Throughput Characterization of Node-based Scheduling in Multihop Wireless Networks: A Novel Application of the Gallai-Edmonds Structure Theorem

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1 Throughput Characterization of Noe-base Scheuling in Multihop Wireless Networks: A Novel Application of the Gallai-Emons Structure Theorem Bo Ji an Yu Sang Dept. of Computer an Information Sciences Temple University Philaelphia, PA {boji, yu.sang}@temple.eu ABSTRACT Maximum Vertex-weighte Matching (MVM) is an important link scheuling algorithm for multihop wireless networks. Uner certain assumptions, it has been shown that if the unerlying network graph is bipartite, MVM not only maximizes the throughput in settings with continuous packet arrivals, but also minimizes the evacuation time (i.e., time to rain all the initial packets) in settings without future packet arrivals. Further, even if the network graph is arbitrary, MVM achieves the best known performance guarantee for the evacuation time among existing online link scheuling algorithms. Also, it empirically exhibits close-to-optimal throughput performance an goo elay performance. However, in an arbitrary network graph the throughput performance of MVM has not been well unerstoo. To that en, in this paper we aim to carry out a systematic stuy of the throughput performance of MVM, assuming single-hop flows an the noe-exclusive interference moel. Inspire by the celebrate Gallai-Emons structure theorem, we introuce a novel topological notion, calle the Gallai-Emons ecomposition factor, an rigorously prove that the efficiency ratio of MVM is no smaller than the Gallai-Emons ecomposition factor of the network graph. Further, we show that if the smallest size of an o cycle in a graph is 2m + 1 for a positive integer m, then the Gallai-Emons ecomposition factor is equal to 2m/(2m + 1). This implies that the Gallai-Emons ecomposition factor is at least 2/3 for an arbitrary graph an is equal to 1 for bipartite graphs. Having these results, the throughput performance of MVM can be well characterize. CCS Concepts Networks Network control algorithms; Network performance analysis; Wireless mesh networks; Permission to make igital or har copies of all or part of this work for personal or classroom use is grante without fee provie that copies are not mae or istribute for profit or commercial avantage an that copies bear this notice an the full citation on the first page. Copyrights for components of this work owne by others than ACM must be honore. Abstracting with creit is permitte. To copy otherwise, or republish, to post on servers or to reistribute to lists, requires prior specific permission an/or a fee. Request permissions from permissions@acm.org. MobiHoc 16, July 04-08, 2016, Paerborn, Germany c 2016 ACM. ISBN /16/07... $15.00 DOI: Keywors Throughput; Noe-base Scheuling; MVM; Wireless Networks; Gallai-Emons Structure Theorem 1. INTRODUCTION Link scheuling is one of the most critical functionalities in multihop wireless networks. The esign of efficient wireless scheuling algorithms is very challenging an has been extensively stuie in the literature (e.g., see [7, 18] an references therein). Among several metrics that are commonly use for evaluating the scheuling performance, the throughput an the evacuation time are two of the most important ones [8,9,11,12]. In settings with continuous packet arrivals, the throughput is measure by the set of traffic loas uner which the network can be stabilize, while in settings without future arrivals, the evacuation time is efine as the time for raining all the initial ata packets in the network. Ieally, we wish to have a scheuling algorithm that both maximizes the throughput an minimizes the evacuation time in respective settings. However, these two objectives state in ifferent settings coul lea to conflicting scheuling ecisions [8,9]. Therefore, it is generally very challenging to esign scheuling algorithms that can achieve provably goo performance in both imensions at the same time. Maximum Vertex-weighte Matching (MVM) is an important scheuling algorithm, which in each scheuling cycle selects a scheule that maximizes the sum of the loas of the scheule noes (see Section 5.1 for the etaile escription). MVM has recently arouse increasing interests ue to its goo performance. First, uner the assumption of single-hop flows an the noe-exclusive interference moel [10, 18, 22], it has been shown in [8, 9] that MVM is both throughput-optimal an evacuation-time-optimal if the unerlying network graph is bipartite. Secon, even if the network graph is arbitrary, MVM achieves the best known guarantee for the evacuation time among existing online scheuling algorithms. Specifically, it has been shown in [12] that MVM has an approximation ratio 1 no greater than 3/2 for the evacuation time. Finally, extensive numerical stuies show that in various scenarios (with ifferent network graphs, arrival processes, an etc.) [8, 9, 11], MVM 1 Uner the noe-exclusive interference moel, the evacuation time problem can be mappe to the well-known ege coloring problem that is generally NP-har (e.g., see [12]). The approximation ratio is the worst-case ratio of the evacuation time of an algorithm to the minimum evacuation time.

2 not only empirically achieves close-to-optimal throughput performance, but also exhibits goo elay performance. However, the throughput performance of MVM has not been well unerstoo. To that en, in this paper we aim to carry out a systematic stuy of the throughput performance of MVM. Throughout the paper, we assume single-hop traffic flows (that traverse only one link) an the noe-exclusive interference moel [10, 18, 22], uner which links sharing a common noe cannot make simultaneous ata transmissions. We summarize our key contributions as follows. First, we introuce a novel topological notion, calle the Gallai-Emons ecomposition factor (Definition 5), which is inspire by the celebrate Gallai-Emons structure theorem in graph theory an the associate ecomposition structure. To the best of our knowlege, this paper, for the first time, evelops such a notion that establishes an interesting connection between the Gallai-Emons structure theorem an the throughput performance of a scheuling algorithm in multihop wireless networks. Secon, we show that the Gallai-Emons ecomposition factor is fully etermine by the smallest size of an o cycle in a graph. Specifically, we prove that if the smallest size of an o cycle in a graph is 2m + 1, where m is a positive integer, then the Gallai-Emons ecomposition factor is equal to 2m/(2m + 1) (Theorem 3). This further implies that the Gallai-Emons ecomposition factor is no smaller than 2/3 for an arbitrary graph an is equal to 1 for bipartite graphs (Corollary 1). Thir, we prove several interesting an important properties of MVM (Propositions 1 an 2). Using these key properties, we rigorously prove that the efficiency ratio of MVM (i.e., the worst-case ratio of the throughput of MVM to that of a throughput-optimal algorithm; see Definition 4) is no smaller than the Gallai-Emons ecomposition factor of the unerlying network graph (Theorem 4). Having these results, the throughput performance of MVM can be well characterize. Together with the result on the evacuation time performance of MVM [12], our finings show that among existing online scheuling algorithms, MVM achieves the most balance performance guarantees in both imensions of throughput an evacuation time. Not only o these results substantially improve our unerstaning of the performance of MVM, but also are instrumental in proviing important guielines for the esign of wireless networks when MVM-type of algorithms are employe. The remainer of this paper is organize as follows. We first iscuss relate work an present our system moel in Sections 2 an 3, respectively. In Section 4, we review the Gallai-Emons structure theorem an introuce the Gallai-Emons ecomposition factor. Then, in Section 5 we escribe the MVM algorithm an prove some key properties of MVM, an in Section 6 we show that the efficiency ratio of MVM is no smaller than the Gallai-Emons ecomposition factor of the unerlying network graph. Finally, we make concluing remarks in Section RELATED WORK As mentione in the introuction, the problem of esigning efficient scheuling algorithms for multihop wireless networks has been extensively stuie. There has been a significant boy of relate work on this topic since the seminal work by Tassiulas an Ephremies [25]. In the following iscussion, we will focus on prior work that is most relevant to this paper. For more general iscussions on this topic, the intereste reaer is referre to [7, 18] an references therein. Since the seminal work of [25], it has been shown that the MaxWeight algorithm an its variants are throughputoptimal in very general settings. However, MaxWeight-type of algorithms generally have a high complexity an require a centralize controller. To that en, several lower-complexity an/or istribute approximation algorithms have been evelope, incluing the greey algorithms (e.g., [14, 17, 26]), ranom access algorithms (e.g., [15, 16]), an those base on the CSMA (Carrier Sensing Multiple Access) techniques (e.g., [13, 20, 21]), which have been shown to be throughputefficient. However, uner the noe-exclusive interference moel none of them can achieve an approximation ratio smaller than 2 for the evacuation time [8, 9]. In stark contrast to the aforementione algorithms that make scheuling ecisions base on the link loas or are loa-agnostic, another class of algorithms take a noe-base approach an make ecisions base on the noe loas [8, 9, 11, 12, 19, 24]. The noe-base approach respects the fact that noes an o cycles with maximum loa are the bottlenecks for achieving small evacuation time, an typically leas to better evacuation time performance [11]. Specifically, it has been shown in [8,9] that a class of noe-base algorithms, incluing MVM, are both throughput-optimal an evacuation-time-optimal in input-queue switches 2. Further, the authors of [12] focus on the evacuation time performance of MVM in arbitrary network graphs an show that MVM achieves an approximation ratio no greater than 3/2. However, the throughput performance of MVM in an arbitrary network graph has not been well unerstoo, which is the focus of this paper. In [11], two noe-base service-balance scheuling algorithms are propose, which have been shown to achieve an efficiency ratio no smaller than 2/3 for the throughput an an approximation ratio no greater than 3/2 for the evacuation time. However, the way these algorithms are esigne reners it ifficult to pursue a fine-graine performance characterization when the network graph belongs to ifferent classes that possess structural properties, while in this paper, the throughput performance of MVM can be well characterize through a novel application of the Gallai-Emons structure theorem an the associate ecomposition structure. Finally, it is worth noting that in [14] another topological notion calle the local-pooling factor is propose to characterize the efficiency ratio of the Greey Maximal Matching (GMM) algorithm. However, there are key ifferences between [14] an this paper. First, the efficiency ratio of GMM in an arbitrary network graph is lower boune by 1/2, which is smaller than 2/3 for MVM as we show in this paper. Secon, the local-pooling factor is generally harer to estimate, while the Gallai-Emons ecomposition factor introuce in this paper is fully etermine by the smallest size of an o cycle in a graph. Moreover, MVM achieves a better approximation ratio for the evacuation time than GMM (3/2 vs. 2). 3. SYSTEM MODEL We consier a multihop wireless network an escribe it as an unirecte, simple graph G = (V, E), where V is the set 2 The results also apply to multihop wireless networks of which the unerlying network graph is bipartite.

3 of vertices an E is the set of eges. A vertex correspons to a wireless noe that can transmit an receive ata packets; an ege correspons to a wireless link between two noes 3. We assume a time-slotte system with one single frequency channel. A time-slot is enote by n {0, 1, 2,... }. For ease of presentation, we assume unit link capacities, i.e., each link can transmit at most one packet per time-slot. However, it is easy to exten the analysis an results to more general moels with heterogeneous link capacities. We assume the noe-exclusive interference moel [10,18,22], uner which links sharing a common noe cannot be activate at the same time. Hence, a feasible scheule correspons to a matching, enote by M, in the unerlying network graph. In this paper, we focus on the link scheuling problem an assume that all the flows are single-hop (i.e., their packets traverse only one link before leaving the network). Let Âl(n) enote the cumulative number of packet arrivals at link l E up to (an incluing) time-slot n. We assume that the arrival processes {Âl(n), n = 0, 1, 2,... } satisfy the strong law of large numbers. That is, with probability one, the  following is satisfie for all l E: lim l (n) n = ˆλ n l, where ˆλ l is the mean arrival rate of link l. Let ˆλ [ˆλ l : l E] be the arrival rate vector. Let ˆD l (n) enote the cumulative number of packet epartures at link l up to time-slot n, an let ˆQ l (n) enote the number of packets waiting to be transmitte over link l (or the queue length of link l) at the beginning of time-slot n. We assume that there are a finite number of packets in the system at the beginning of timeslot 0. Also, we assume that packets arrive (respectively, epart) at the beginning (respectively, en) of each timeslot. By convention, we set Âl(0) = ˆD l ( 1) = 0. Then, the queueing ynamics of link l is given by ˆQ l (n) = ˆQ l (0) + Âl(n) ˆD l (n 1). (1) Since the scheuling algorithm of interest (i.e., MVM) takes a noe-base approach, we introuce some aitional notations associate with the noes. First, let L(i) enote the set of links that are incient to noe i V, i.e., L(i) = {l E noe i is an en noe of link l}. Then, let Q i(n) enote the workloa (i.e., number of packets to transmit or receive) at noe i V at the beginning of timeslot n, i.e., Q i(n) ˆQ l L(i) l (n). Similarly, let A i(n) an D i(n) enote the cumulative workloa arrivals an epartures at noe i up to time-slot n, respectively, i.e., A i(n) = l L(i) Âl(n) an D i(n) = ˆD l L(i) l (n). Hence, the queueing ynamics of noe i is given by Q i(n) = Q i(0) + A i(n) D i(n 1). (2) Let λ [λ i : i V ] enote the noe arrival rate vector, where λ i = ˆλ l L(i) l is the mean arrival rate of noe i. Let M enote the set of matchings in G. By slightly abusing the notations, for matching M M, let M l = 1 if link l is inclue in M, an M l = 0 otherwise. In each timeslot, a scheuling algorithm will select a matching in M as a feasible scheule. Let H M (n) be the number of time-slots in which matching M is selecte as a scheule up to time-slot 3 Throughout the paper, we interchangeably use the terms vertex an noe ; similar for ege an link. However, we ten to use the terms vertex an ege in the context of a graph, an use the terms noe an link in the context of a multihop wireless network. n. Hence, we have the following equations: D i(n) = n M l (H M (τ) H M (τ 1)), (3) M M M M τ=1 l L(i) H M (n) = n. (4) Next, we give the efinition of network stability. Definition 1. The network is rate stable uner arrival rate vector ˆλ = {ˆλ l : l E} if with probability one, the following is satisfie for all l E: lim n ˆD l (n) n = ˆλ l. (5) D Note that (5) is equivalent to lim i (n) n = λ n i for all i V. Rate stability is a weak version of stability an only implies that the eparture rate is equal to the arrival rate [3]. We focus on rate stability for ease of presenting our main ieas only. Our analysis follows similarly for stronger versions of stability (e.g., positive recurrence type of stability) if stronger assumptions on the arrival processes are mae [2]. We present more efinitions as follows. Definition 2. The throughput region (or stability region) of a scheuling algorithm, enote by Λ, is efine as the set of arrival rate vectors λ for which the network is rate stable uner this scheuling algorithm. Definition 3. The optimal throughput region, enote by Λ, is efine as the union of the throughput regions of all possible scheuling algorithms. Definition 4. The efficiency ratio of a scheuling algorithm, enote by γ, is efine as the largest fraction of the optimal throughput region Λ that is containe in the throughput region Λ of this scheuling algorithm, i.e., γ sup{γ γλ Λ}. (6) Throughout the paper, we will use the efficiency ratio to measure the throughput performance of MVM. Clearly, we have γ [0, 1] for any scheuling algorithm. Particularly, a scheuling algorithm is sai to be throughput-optimal or achieve throughput optimality if its efficiency ratio is equal to 1. To assist our analysis, we efine the following region: Ψ {λ λ i 1 for all i V }. (7) Note that Ψ is an outer boun of the optimal throughput region (i.e., Λ Ψ), since uner any scheuling algorithm, at most one packet can be transmitte to or from a noe. 4. GALLAI-EDMONDS STRUCTURE THE- OREM AND GALLAI-EDMONDS DECOM- POSITION FACTOR In this section, we will introuce a new topological notion of a graph, calle the Gallai-Emons ecomposition factor, which is inspire by the well-known Gallai-Emons structure theorem an the associate ecomposition structure. This notion is novel in the sense that it can be use to characterize the throughput performance of MVM. Specifically, we will later show that the efficiency ratio of MVM is no smaller than the Gallai-Emons ecomposition factor

4 of the unerlying network graph (Section 6). Moreover, the Gallai-Emons ecomposition factor is fully etermine by the smallest size of an o cycle in a graph. Specifically, we show that if the smallest size of an o cycle containe in a graph is 2m+1 for m {1, 2,... }, then the Gallai-Emons ecomposition factor of this graph is equal to 2m/(2m + 1). This further implies that the Gallai-Emons ecomposition factor is no smaller than 2/3 for an arbitrary graph an is equal to 1 for bipartite graphs. 4.1 Gallai-Emons Structure Theorem We begin with a review of two important results in graph theory: the Tutte-Berge formula an the Gallai-Emons structure theorem. We first give the following basic efinitions. Recall that a matching is a subset of eges without common vertices. A matching M is sai to miss a vertex v if v is not matche by M. A maximal matching is a matching M with the following property: if an ege l not in M is ae to M, then M {l} is no longer a matching. A maximum matching (also known as maximum-carinality matching) is a matching that contains the largest number of eges. A perfect matching (also known as 1-factor) is a matching that matches all the vertices of a graph. A near-perfect matching is a matching that misses exactly one vertex. A graph is calle factor-critical if for every vertex of this graph, there exists a near-perfect matching that misses only that vertex. Any factor-critical graph must be connecte, non-bipartite, an of o size [27]. Next, we give some aitional notations. Let enote the carinality of a set. For a graph G = (V, E), we let G U enote the subgraph inuce by a vertex-subset U V, let θ(g) enote the size of a maximum matching in graph G, an let o(g U ) enote the number of o components (i.e., connecte components with an o number of vertices) in G U. Note that an o component coul consist of exactly one vertex, which is then calle an isolate vertex. We now state the Tutte-Berge formula [27] in Theorem 1. Theorem 1 (Tutte-Berge Formula). The size of a maximum matching in a graph G = (V, E) satisfies: 1 θ(g) = min U V 2 ( V + U o(g V \U )). (8) The Tutte-Berge formula is a generalization of Tutte s theorem 4. While the Tutte-Berge formula provies a characterization of the size of a maximum matching in an arbitrary graph, it oes not reveal any structural information of the graph from the matching point of view. The Gallai-Emons structure theorem briges this gap an uncovers interesting an important properties of maximum matchings in an arbitrary graph. In orer to state this theorem, we introuce a canonical ecomposition, which lies in the heart of the Gallai-Emons structure theorem. Uner this canonical ecomposition, the vertex set V can be partitione into three isjoint vertex-subsets: R {v V there exists a maximum matching missing v}, S {v V v is not in R, but has a neighbor in R}, T V \(R S). 4 Tutte s (1-factor) theorem gives a necessary an sufficient conition for the existence of a perfect matching in an arbitrary graph. It is a generalization of Hall s marriage theorem from bipartite to arbitrary graphs. (9) T S R (a) Decomposition Structure T S R (b) A Maximum Matching Figure 1: An illustration of the Gallai-Emons ecomposition [1]. (a) presents the ecomposition structure, an (b) shows a maximum matching, enote by thick blue eges. The above vertex-partition (R, S, T ) is also known as the Gallai-Emons ecomposition. The ecomposition is well efine even if the graph is not connecte. An illustration of the Gallai-Emons ecomposition is provie in Fig. 1a. Having efine the above ecomposition, we are now reay to restate the Gallai-Emons structure theorem [27]. Theorem 2 (Gallai-Emons Structure Theorem). Given a graph G = (V, E), let (R, S, T ) be the ecomposition efine in (9). Then, there exist the following properties: i) all the components in G R are factor-critical an thus are of o size; ii) all the components in G T have a perfect matching an thus are of even size; iii) every maximum matching in G contains a perfect matching in each component of G T an a near-perfect matching in each component of G R, an matches all the vertices in S with vertices in istinct components of G R (see Fig. 1b for an example of a maximum matching); iv) choosing U = S achieves the minimum on the right sie of the Tutte-Berge formula (Eq. (8)), i.e., the size of a maximum matching is θ(g) = 1 ( V + S 2 o(g V \S )) = 1 ( V + S o(gr)). 2 It is remarkable that the Gallai-Emons structure theorem is the founation of the first polynomial-time algorithm (i.e., the famous blossom algorithm evelope by Emons [6]) for computing a maximum matching in an arbitrary graph. This theorem reveals an interesting graph structure that leas to many important consequences in graph theory. Nevertheless, as pointe out in [27], this powerful result oes not seem to have reache its full potential yet. To the best of our knowlege, this paper, for the first time, applies this theorem in characterizing the throughput performance of a link scheuling algorithm in multihop wireless networks. We hope that the results of this paper will serve as an invitation of more efforts towars fully utilizing the power of this beautiful theorem in networking research. 4.2 Gallai-Emons Decomposition Factor We are now reay to introuce the Gallai-Emons Decomposition factor. We first give some aitional notations. For any vertex-subset U V, let (R(U), S(U), T (U)) enote the Gallai-Emons ecomposition of its inuce subgraph G U. Recall that G R(U) enotes the subgraph inuce by

5 R(U) an that all the components of G R(U) are o. Let µ(g R(U) ) enote the number of components of size larger than one (i.e., components that are not isolate vertices) in G R(U). Then, the Gallai-Emons ecomposition factor of a graph is efine below. Definition 5. For a given graph G = (V, E), consier any vertex-subset U V an the Gallai-Emons ecomposition (R(U), S(U), T (U)) of the inuce subgraph G U. The Gallai-Emons ecomposition factor of graph G, enote by σ, is the supremum of all σ 0 such that for any U V, the number of o components of size larger than one in G R(U) is no greater than (1 σ) U, i.e., σ sup{σ 0 For any U V, the following is satisfie: µ(g R(U) ) (1 σ) U )}. (10) Remark: We highlight that although the Gallai-Emons structure theorem is a well-known result in graph theory, to the best of our knowlege, the Gallai-Emons ecomposition factor is a new topological notion evelope in this paper for the first time, which epens on the Gallai-Emons ecomposition of the subgraphs inuce by the vertex-subsets. Interestingly, it turns out that the Gallai-Emons ecomposition factor is closely relate to the throughput performance of MVM. Specifically, we will show that the efficiency ratio of MVM is no smaller than the Gallai-Emons ecomposition factor of the network graph (Theorem 4). In the sequel, we show that if the smallest size of an o cycle in a graph is 2m + 1 for m {1, 2,... }, then the Gallai-Emons ecomposition factor of this graph is equal to 2m/(2m + 1) (Theorem 3). As a consequence, the Gallai-Emons ecomposition factor is no smaller than 2/3 for an arbitrary graph an is equal to 1 for bipartite graphs (Corollary 1). We present examples of graphs with ifferent Gallai-Emons ecomposition factors in Fig. 2. Theorem 3. Consier an arbitrary graph G = (V, E). Suppose the smallest size of an o cycle in G is 2m + 1 for m {1, 2,... }. Then, the Gallai-Emons ecomposition factor of G is equal to 2m/(2m+1), i.e., σ = 2m/(2m+1). Proof. The proof is straightforwar. Suppose the smallest size of an o cycle in graph G is 2m + 1. We first show that the Gallai-Emons ecomposition factor is no smaller than 2m/(2m + 1). Consier any vertex-subset U V an the Gallai-Emons ecomposition (R(U), S(U), T (U)) of G U. Recall that µ(g R(U) ) enotes the number of o components of size larger than one in G R(U). From Property i) of the Gallai-Emons structure theorem, we know that all the components in G R(U) are factor-critical an thus must be a non-bipartite graph of o size [27]. Since any o cycle containe in graph G has a size no smaller than 2m + 1, then every o component in G R(U) must contain at least 2m + 1 vertices. Hence, we must have µ(g R(U) ) U /(2m + 1), which implies that µ(g R(U) ) (1 σ) U for any σ 2m/(2m + 1). Therefore, it is implie from Definition 5 that the Gallai-Emons ecomposition factor must be no smaller than 2m/(2m + 1), i.e., σ 2m/(2m + 1). To show that the Gallai-Emons ecomposition factor is no greater than 2m/(2m + 1), we only nee to consier U as the vertex-subset that consists of all the vertices in an o cycle of size 2m + 1. In this case, we have R(U) = U an thus µ(g R(U) ) = U /(2m + 1). Then, we must have (a) σ = 2/3 (b) σ = 4/5 (c) σ = 1 Figure 2: Examples of graphs with ifferent Gallai-Emons ecomposition factor σ. This factor σ is fully etermine by the smallest size of an o cycle in the graph. σ 2m/(2m+1), because for any σ greater than 2m/(2m+ 1), the conition in (10) will not be satisfie for U uner consieration. This completes the proof. Corollary 1. The Gallai-Emons ecomposition factor is no smaller than 2/3 for an arbitrary graph an is equal to 1 for bipartite graphs. Proof. The proof follows immeiately from Theorem 3, since any o cycle containe in an arbitrary graph has a size no smaller than 3 (i.e., m = 1), an a bipartite graph oes not contain any o cycles (i.e., m = ). 5. THE MVM ALGORITHM In this section, we first escribe the operations of the Maximum Vertex-weighte Matching (MVM) algorithm (Subsection 5.1), an then prove several interesting properties of MVM (Subsection 5.2), which will play a key role in analyzing the throughput performance of the MVM algorithm. 5.1 Algorithm Description For a graph G = (V, E), we assume that each vertex i V is assigne a positive weight, enote by w i > 0. We will later escribe how to assign the vertex weights. Let V (M) enote the set of vertices matche by matching M. Then, the weight of a matching M is efine as the sum of the weights of all the vertices matche by M, i.e., w(m) = i V (M) wi. A matching M is calle a Maximum Vertex-weighte Matching (MVM) if M achieves the largest weight over all the matchings in G, i.e., w(m ) w(m) for any matching M in G. It has been shown in [23] that an MVM can be foun in O( E V 1/2 log V )-time, lower than O( E V )-complexity of computing its ege-weighte counterpart Maximum Weighte Matching (MWM). Now, consier the multihop wireless network we escribe in Section 3. Recall that Q i(n) is the queue length of noe i at the beginning of time-slot n. In each time-slot n, we assign the weight of a noe i as its queue length, i.e., w i = Q i(n) for all i V. Base on the assigne noe weights, we compute an MVM. The links of this matching will be activate to transmit packets. We assume that if a link has a zero queue length, then the corresponing ege will not be consiere when computing an MVM in that time-slot. 5.2 Properties of MVM The MVM algorithm an its variants have been stuie in several prior work in the literature (e.g., [8, 9, 11, 12, 19, 23]), which shows that MVM exhibits interesting an useful properties. In this section, we present an prove some new properties of MVM, which will be the key to characterizing the efficiency ratio of the MVM algorithm.

6 Consier a graph G = (V, E) with vertex weights w i s. Let H k enote the set of the k strictly heaviest vertices for k {1, 2,..., V }, i.e., w i > w j for any i H k an any j V \H k. We show in Proposition 1 that an MVM maximizes the number of matche vertices in set H k. Proposition 1. For any k {1, 2,..., V } such that set H k exists, an MVM, enote by M, maximizes the number of matche vertices in H k over all the matchings, i.e., V (M ) H k V (M) H k for any matching M in G. Proof. We first provie some efinitions an notations that will be use in the proof. Define a path in a graph as a sequence of connecte eges. The length of a path is the number of eges on the path. For any two ege-subsets E 1 E an E 2 E, let E 1 E 2 (E 1\E 2) (E 2\E 1) enote their symmetric ifference, i.e., the set of eges that are in one of the ege-subsets, but not in both of them. As in [8, 9], we efine augmenting path an absorbing path. Consier a matching M an a vertex i that is misse by M. A path P is calle an M-augmenting path if it has the following properties: i) it has an o length; ii) its every alternate ege is in M; iii) it starts from vertex i an ens at another vertex j misse by M. Similarly, a path P is calle an M-absorbing path if it has the following properties: i) it has an even length; ii) its every alternate ege is in M; iii) it starts from vertex i an ens at a vertex r, which is matche by M an has a smaller weight than vertex i, i.e., w r < w i. It is easy to verify that both M P an M P are also a matching an have a weight strictly larger than that of the original matching M because w(m P ) = w(m)+w i+w j > w(m) an w(m P ) = w(m) + w i w r > w(m). We procee the proof by contraiction. Suppose matching M is an MVM an there exists a matching M such that V (M) H k > V (M ) H k for some k {1, 2,..., V }. This implies that there exists a vertex in H k, say i, which is matche only by M but not by M. A little thought gives that in the symmetric ifference M M, vertex i must be one en of a path P whose eges alternate between M an M. Note that path P cannot have an o length ue to the following reason. Suppose path P has an o length, then the other en of P must be another vertex that is misse by M, which makes P an M -augmenting path. Then, matching M P will have a weight larger than that of M, which contraicts the fact that M is an MVM. Therefore, path P must have an even length an must en at another vertex, say j, that is only matche by M but not by M. Note that vertex j must have a weight no smaller than that of vertex i, i.e., w j w i, otherwise P woul be an M - absorbing path, an matching M P will have a weight larger than that of M, which again leas to a contraiction. Therefore, vertex j is also in H k. Base on the argument above, we know that path P must have an even length; both of its ens vertices i an j are in H k ; vertex i is only matche by M, an vertex j is only matche by M. Now, we construct a new matching M 1 = M P. By comparing M 1 with M, we make the following two observations: i) V (M 1) H k = V (M) H k ; ii) V (M 1)\V (M ) H k = V (M)\V (M ) H k 1. The first observation means that M 1 an M match the same number of vertices in H k. This is true because M 1 an M match exactly the same set of vertices except for i an j, an both vertices i an j are in H k. The secon observation means that compare to M, matching M 1 has one more common vertex matche by M in H k. This is true ue to the following reason. First, matching M 1 an M have the same number of common vertices matche by M in H k \{i, j}, because M 1 an M match exactly the same set of vertices except for i an j. Secon, vertex j is matche by both M 1 an M, but is misse by M, an vertex i is matche by M, but is misse by both M 1 an M. Hence, matching M 1 has one more common vertex in H k (i.e., vertex j) matche by M than matching M. Thanks to these observations, we can repeat the argument above by constructing new matchings M 2, M 3,... in a similar way as M 1 is constructe, until we have a matching M such that V (M )\V (M ) H k = 0. This must happen within H k rouns ue to the secon observation. This implies that all the vertices in H k matche by M must also be matche by M, an thus V (M ) H k V (M ) H k. Also, we have V (M ) H k = V (M) H k from the first observation. Hence, we have V (M) H k V (M ) H k. This leas to a contraiction an completes the proof. Remark: Proposition 1 generalizes Lemma 6 of [23], which states that if there exists a matching that matches all the k heaviest vertices, then an MVM also matches all of them. Next, in Proposition 2 we present a key property of MVM in multigraphs, where more than one ege, calle multi-ege, is allowe between two vertices. Although the iscussions in Section 4 are focuse on simple graphs, the results there also apply to multigraphs. In particular, the Gallai-Emons ecomposition factor of a multigraph is equal to that of the corresponing simple graph. Similarly, Proposition 1 also applies to multigraphs. We use G = (V, E) to enote a multigraph, with V an E being the set of vertices an the set of multi-eges, respectively. Let G(i) enote the egree of vertex i in G. For a vertex-subset U V, let G U enote the subgraph of G inuce by U, an let (R(U), S(U), T (U)) enote the Gallai-Emons ecomposition of G U. We use G R(U) to enote the subgraph inuce by R(U), use I(G R(U) ) to enote the set of isolate vertices in G R(U), an use µ(g R(U) ) to enote the number of o components of size larger than one in G R(U). Note that in any time-slot, the network together with the present packets can be represente by a multigraph, where each multi-ege correspons to a packet. Then, uner the MVM algorithm we specifie, the weight of a vertex is equal to the vertex egree, i.e., w i = G(i) for i V. Let ν(g) = V 1. We now state Proposition 2 below. V Proposition 2. Consier a multigraph G = (V, E) with maximum vertex egree an a vertex-subset U V. Suppose the following conitions are satisfie: i) every isolate vertex in G R(U) has a egree no smaller than a ν(g)-fraction of the maximum vertex egree, i.e., G(i) ν(g) for any i I(G R(U) ); ii) every vertex in U has a larger egree than a vertex not in U, i.e., G(i) > G(j) for any i U an any j / U. Then, an MVM matches at least a σ -fraction of the vertices in U, where σ is the Gallai-Emons ecomposition factor. The proof of Proposition 2 follows immeiately from a property of MVM (Proposition 1) an an important result relate to the Gallai-Emons ecomposition (Proposition 3). We present an prove Proposition 3 below.

7 Proposition 3. Consier a multigraph G = (V, E) with maximum vertex egree an a vertex-subset U V. Suppose every isolate vertex in G R(U) has a egree no smaller than a ν(g)-fraction of the maximum vertex egree, i.e., G(i) ν(g) for any i I(G R(U) ). Then, there exists a matching in G that matches every vertex in U, except for at most one vertex from each o component of size larger than one in G R(U). We restate two lemmas (Lemma 6 of [11] an Lemma of [8]) that will be use in the proof of Proposition 3. Lemma 1. Consier a multigraph G = (V, E) with maximum vertex egree an a vertex-subset W V. Suppose the following conitions are satisfie: i) every vertex in W has a egree no smaller than a ν(g)- fraction of the maximum vertex egree, i.e., G(i) ν(g) for any i W ; ii) G W is bipartite. Then, there exists a matching M in G such that every vertex of U is matche by M. Lemma 2. Consier a bipartite graph G = (V 1 V 2, E), where (V 1, V 2) is the vertex partition, an E is the set of eges. Let V 1 V 1 an V 2 V 2 be a vertex-subset of V 1 an V 2, respectively. Suppose there exist two matchings M 1 an M 2 such that M 1 matches all the vertices in V 1, an M 2 matches all the vertices in V 2. Then, there exists a matching M that matches all the vertices in V 1 V 2. We provie the proof of Proposition 3 below an present an illustration of the key components of the proof in Fig. 3. Proof of Proposition 3. Consier a vertex-subset U V. Note that G U enotes the subgraph inuce by U. Recall that (R(U), S(U), T (U)) enotes the Gallai-Emons ecomposition of G U, an I(G R(U) ) enotes the set of isolate vertices in G R(U). Suppose the conition of Proposition 3 is satisfie: every vertex in I(G R(U) ) has a egree no smaller than a ν(g)-fraction of the maximum vertex egree, i.e., G(i) ν(g) for any i I(G R(U) ). We want to show that there exists a matching in G that matches every vertex in U, except for at most one vertex from each o component of size larger than one in G R(U). First, we want to show that there exists a matching in G that matches all the vertices in I(G R(U) ) S(U). To prove this, there are three steps: 1) we use Lemma 1 to show that there exists a matching M 1 in G that matches all the vertices in I(G R(U) ); 2) we use Property iii) of Theorem 2 to show that there exists another matching M 2 in G that matches all the vertices in S(U); 3) we construct a bipartite graph from matchings M 1 an M 2, an then use Lemma 2 to show that there exists a matching M 3 in G that matches all the vertices in I(G R(U) ) S(U). Step 1). First, note that the first conition of Lemma 1 is satisfie for I(G R(U) ), i.e., G(i) ν(g) for any i I(G R(U) ). The secon conition of Lemma 1 is also satisfie for I(G R(U) ) ue to the fact that all the vertices in I(G R(U) ) are the isolate vertices in G R(U) an thus the subgraph inuce by I(G R(U) ) is bipartite. Hence, it is implie from Lemma 1 that there exists a matching M 1 in G that matches all the vertices in I(G R(U) ). a c e g h b v f T (U) S(U) R(U) Figure 3: An illustration of the key components in the proof of Proposition 3. Suppose the inuce subgraph G U is as in Fig. 1a. Then, the set of isolate vertices in G R(U) is I(G R(U) ) = {b, }. Matchings M 1, M 2, an M 4 are as shown in the above figure. Vertex a is not in U. The constructe bipartite graph G = (V 1 V 2, E) has V 1 = {b,, f, h}, V 2 = {a, c, e, g}, an E = M 1 M 2. One option of matching M 3 that matches all the vertices in I(G R(U) ) S(U) = {b, c,, e, g} is M 3 = {(a, b), (c, ), (e, f), (g, h)}, where (a, b) enotes the ege between vertices a an b. Then, matching M = M 3 M 4 matches every vertex in U, except for at most one vertex from each o component of size larger than one in G R(U). In this case, only vertex v is misse by M. (An alternative matching M 3 coul be M 3 = {(c, b), (e, ), (g, h)}. Then, two vertices f an v are misse by the resultant M.) Step 2). From Property iii) of Theorem 2, we know that every maximum matching in G U contains a perfect matching in each component of G T (U) an a near-perfect matching in each component of G R(U), an matches all the vertices in S(U) with vertices in istinct components of G R(U). Let M be a maximum matching in G U. Then, the above property is satisfie for M. Let M 2 M be the set of eges of M that are incient to vertices in S(U). Clearly, set M 2 is also a matching in G an matches all the vertices in S(U). Step 3). Let V 1 be the union of I(G R(U) ) an the vertices in R(U) that are matche by M 2, i.e., V 1 = I(G R(U) ) (V (M 2) R(U)). Let V 2 be the union of S(U) an the vertices not in U that are matche by M 1, i.e., V 2 = S(U) (V (M 1) (V\U)). Let E be the union of matchings M 1 an M 2, i.e., E = M 1 M 2. We construct a graph G = (V 1 V 2, E), which is a bipartite graph with (V 1, V 2) being the vertex partition. Then, the conitions in Lemma 2 are satisfie. Specifically, let V 1 = I(G R(U) ) V 1 an V 2 = S(U) V 2. Clearly, M 1 an M 2 are also matchings in G. Moreover, M 1 matches all the vertices in V 1, an M 2 matches all the vertices in V 2. Therefore, it is implie from Lemma 2 that there exists a matching M 3 in G that matches all the vertices in V 1 V 2 = I(G R(U) ) S(U). Next, we want to show that there exists a matching M in G that matches all the vertices in U, except for at most one vertex from each o component of size larger than one in G R(U). This is a straightforwar consequence of the above iscussion. To see this, let M 4 = M \M 2 be the set of eges of M that are not incient to vertices in S(U). It is easy to see that M 4 is also a matching in G U (an in G). From Properties iii) of of Theorem 2, we know that M 4 contains a perfect matching in each component of G T (U) an a near-perfect matching in each component of G R(U). In other wors, matching M 4 matches all the vertices in the M 1 M 2 M 4

8 union of T (U) an R(U), except for at most one vertex from each of the o components (incluing the set of isolate vertices I(G R(U) )) in G R(U). Let M = M 3 M 4. It is easy to see that M is also a matching in G, an that M matches all the vertices in U, except for at most one vertex from each o component of size larger than one in G R(U). This completes the proof. Remark: Property iv) of the Gallai-Emons structure theorem tells us that a maximum matching in G U misses o(g R(U) ) S(U) vertices, each of which is from istinct o components in G R(U) an can potentially be an isolate vertex. However, Proposition 3 presents a stronger result uner the conition that every isolate vertex in G R(U) has a egree close to the maximum vertex egree. We now use Propositions 1 an 3 to prove Proposition 2. Proof of Proposition 2. Consier any U V such that the conitions in Proposition 2 are satisfie. From Proposition 3 an the first conition of Proposition 2, we know that there exists a matching M in G that matches all the vertices in U, except for at most one vertex from each o component of size larger than one in G R(U). From the secon conition of Proposition 2, we know that U is the set of the U heaviest vertices, i.e., U = H U. Then, it follows from Proposition 1 that an MVM matches at least the same number of vertices in U as M oes, i.e., an MVM matches at least U µ(g R(U) ) vertices in U. From the efinition of Gallai-Emons ecomposition factor (Definition 5), we have U µ(g R(U) ) σ U. Remark: Proposition 1 only shows that MVM maximizes the number of matche vertices among the k heaviest vertices (i.e., H k ), whereas Proposition 2 allows us to quantify the fraction of the matche vertices in such set if certain conitions are satisfie. This nice property of MVM is the key to proving our main result in the next section. 6. THROUGHPUT PERFORMANCE In this section, we focus on analyzing the throughput performance of the MVM algorithm. A main result of this paper is presente in Theorem 4. Theorem 4. The efficiency ratio of the MVM algorithm is no smaller than the Gallai-Emons ecomposition factor of the unerlying network graph, i.e., γ σ. Our analysis will employ the flui limit techniques [2, 4], which not only help simply the analysis through eliminating the unnecessary ranomness in the original stochastic network, but also reveal aitional properties of the system ynamics that will be useful in the analysis. We begin by constructing the flui moel as in [2,4]. Recall that A i(n) an D i(n) enote the cumulative workloa arrivals an epartures of noe i up to time-slot n, respectively, Q i(n) enotes the queue length of noe i at the beginning of time-slot n, an H M (n) enotes the cumulative number of time-slots in which matching M M is chosen as a scheule up to time-slot n. For processes Y = Q, D, H, we exten their omain from iscrete time to continuous time by setting Y (t) = Y ( t ). Then, we construct the Markov process X = {X(t) : t 0} as X(t) = (Q(t), D(t), H(t)), which represents the system ynamics. Following the same argument as in the proof of Theorem 4.1 of [4], we can show that for almost all sample paths an for all positive sequences x r, there exists a subsequence x rj with x rj as j such that the following convergence hols uniformly over compact intervals of time t: A i(x rj t) x rj λ it for all i V, (11) Q i(x rj t) x rj q i(t) for all i V, (12) D i(x rj t) x rj i(t) for all i V, (13) H M (x rj t) x rj h M (t) for all M M. (14) Then, the flui moel equations of the system are: q i(t) = q i(0) + λ it i(t) for all i V, (15) i(t) = M t l hm (t) for all i V, (16) t M M M M l L(i) h M (t) = t. (17) Any limit (q,, h) satisfying the above flui moel equations is calle a flui limit. Note that functions q i(t), i(t), an h i(t) are absolutely continuous an are ifferentiable at almost all times t 0, which are calle regular times. Taking the erivative of both sies of (15) an substituting (16) into it, we obtain that the following is satisfie for all i V : qi(t) = λi t M M l L(i) M l hm (t). (18) t Next, we borrow some efinitions an results from [5], which establish an important relationship between the original stochastic network an the flui moel. Definition 6. The flui moel is weakly stable if for every flui moel solution (q,, h) with q(0) = 0, the following is satisfie: q(t) = 0 for all regular times t 0. Lemma 3. A network is rate stable if the associate flui moel is weakly stable. Now, we present the proof of Theorem 4. Proof of Theorem 4. We want to show that uner any arrival rate vector λ strictly insie σ Λ, the MVM algorithm stabilizes the network. Note that λ is also strictly insie σ Ψ (i.e., λ i < σ for all i V ) ue to Λ Ψ. We efine ɛ min i V (σ λ i). Then, we must have ɛ > 0. Due to Lemma 3, it suffices to show that the flui moel is weakly stable. We will use the stanar Lyapunov analysis by choosing the following Lyapunov function: V (q(t)) = max qi(t). (19) i V Note that V (q(t)) is a non-negative function with V (0) = 0. Since we assume that there are a finite number of initial packets in the network, we have q(0) = 0 in the flui limits. In orer to show q(t) = 0 for all regular times t 0, it suffices to show the following: if V (q(t)) > 0 for t > 0, then V (q(t)) has a negative rift an ecreases to 0 at least at a given rate. Then, the flui moel is weakly stable accoring to Definition 6. Therefore, we want to show that for all

9 regular times t > 0, the Lyapunov function V (q(t)) has a negative rift, i.e., V (q(t)) ɛ, whenever V (q(t)) > 0. t We first efine some useful notions in the flui limits. For any fixe time t, a noe i V is calle a critical noe in the flui limits if it has the largest queue length in the flui limits, i.e., q i(t) = q max max j V q j(t). We use C to enote the set of critical noes in the flui limits at time t, i.e., C {i V q i(t) = q max}. (20) Further, let L enote the set of critical noes that have the largest queue-length erivative at time t, i.e., L {i C qi(t) = max qj(t)}. (21) t j C t Next, we will use Proposition 2 to show that in every timeslot corresponing to a small time interval aroun scale time t, at least a σ -fraction of the noes in L will be scheule by the MVM algorithm. We show that the conitions of Proposition 2 are satisfie for L. Following a similar argument to that in [14] for GMM, we show that Conition ii) of Proposition 2 is satisfie. Since the noes in L have the largest queue-length erivative among the noes in C at time t, an q i(t) s are absolutely continuous, there exists a small δ 1 > 0 such that the noes in L will have a queue length strictly larger than any other noes uring interval (t, t + δ 1), i.e., the following is satisfie for any time τ (t, t + δ 1): min qi(τ) > max q j(τ). i L j V \L This also implies that all the noes in L will have a queue length strictly larger than any other noes in the original stochastic network within all the time-slots corresponing to the scale time interval (t, t + δ 1) in the flui limits. Following a similar argument to that in [8,9,11], we show that Conition i) of Proposition 2 is satisfie. Let ˆq max be the largest queue length of the noes not in C in the flui limits at time t, i.e., ˆq max = max i / C q i(t). Then, we have ˆq max < q max. Choosing β small enough such that ˆq max < q max 3β an β < 1 qmax, we have 2 V 1 q max β > V 1 (q max + β). (22) V Recall that q i(t) s are absolutely continuous. Hence, there exists a small δ (0, δ 1] such that for all i C an for all times τ (t, t + δ), the queue lengths in the flui limits satisfy q i(τ) (q max β/2, q max + β/2). Let x rj be a positive subsequence for which the convergence to the flui limits hols. For large enough j, we have Qi(x rj τ)/x rj q i(τ) < β/2 for all τ (t, t+δ). Define a set of consecutive time-slots in the original stochastic system as N { x rj t, x rj t + 1,..., x rj (t + δ) }, which correspons to the scale time interval (t, t + δ) in the flui limits. Hence, for all i C an for all time-slots n N, the queue lengths in the original stochastic network satisfy Q i(n) (x rj (q max β), x rj (q max+ β)). Then, it is implie from (22) that all the noes in C V 1 have a queue length no smaller than a -fraction of the V largest queue length in all time-slots n N. Conition i) of Proposition 2 is also satisfie for L ue to L C. Then, it is implie from Proposition 2 that in every timeslot of N in the original stochastic system, at least a σ - fraction of the noes in L will be scheule by MVM. Accoring to the pigeonhole principle, there must exist a noe i L such that i has been scheule by MVM for at least a σ -fraction of the time-slots in N, i.e., M l (H M (x rj (t + δ)) H M (x rj t)) M M l L(i ) (23) σ ( x rj (t + δ) x rj t + 1). Therefore, the following is satisfie: M l hm (t) t M M l L(i ) = lim (a) δ 0 M M l L(i ) = lim (b) lim δ 0 j M M l L(i ) lim δ 0 lim j =σ, M l h M (t + δ) h M (t) δ M l (H M (x rj (t + δ)) H M (x rj t)) x rj δ σ ( x rj (t + δ) x rj t + 1) x rj δ (24) where (a) an (b) are from (14) an (23), respectively. Then, we have qi (t) t λi σ ɛ from (18). Since noe i has the largest queue-length erivative among noes in C, we have qi(t) qi (t) ɛ for all i C. Therefore, the flui moel is weakly stable accoring to Defini- t t tion 6. This completes the proof by applying Lemma 3. Theorem 5. Suppose the smallest size of an o cycle in the unerlying network graph is 2m + 1, where m is a positive integer. Then, the efficiency ratio of MVM is no smaller than 2m/(2m + 1), i.e., γ 2m/(2m + 1). Moreover, the efficiency ratio of MVM is no smaller than 2/3 in general, i.e., γ 2/3, an MVM is throughput-optimal if the unerlying network graph is bipartite. Proof. The proof follows immeiately from Theorems 3 an 4 an Corollary 1. Remark: Together with the result on the evacuation time performance of MVM [12], Theorem 5 shows that MVM achieves the most balance performance guarantees in both imensions of throughput an evacuation time among existing online scheuling algorithms. Also, our new approach offers an alternative means to that of [8,9] for proving throughput optimality of MVM in bipartite graphs. 7. CONCLUSION In this paper, we carrie out a systematic stuy of the throughput performance of MVM through a novel application of the Gallai-Emons structure theorem. We showe that the efficiency ratio of MVM can be well characterize by the Gallai-Emons ecomposition factor introuce in this paper. Not only o the results of this paper substantially improve our unerstaning of the throughput performance of MVM, but also provie useful insights for guiing the esign of wireless networks when MVM-type of algorithms are employe. We also hope that our finings will she light on the full utilization of the beautiful Gallai-Emons structure theorem in networking research. Despite the importance of the results we obtaine in this paper, in some cases the erive lower bouns of the efficiency ratio coul be loose ue to the following reasons.