS problems are and . respectively. The proposed algorithms are implemented in MATLAB,except that we use the C++ program Blossom V for mini-mum cost perfect matching [36] and Gurobi [42] for linearprogramming.We evaluate the optimal algorithms OPT-FD-MTFS andOPT-HD-MTFS as well as three approximation algorithms—F WC-FAO, F WC-LSLO, and PDS, for 10 backhaul net-works with × relay BSs. OPT-FD-MTFS works forfull-duplex scheduling under the NI model while OPT-HD-MTFS works for half-duplex scheduling of uniform orthog-onal backhaul networks. F WC-FAO and F WC-LSLO aregenerally applicable for any combination of half-duplex/full-duplex, NI/PI model and MAX-SU-SM/REAL-SU-SM modelwhile PDS only works for half-duplex scheduling under theNI model.We place 1, × or × macro BSs in each network. Themacro BSs and the relay BSs have the same number of RFchains r B and r M respectively, while r B and r M range from1 to 5. B. Optimal Algorithms Both full-duplex and half-duplex optimal schedules can becomputed efficiently for uniform orthogonal backhaul net-works. Surprisingly, for such networks, the max-min through-put of both OPT-HD-MTFS and OPT-FD-MTFS schedulesare usually the same. We believe that the close performanceof max-min throughput for both half-duplex and full-duplexscheduling is due to the good connectivity of the backhaulnetwork which allows plenty of scheduling possibilities. Asimple network in Fig. 14 shows that the performance gapcan be large. The max-min throughput θ ∗ of OPT-FD-MTFS for theREAL-SU-SM model is shown in Fig. 15 for various numberof macro BSs and RF chains. Generally, θ ∗ increases withthe number of RF chains of relay BS ( r M ) and of macro BS( r B ), as well as the number of macro BSs. When the numberof macro BSs and r B are fixed, θ ∗ gradually saturates despitethe increase of r M . In such cases, the bottleneck is at thelinks between macro BSs and relay BSs. To achieve higherperformance in θ ∗ , we need to increase all three variables.Yet, adding macro-BSs would be very costly. Adding moreRF chains to each macro-BS while increasing the relay BSsthat are neighbors to these macro BSs seems like a more cost-effective approach. Moreover, the average throughput per relayBS is from 1x to 1.96x of the max-min throughput. This showsthat in a dense network, we can achieve a rather equal distribu-tion of throughput among relay BSs. As expected, θ ∗ of OPT-FD-MTFS for the MAX-SU-SM model is greater than or equalto that of the REAL-SU-SM model. The difference increaseswith r M and r B (Tab. II), which shows that multiple RF chainsare especially beneficial to a rich multi-path channel. TABLE IIT HE RATE OF θ ∗ OF MAX-SU-SM TO THAT OF REAL-SU-SM.Avg. rate r M = 1 r M = 2 r M = 3 r M = 4 r M = 5 r B = 1 r B = 2 r B = 3 r B = 4 r B = 5 The distributions of execution time of OPT-FD-MTFS andOPT-HD-MTFS for uniform orthogonal backhaul networks areshown in Fig. 16. OPT-HD-MTFS achieves almost the sameperformance in max-min throughput and network throughputas OPT-FD-MTFS, yet it runs much faster than the latter, byshortening the execution time by 27% on average and by 79%in the best case. The reason is due to the step of mergingRF chains in the OPT-HD-MTFS (same as PDS) algorithmwhich leads to a smaller (in terms of vertices and arcs) graphon which matching is performed. Recall that in general cases,the HD-MTFS problem is NP-hard. In addition, we observefrom Fig. 16 that the execution time increases with the numberof macro BSs for both algorithms. In addition, the executiontime of OPT-FD-MTFS also increases with the number of RFchains at BSs due to the growth of the graph for matching. C. Full-Duplex Approximation Algorithm If there is mutual interference between links in a back-haul network, we cannot use the optimal full-duplex MTFSscheduling algorithm. However, two fractional weighted col-oring based approximation algorithms proposed in §VI canbe applied. Because the MTFS problem is NP-hard under thePI model, we use the performance of OPT-FD-MTFS as anupper bound. Fig. 17 shows the results for the REAL-SU-SMmodel. We observe that mmWave backhaul networks are noise-limited instead of interference-limited. On average, there are611 directional links in an evaluated backhaul network, amongwhich only 21 pairs of links are interfering, although we m ean m a x - m i n t h r oughpu t ( G bp s ) (a) 1 macro BS (b) 2 macro BSs (c) 4 macro BSsFig. 15. Max-min throughput of the OPT-FD-MTFS algorithm for the REAL-SU-SM model and for different number of RF chains. number of macro BSs e x e c u t i on t i m e ( s e c ) (a) OPT-FD-MTFS number of macro BSs (b) OPT-HD-MTFSFig. 16. Execution time of optimal algorithms for uniform orthogonalbackhaul networks. Fig. 17. Max-min throughput of full-duplex approximation algorithms forREAL-SU-SM and PI model normalized to that of OPT-FD-MTFS, and thelower bounds of the approximation ratios. Median, and percentilesare shown in the errorbars. choose a relatively large beamwidth of ◦ . Despite consider-ing the interference, both algorithms achieve on average morethan of the optimal max-min throughput for the idealinterference-free case. In general, F WC-FAO outperformsF WC-LSLO in terms of max-min throughput. Besides, thetheoretical approximation ratios of Theorem 6 significantlyunderestimate the actual performance of the F WC algorithms.The results for the MAX-SU-SM model are omitted as theyare similar.The execution time of the F WC algorithms and OPT-FD-MTFS are shown for two SU-SM models in Fig. 18. In general,it takes OPT-FD-MTFS less than 100 seconds to schedule abackhaul network with 100 relay BSs and the execution time -2 -1 e x e c u t i on t i m e ( s e c ) Fig. 18. Execution time comparison of approximation algorithms and OPT-FD-MTFS for full-duplex scheduling. Median, and percentiles areshown in the errorbars. (R) and (M) stands for REAL-SU-SM and MAX-SU-SM, respectively. even decreases with r M . Thus, it is practical to compute theoptimal schedule for full-duplex backhauls if interference canbe ignored. The approximation algorithms are more efficientthan OPT-FD-MTFS when r M is small. Yet the execution timegoes up quickly with r M , especially for the REAL-SU-SMmodel. The reason is due to the large number of vertices inthe conflict graph | V ( C ) | which is equal to the number of arcsin the expanded network H (see §VI-A). A F WC algorithmneeds to solve a linear program of | V ( C ) | + 1 variables.For example, with r M = 5 and the REAL-SU-SM model,the linear program has about 30,000 variables, which takes along time to solve. For future work it would be interestingto investigate how to shrink the conflict graph, in order toimprove the runtime. D. Half-Duplex Approximation Algorithms F WC-FAO, F WC-LSLO and PDS are 3 approximationalgorithms for half-duplex MTFS scheduling. The first twowork for all cases while PDS only works for the NI model.We show in §IX-B that the optimal max-min throughput ofthe half-duplex MTFS problem is the same or very close tothat of full-duplex MTFS for uniform orthogonal backhaulnetworks. Therefore, we use the max-min throughput of OPT-FD-MTFS as the reference for the evaluation of half-duplexapproximation algorithms. Fig. 19(a) and 19(b) show the (a) PI and REAL-SU-SM (b) NI and REAL-SU-SMFig. 19. Max-min throughput of half-duplex approximation algorithms nor-malized to that of OPT-FD-MTFS, and the lower bounds of the approximationratios. Median, and percentiles are shown in the errorbars. -2 -1 e x e c u t i on t i m e ( s e c ) Fig. 20. Execution time of the approximation algorithms for half-duplexscheduling. Median, and percentiles are shown in the errorbars. (R)and (M) stands for REAL-SU-SM and MAX-SU-SM, respectively. results for the PI and NI models assuming the REAL-SU-SM model. All three algorithms attain far better performancethan the theoretical lower bounds. The two F WC algorithmshave similar performance. Under the NI model, PDS has thebest max-min throughput, being higher than on average.The performance of PDS is even better for the MAX-SU-SMmodel. For example, it is guaranteed to reach the optimal whena backhaul network is uniform orthogonal.Fig. 20 displays the time efficiency of the three approxi-mation algorithms. They are all relatively efficient, requiringno more than two minutes. In comparison, F WC algorithmsrun faster because we use the property that a directed networkcan be sparsely expanded under the condition of half-duplex scheduling, which leads to a small conflict graph. We againobserve the trend that the execution time of F WC goesup with r M while that of PDS goes down. In addition, theexecution time of the REAL-SU-SM model is larger than thatof the MAX-SU-SM model. This is due to a larger conflictgraph for F WC and an increase in time for maximum weightmatching for PDS.In summary, the evaluation shows that a mmWave backhaulnetwork is generally noise-limited even for a relatively largebeamwidth of ◦ . The optimal max-min throughput in prac-tical backhaul networks is quite similar for both full-duplexand half-duplex scheduling. PDS is an ideal approximationalgorithm for half-duplex scheduling under the NI model asit achieves near optimal performance within practical time.Finally, the two F WC algorithms have similar max-minthroughput. They are competitive in execution time for smallbackhaul networks with a small number of RF chains andhalf-duplex scheduling.X. C ONCLUSION In this article, we studied the scheduling of mmWave back-haul networks assuming a general system model of multiplemacro BSs, relay BSs and RF chains as well as interferencebetween links and realistic single-user spatial multiplexing.Under the assumption of full-duplex radios and interference-free links, we found an optimal joint routing and schedulingmethod— schedule-oriented optimization based on matchingtheory. It can solve any problem formulated as a linearprogram whose variables are data stream activation durationsand QoS metrics. The method is demonstrated to be efficientin practice, capable of solving the maximum throughput fairscheduling (MTFS) problem within a few minutes for abackhaul network of 4 macro BSs, 100 relay BSs and 5RF chains at each node. However, for the more realisticassumption of half-duplex radios or pairwise link interference,we proved that the MTFS problem is NP-hard. Subsequently,the paper proposed a number of approximation algorithmswith provable performance bounds for the MTFS problem. ThePDS algorithm works for half-duplex scheduling under the NI(no interference) model. It achieves the optimal performancefor uniform orthogonal backhaul networks and about 80%of the optimum for general backhaul networks. The F WCalgorithms adapted to our problem are more general than PDSas they support any combination of full-duplex/half-duplex,REAL-SU-SM/MAX-SU-SM model and PI/NI model. Theirperformance is in general more than half of the optimum.In summary, the paper presents optimal and approximationalgorithms that are highly practical for scheduling mmWavecellular networks. A CKNOWLEDGEMENT This work has been performed in the context of the DFGCollaborative Research Center (CRC) 1053 MAKI and theLOEWE center emergenCITY. It was also supported in partby the Minister of Science and Technology of Taiwan underGrant 104-2911-I-011-503 and the Region of Madrid throughTAPIR-CM (S2018/TCS-4496). A PPENDIX A. Proof of Theorem 1Proof. The proof applies the technique used in [43] for prov-ing that fractional edge coloring can be solved in polynomialtime by the ellipsoid algorithm. Specifically, a linear programis solvable in polynomial time if the separation problem of itsdual problem can be solved in polynomial time. The separationproblem of a linear program J is to determine whether a givensolution satisfies all constraints of J or a violated constraintis identified.If we can solve both linear programs of (1) and (2) inpolynomial time, then we can solve the MTFS problem inpolynomial time. We first prove that (1) can be solved inpolynomial time. The dual of (1) ismin q (10a)s.t. p T A M − q T ≤ T (10b) p T = 1 (10c) p ≥ . (10d)Let D be the directed network. Given a solution ( p , q ) , (10c)and (10d) can be checked in polynomial time, since the totalnumber of constraints in (10c) and (10d) is | M ( D ) | + 1 and p contains | M ( D ) | elements.To check (10b), we use the polynomial-time maximumweighted simple b -matching algorithm [38, Chap. 33]. Aconstraint of (10b) is of the form p T a Mk ≤ q , where a Mk isthe k -th column of A M (corresponding to a simple b -matchingof D ). Define a weight function w : E ( D ) (cid:55)→ R . We set theweights to each arc e = ( v i , v j ) l ∈ E ( D ) ( e is the l -th arcfrom vertex v i to vertex v j ): w ( e ) = (cid:40) c ( e )( p j − p i ) if v i ∈ M ( D ) c ( e ) p j otherwise. (11)Then we perform maximum weighted simple b -matchingon D . Let the maximum weight be w = max k p T a Mk . If w ≤ q , then ( p , q ) satisfies (10b). Otherwise it gives a violatedconstraint.According to Theorem 3.10 in [39], for a linear program J , if we can solve the separation problem of its dual J ∗ in polynomial time, then we can solve both J and J ∗ inpolynomial time with the ellipsoid algorithm. This proves that(1) can be solved in polynomial time.Similarly, we next prove that (2) can be solved in polyno-mial time. The dual of (2) ismin θ ∗ p T + q (12a)s.t. p T A M + q T ≥ c T (12b) p ≤ . (12c)Given a tuple ( p , q ) , we set the following weights to each arc e = ( v i , v j ) l ∈ E ( D ) w ( e ) = (cid:40) c ( e )( p i − p j ) if v i ∈ M ( D ) c ( e )(1 − p j ) otherwise. (13)Then we perform maximum weighted simple b -matching on D . Depending on whether the maximum weight satisfies w = max k ( c k − p T a Mk ) ≤ q , the constraints of (12b) are satisfiedor a violated one is identified. With the same argument asabove, (2) can be solved in polynomial time. This completesthe proof. B. Proof of Theorem 2Proof. As is well-known that it is NP-hard to find the fractional chromatic number χ f ( G, ) (minimum fractionalweighted coloring assuming each vertex has weight 1) for anarbitrary graph G [39]. Given a graph G , we create a directednetwork D as follows. D has | V ( G ) | vertices and | V ( G ) | arcs. For each v ∈ V ( G ) , we create a pair of vertices v B and v M representing a macro BS and a relay BS, and an arc ( v B , v M ) in D . For each edge { u, v } ∈ E ( G ) , we specifythat the two arcs ( u B , u M ) and ( v B , v M ) in D interfere witheach other. In addition, we assume that every vertex in D has one RF chain and every arc in D have unit capacity.Then it is obvious, that the optimal max-min throughput θ ∗ = 1 /χ f ( G, ) . This proves that the MTFS problem is NP-hard under the PI model. This result applies for both half-duplex and full-duplex scheduling as it makes no differencewhen the RF chain number is one. C. Proof of Lemma 1Proof. We reduce the satisfiability (SAT) problem [44], whichis NP-hard, to the MWHS problem on a DAG. Let Z = C ∧ · · · ∧ C K be a boolean expression to satisfy. Z consistsof K clauses and each clause C k is of the form y ∨ · · · ∨ y J ,where k ∈ { . . . K } (cid:44) [1 : K ] . Note J is the numberof literals in C k and dependent on k . Suppose Z containsin total L boolean variables x . . . x L , then the literals y j ∈{ x , ¬ x . . . x L , ¬ x L } for j ∈ [1 : J ] . We construct a directednetwork D as follows. Note, D is a strict digraph. Let W, Q be two disjoint vertex sets with W = W ∪ · · · ∪ W L and Q = { q . . . q K } , where W l = { p l , n l , r l } , for l ∈ [1 : L ] .Let V ( D ) = W ∪ Q , so D has L + K vertices. Next, weconstruct the arc set E ( D ) . For each clause C k , we define thearc set E k (cid:44) { ( p l , q k ) | ∃ y j in C k such that y j = x l }∪{ ( n l , q k ) | ∃ y j in C k such that y j = ¬ x l } . In addition, for each variable x l , we define the arc set A l (cid:44) { ( r l , p l ) , ( r l , n l ) } . The arc set of D is E ( D ) = E ∪ · · · ∪ E K ∪ A ∪ · · · ∪ A L . The weight is set as w ( e ) = 1 , ∀ e ∈ E ( D ) . We define the RFchain number function r as: r ( v ) (cid:44) (cid:40) max { deg + ( v ) , } , if v = p l or v = n l , , otherwise,where deg + ( v ) is the outdegree of vertex v . Obviously, D isa DAG. An example for constructing D from a SAT problemis shown in Fig 21. To complete the proof, we need to show: Fig. 21. The DAG directed network D for Z = ( x ∨ x ) ∧ ( x ∨ ¬ x ) .The number in a vertex v is r ( v ) . w ( e ) = 1 for each arc e . Claim: Z is satisfiable if and only if D has a half-duplexsubgraph with total weight of K + L .Now we prove the claim. Suppose Z is satisfiable. We willselect a set of arcs E ⊆ E ( D ) . For each variables x l = true,we add to E all arcs leaving p l and the arc ( r l , n l ) . For eachvariable x l = false, we add to E all arcs leaving n l and thearc ( r l , p l ) . E satisfies the degree and half-duplex constraintson each vertex w ∈ W . Since Z is satisfied, for each k , thereis at least one arc in E that has one end in W and the otherend at q k . We remove arcs from E that are incident to Q untileach q k is incident to exactly one arc. Now E is a half-duplexsubgraph of D with total weight K + L .Conversely, suppose E is a half-duplex subgraph of D , thenthe maximum weight of arcs in E that are between W and Q is K and the maximum weight of arcs in E that are betweenvertices in W is L . If D has a half-duplex subgraph E withtotal weight K + L , then there are exactly L arcs betweenvertices in W , one for each W l . If there is an arc ( r l , p l ) ∈ E ,we set x l = false, otherwise, if there is an arc ( r l , n l ) ∈ E ,we set x l = true. With this assignment Z is satisfied, since Q is incident to exactly K arcs in E . Thus, the MWHS problemis NP-hard on a general directed network that is a DAG. D. Proof of Theorem 3Proof. Similar to Lemma 1, we prove by reducing the SATproblem [44], which is NP-hard, to the full-duplex MTFSproblem on a directed network. Let Z = C ∧ · · · ∧ C K be a boolean expression to satisfy. Z consists of K clausesand each clause C k is of the form y ∨ · · · ∨ y J , where k ∈ { . . . K } (cid:44) [1 : K ] . Note J is the number ofliterals in C k and dependent on k . Suppose Z contains intotal L boolean variables x . . . x L , then the literals y j ∈{ x , ¬ x . . . x n , ¬ x n } for j ∈ [1 : J ] . We construct adirected network D as follows. Note D is a strict digraph. Theconstruction is more complex than in the proof of Lemma 1,which is necessary for the transformation between a SATproblem and an optimal schedule. Let W, Q be two disjointvertex sets with W = W ∪ · · · ∪ W L and Q = { q , . . . , q k } ,where W l = { p (1) l , p (2) l , n (1) l , n (2) l , r (1) l , r (2) l , r (3) l , r (4) l } , for l ∈ [1 : L ] . Let V ( D ) = W ∪ Q , so D has L + K vertices.Next, we construct the arc set E ( D ) . Let the L vertices r ( m ) l , ∀ m ∈ [1 : 4] be macro BSs and all the other verticesbe relay BSs. For each clause C k , we define the arc set E k (cid:44) { ( p (1) l , q k ) , ( p (2) l , q k ) | ∃ y j in C k such that y j = x l }∪{ ( n (1) l , q k ) , ( n (2) l , q k ) | ∃ y j in C k such that y j = ¬ x l } , and set c ( e ) = 1 , ∀ e ∈ E k . In addition, for each variable x l ,we define the arc set A l (cid:44) { ( r (1) l , p (1) l ) , ( r (1) l , n (1) l ) , ( r (2) l , p (2) l ) , ( r (2) l , n (2) l ) , ( r (3) l , p (2) l ) , ( r (3) l , n (1) l ) , ( r (4) l , p (1) l ) , ( r (4) l , n (2) l ) } . The capacity of all arcs leaving macro BSs is set to c ( e ) = K/ , ∀ e ∈ A l . The reason for choosing the value K/ isthat it is a sufficiently large capacity such that the constructedschedule S in the following achieves the optimal max-minthroughput of 1. The arc set of D is E ( D ) = E ∪ · · · ∪ E K ∪ A ∪ · · · ∪ A L . We define the RF chain number function r as: r ( v ) (cid:44) (cid:40) max { deg + ( v ) , } , if v = p ( m ) l or v = n ( m ) l , , otherwise,where deg + ( v ) is the outdegree of the vertex v . An examplefor constructing D from a SAT problem is shown in Fig. 22.To complete the proof, we need to show: Claim: Z is satisfiable if and only if D has a unit time half-duplex schedule that achieves the max-min throughput θ = 1 and the network throughput α = 4 L ( K/ . Fig. 22. The directed network D for Z = ( x ∨ x ) ∧ ( x ∨ ¬ x ) . Thenumber in a vertex v is the value r ( v ) . Vertices r ( j ) i are macro BSs. Thethick arcs e have capacity c ( e ) = K/ and the thin arcs e (cid:48) havecapacity c ( e (cid:48) ) = 1 . Now we prove the claim. Suppose Z is satisfiable, we createa unit time half-duplex schedule S that consists of two slots S , S ⊆ E ( D ) , each with length 0.5. We first create the arcset S , For each variable x l , we define the arc set E (1) l (cid:44) (cid:40) δ + ( p (1) l ) ∪ A (1) l , if x l = true ,δ + ( n (1) l ) ∪ B (1) l , otherwise, where δ + ( v ) is the set of arcs that leave vertex v , A (1) l = { ( r (1) l , n (1) l ) , ( r (2) l , p (2) l ) , ( r (3) l , p (2) l ) , ( r (4) l , n (2) l ) } and B (1) l = { ( r (1) l , p (1) l ) , ( r (2) l , n (2) l ) , ( r (3) l , p (2) l ) , ( r (4) l , n (2) l ) } . Initially, S = E (1)1 ∪ · · · ∪ E (1) L .S satisfies the degree and half-duplex constraints on eachvertex w ∈ W . Since Z is satisfied, for each k , there is atleast one arc in S that has one end in W and the other endat q k . We remove arcs from S that are incident to Q untileach q k is incident to exactly one arc. S is symmetric to S in the sense that it can be created from S : S is obtained by scanning the arcs in S and replacingeach occurrence of r (1) l and r (2) l , r (3) l and r (4) l , p (1) l and p (2) l , n (1) l and n (2) l with each other.It is obvious that the schedule S gives the max-min through-put θ = 1 and the network throughput α = 4 L ( K/ .Conversely, suppose that a unit time half-duplex schedule S (cid:48) achieves the max-min throughput θ = 1 and the networkthroughput α = 4 L ( K/ . The network throughput α is maximum since D has in total L single-RF-chain macroBSs and each arc leaving a macro BS has capacity K/ . So each macro BS must be always active as a sender in S (cid:48) . In addition, since θ = 1 , each relay BS q k achieves thethroughput at least one. Since each q k has single RF chain andany incoming arc to it has capacity one, q k must be alwaysactive as a receiver in S (cid:48) . We pick an arbitrary slot S from S (cid:48) . Since among vertices of W l , 4 macro-BS-to-relay-BS arcsare active at any time, it is impossible to have any pair ofvertices p ( m ) l and n ( m (cid:48) ) l ( m, m (cid:48) ∈ { , } ) active as senders atthe same time. Otherwise, a macro BS must be inactive whichis contradictory to the property of being always active. Finally,we can set the variables x l as follows: if none of the 4 vertices p ( m ) l and n ( m (cid:48) ) l is active as a sender, we set x l arbitrarily; ifone or two of the vertices p ( m ) l are active as senders, we set x l = true; otherwise one or two of the vertices n ( m (cid:48) ) l must beactive as senders, we set x l = false. With this assignment Z is satisfied. Thus, half-duplex MTFS problem is NP-hard fora general directed network. E. Proof of Theorem 4 To prepare the proof of Theorem 4, let us first prove thefollowing lemma. Lemma 2. Assume that an undirected loopless multigraph G has the property that between any pair of vertices u, v ∈ V ( G ) , there are either R ∈ N edges of the same weight w ( { u, v } ) or zero edges. The maximum weight biparite sub-graph J ⊆ G such that each vertex v ∈ V ( J ) has degree deg J ( v ) ≤ R , can be found in polynomial time as follows: Create a simple graph G (cid:48) for G : between each pair ofvertices u, v ∈ V ( G ) , if there are R edges, we remove R − of them. Let the resulting graph be G (cid:48) . Find the maximum weight matching M of G (cid:48) with theweight function w . J is a graph whose edge set is themultiset ( M, R ) , i.e., R -time repetition of M . Proof. Since M is a matching of G (cid:48) , then M is a bipartitegraph such that deg M ( v ) = 1 , ∀ v ∈ V ( M ) . Since J is agraph whose edge set is the multiset ( M, R ) , J is a bipartitesubgraph of G such that deg J ( v ) ≤ R, ∀ v ∈ V ( J ) .Let K be a bipartite subgraph of G such that deg K ( v ) ≤ R, ∀ v ∈ V ( K ) . Since K is bipartite, its vertices have abipartition [ U, V ] . We assume without loss of generality that | U | ≥ | V | . Then we add | U | − | V | new vertices to V , andadd edges between U and V to K until we get a R -regularbipartite graph K (cid:48) . Regular means that each vertex has thesame degree, deg K (cid:48) ( v ) = R, ∀ v ∈ V ( K (cid:48) ) . Since K (cid:48) is a R -regular bipartite graph, any subset S ⊆ U is connectedwith at least | S | vertices in V according to the pigeonholeprinciple. Then according to the Hall’s marriage theorem [45], K (cid:48) contains a matching N with cardinality | U | . Removing N from K (cid:48) , we get a ( R − -regular bipartite graph. Inductively,we have proved that K (cid:48) can be decomposed into R matchings.Therefore, K , a subgraph of K (cid:48) , can be decomposed into atmost R matchings. Each matching is a subgraph of G (cid:48) . Since M is a maximum weight matching of G (cid:48) , J is a maximumweight bipartite subgraph of G such that each vertex v ∈ V ( J ) satisfies deg J ( v ) ≤ R . The algorithm is polynomial-timebecause the maximum weight matching on a graph can besolved in polynomial time [35]. Proof of Theorem 4:Proof. The linear program formulation of the half-duplexMTFS problem is as follows.max θ (14a)s.t. L M t S ≥ θ (14b) T t S = 1 and t S ≥ , (14c)max c T t S (15a)s.t. L M t S ≥ θ ∗ (15b) T t S = 1 and t S ≥ , (15c)We prove by solving (14) and (15), which give the optimalschedule for the half-duplex MTFS problem. The method issimilar to that of Alg. 1 and Alg. 2.The first step is to find an initial basic feasible solution to(14). We use the method for the full-duplex MTFS problemin §IV-B. Suppose the result is a schedule S . Then we define S (cid:48) to be R copies of S running in parallel. Obviously, S (cid:48) isan initial basic feasible solution to (14).To compute the max-min throughput, Alg. 1 and Alg. 2need to be modified. In Line 4 of both Alg. 1 and Alg. 2,we replace "Do max weight simple b -matching on D " with"Solve the MWHS problem on D ".The MWHS problem on D can be solved as follows. Let D (cid:48) be a subgraph of D that contains only positive arcs. Thesolution of MWHS on D is the same as that on D (cid:48) . Note that D (cid:48) satisfies the condition that if there is an arc ( u, v ) ∈ E ( D (cid:48) ) ,no opposite arcs ( v, u ) are contained in D (cid:48) . The reason isas follows. If both u, v are relay BSs and ( u, v ) ∈ E ( D (cid:48) ) ,then w (( v, u )) = − w (( u, v )) < and the ( v, u ) arcs will beremoved. If u is a macro BS, then ( v, u ) are not contained in D (cid:48) . Moreover, between any two vertices in D (cid:48) , there are either R equivalent arcs (same head, tail and weight) or zero arcs.So, D (cid:48) can be considered as a weighted undirected looplessmultigraph of the uniform edge multiplicity R . Because ahalf-duplex subgraph of D (cid:48) must be a bipartite subgraph withdegree constraint R , the maximum weight bipartite subgraph B ⊆ D (cid:48) with degree constraint R has weight greater than orequal to that of the maximum weight half-duplex subgraphof D (cid:48) . From Lemma 2, B is also a half-duplex subgraph of D (cid:48) . So it is also the maximum weight half-duplex subgraphof D (cid:48) and D . Therefore, the optimal schedule for the half-duplex MTFS problem S ∗ consists of R copies of the sameschedule S executed in parallel since each iteration in Alg. 1and Alg. 2 produces such a schedule. S must be a unit timeschedule for D assuming that each node has one RF chain.Consequently, the optimal schedule for the half-duplex MTFSproblem is obtained by the algorithm in Theorem 4. F. Proof of Theorem 5 Fig. 23. Arcs of J e entering and leaving K e . Proof. Let an arbitrary half-duplex subgraph of the directednetwork D be J . Obviously, J corresponds to a certainmatching J e (which are the data streams scheduled in atimeslot) in the expanded network H MFD as it is fully expanded.We need to prove that J e is equivalent to J s which is amatching of the sparsely expanded network H MHD .Initially, we set all vertices of J as untagged and let J s = ∅ .Starting from an untagged vertex v of J (suppose it has RFchain number r ( v ) ), we find the maximal induced subgraphwith r ( v ) RF chains K , which is defined as a connected (twovertices are connected if there is an arc between them) inducedsubgraph of J that has the largest number of vertices of exactly r ( v ) RF chains. K is a bipartite graph with maximum vertexdegree of at most r ( v ) because the vertices in K can be dividedinto the sender and receiver sets.Let the link network of D be L . According to the K˝onig’sTheorem [45] for the edge coloring of bipartite graphs, K can be decomposed into at most r ( v ) matchings in G , where G is the induced subgraph of L by the vertex set V ( K ) . Inthe sparsely expanded network H MHD , G is expanded into G s which is r ( v ) copies of G . Suppose that K corresponds to agraph K e ⊆ H MFD . By rearranging senders and receivers, K e isequivalent to a K s ⊆ G s . We add K s to J s . Suppose that J e has an arc e that goes from a vertex u ( i ) out of K e to a vertex w ( j ) inside K e (see Fig. 23). Then in K s there is at least onevertex w ( k ) ( k may be different from j ) which is unconnected.We map ( u ( i ) , w ( j ) ) in J e to ( u ( i ) , w ( k ) ) and add the latter arcto J s . This works in the same way for outgoing arcs. After we have processed all arcs entering and leaving K e , we tagall vertices in K . Then we go on to process untagged verticesin J . After we have tagged all vertices, we get a matching J s ⊆ H MHD . G. Proof of Theorem 6 Before proving Theorem 6, we need to first prove twolemmas: Lemma 3. Let D be a directed network and L be thecorresponding link network. We have Q ⊆ P ⊆ α ∗ Q . Forany t ∈ Q , the coloring of ( C, t ) by F WC-FAO has weightat most 1. Furthermore, • α ∗ ≤ max (cid:16) , max l ∈ E ( L ) (cid:0) (cid:80) l (cid:48) | intf( l (cid:48) ,l )=1 d ( l (cid:48) ) (cid:1)(cid:17) + 2 forthe case of full-duplex network, PI and REAL-SU-SMmodel. • α ∗ ≤ max l ∈ E ( L ) (cid:0) (cid:80) l (cid:48) | intf( l (cid:48) ,l )=1 d ( l (cid:48) ) (cid:1) + 2 for the caseof full-duplex network, PI and MAX-SU-SM model. • α ∗ ≤ max l ∈ E ( L ) (cid:0) r ( l )+ (cid:80) l (cid:48) | intf( l (cid:48) ,l )=1 d ( l (cid:48) ) (cid:1) for the caseof half-duplex network and PI model where r ( l ) = r ( u )+ r ( v ) for l = ( u, v ) ∈ E ( L ) .Proof. Let H be the expanded network of D . Let e be an arcin H . Correspondingly, e is a vertex in the conflict graph C .We check the neighbors of e in C . If any independent set of C contained in e and its neighbors, has a size at most N , thenthe inductive independence number of C is α ∗ ≤ N , where α ∗ is defined to be the maximum size of any independent setof C contained in some V i , for ≤ i ≤ n . Q ⊆ P ⊆ α ∗ Q follows directly from Corollary 5.2 of [15]. By Theorem 5.1of [15], for any t ∈ Q , the coloring of ( C, t ) by F WC-FAOhas weight at most max ≤ i ≤ n { t ( V i ) } ≤ . Now we look atfour backhaul network modelings. Fig. 24. The neighbors of a vertex e = ( u ( j ) , v ( k ) ) i ∈ V ( C ) . A rectangleis a complete graph that contributes only one vertex to the independent setwhile a parallelogram may contribute multiple.Fig. 25. The neighbors of a vertex e = ( u (1) , v (1) ) ∈ V ( C ) . There are four possibilities of H depending on the modeling. 1. We check the case of full-duplex, PI and REAL-SU-SMmodel. Let e = ( u ( j ) , v ( k ) ) i be an arc in H RFD . The neighborsof e in C is shown in Fig. 24. The arcs incident to u ( j ) form a complete graph in C , so they contributes only onevertex to the independent set. This applies also be the arcsincident to v ( j ) . If there are no links that are interfering with ( u, v ) . Then the independent set can be extended by one dueto the expanded arcs of ( u, v ) i . Otherwise, it can be extendedby (cid:80) l (cid:48) | intf( l (cid:48) ,l )=1 d ( l (cid:48) ) because each interfering link l (cid:48) maycontribute at most d ( l (cid:48) ) . Because these arcs are in conflict withthe expanded arcs of ( u, v ) i , the latter does not contribute tothe independent set.2. We check the case of full-duplex, PI and MAX-SU-SMmodel. The neighbors of e = ( u ( j ) , v ( k ) ) is similar to Fig. 24except that we don’t have the neighbors that are the expandedarcs of a data stream.3. We check the case of half-duplex, PI and REAL-SU-SMmodel. Let e = ( u (1) , v (1) ) be an arc in H RHD without lossof generality. The neighbors of e in C is shown in Fig. 25.The arcs incident to u (1) , entering u ( j ) , incident to v (1) andleaving v ( k ) for any j (cid:54) = 1 and any k (cid:54) = 1 , each contributesone vertex to the independent set. If an expanded arc, say ( u ( l ) , v ( m ) ) is add to the independent set, then we need toremove two vertices from the independent set that belong tothe arcs entering u ( l ) and those leaving v ( m ) . So it is not worthto do that. The increase of the independent set due to theinterfering links is the same as that of full-duplex scheduling.4. The case of half-duplex, PI and MAX-SU-SM model isthe same as 3. Lemma 4. Let D be the directed network and L be thecorresponding link network. We have Q (cid:48) ⊆ P ⊆ β ∗ Q (cid:48) . Forany t ∈ Q (cid:48) , the coloring of ( C, t ) by F WC-LSLO has weightat most 1. Furthermore, • β ∗ ≤ max (cid:16) , max l ∈ E ( L ) (cid:0) (cid:80) l (cid:48) | l (cid:48) We prove for the first sorting method (FAO). Thesecond (LSLO) can be proved in the same way. The optimalmax-min throughput θ ∗ can be obtained by solving the linearprogrammax θ (16a)s.t. (cid:88) e ∈ δ − H ( U ( v )) c ( e ) t e − (cid:88) e ∈ δ + H ( U ( v )) c ( e ) t e ≥ θ ∀ v ∈ M ( D ) (16b) t ∈ P (16c)and the approximate max-min throughput θ is obtained bysolving (4) with Q ◦ = Q . Suppose that the optimal solutionto (16) is ( θ ∗ , t ∗ ) , we will see that ( θ ∗ /α ∗ , t ∗ /α ∗ ) is asolution to (4). Since t ∗ ∈ P , we have t ∗ /α ∗ ∈ P/α ∗ ⊆ Q due to Lemma 3. In addition, we verify that ( θ ∗ /α ∗ , t ∗ /α ∗ ) satisfies (4b). Therefore θ ≥ θ ∗ /α ∗ . In addition, because theschedule S produced by F WC-FAO without the scaling stepsatisfies t ∈ Q , we have that the length of S is at most 1due to Lemma 3. The scaling step makes the schedule lengthexactly one and the final max-min throughput θ (cid:48) ≥ θ . H. Proof of Theorem 7Proof. Assuming REAL-SU-SM model, let r D e max be the max-imum number of RF chains of any vertex in D e . Given D e , let the optimal max-min throughput of the half-duplexMTFS problem be θ min when all vertices of D e have 1 RFchain (each link l has one data stream of the largest capacity c ( l ) ). Let the optimal max-min throughput be θ max whenall vertices of D e have r D e max and each link has r D e max datastreams of the largest capacity c ( l ) . According to Theorem 4, θ min ≤ θ ≤ θ ∗ ≤ θ max = r D e max θ min . We have θ ≥ θ min = θ max r Demax ≥ θ ∗ r Demax . r D e max = max( r Mmax , max v ∈ B ( D ) m ( v )) . Inaddition, m ( v ) ≤ d min − .Assuming MAX-SU-SM model, D and D e has the sameoptimal max-min throughput for the MTFS problem θ ∗ . Let r D e min and r D e max be the minimum and maximum number of RFchains of any vertex in D e . Given D e , let the optimal max-min throughput of the half-duplex MTFS problem be θ min and θ max when all vertices of D e have r D e min and r D e max number ofRF chains, respectively. 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