Ashwin Ganesan This work was carried out while the author was with the ECE Department, University of Wisconsin at Madison, WI 53706, USA. This work was presented in part at the International Conference on Networks and Communications (GraphHoc and NetCoM), Chennai, India, December 2009 [5], and at the International Conference on Recent Trends in Graph Theory and Combinatorics (an ICM Satellite Conference), Cochin, India, August 2010 [6]. This paper is a revised version of the Technical Report [8]. The author is with the Department of Mathematics, Amrita School of Engineering, Amrita University, Coimbatore, Tamil Nadu, India. Correspondence address: ashwin.ganesan@gmail.com.
Abstract

Given a wireless network where some pairs of communication links interfere with each other, we study sufficient conditions for determining whether a given set of minimum bandwidth quality-of-service (QoS) requirements can be satisfied. We are especially interested in algorithms which have low communication overhead and low processing complexity. The interference in the network is modeled using a conflict graph whose vertices correspond to the communication links in the network. Two links are adjacent in this graph if and only if they interfere with each other due to being in the same vicinity and hence cannot be simultaneously active. The problem of scheduling the transmission of the various links is then essentially a fractional, weighted vertex coloring problem, for which upper bounds on the fractional chromatic number are sought using only localized information. We recall some distributed algorithms for this problem, and then assess their worst-case performance. Our results on this fundamental problem imply that for some well known classes of networks and interference models, the performance of these distributed algorithms is within a bounded factor away from that of an optimal, centralized algorithm. The performance bounds are simple expressions in terms of graph invariants. It is seen that the induced star number of a network plays an important role in the design and performance of such networks.

Index terms — flow admission control, quality-of-service (QoS), distributed algorithms, interference, wireless networks, conflict graph, link scheduling; fractional chromatic number; induced star number.

1 Introduction

In recent years there has been an increasing interest in using data networks to support a wide variety of applications, each requiring a different Quality-of-Service (QoS). For example, real-time applications such as voice, video and industrial control are time-sensitive and require that the delay be small, while for other data applications the sender may require that a constant, minimum bit-rate service be provided. In the simplest and lowest level of service, such as the one provided in the Internet Protocol service model, the network makes a best-effort to deliver data from the source to destination, but it makes no guarantees of any kind, so it is possible that packets can get dropped, delayed, or delivered out of order. However, this basic level of service is insufficient for many data applications such as video-conferencing that also have a minimum bandwidth requirement. We consider in this work applications requiring a minimum bandwidth quality-of-service.

More formally, the wireless network model and desired QoS are specified as follows. Let V be a set of nodes and be a set of communication links. Each link makes a demand to transmit information from to at a rate of b/s. The total bandwidth of the shared wireless medium available for the communication network is  b/s. The main problem studied here is to determine whether a set of demands can be satisfied. Of course, if all the links can be simultaneously active, the set of demands can be satisfied as long as each individual demand is at most . However, due to interference effects nodes in the same vicinity contend for the shared wireless medium and hence cannot be active at the same time. For example, in IEEE 802.11 MAC protocol-based networks, any nodes adjacent to node or to node are required to be idle while the communication takes place. (The reason for this constraint is that when the communication from to takes place, also sends acknowledgement of the receipt of information back to , so nodes adjacent to cannot receive information from other nodes during this time due to interference.) We will use a more general interference model which includes this special case and which is still tractable enough for further study.

The interference in the network is modeled using a conflict graph. Given a network graph , define its conflict graph to be , where two links and are adjacent in the conflict graph if and only if they interfere with each other due to being in the same vicinity and hence cannot be simultaneously active. This interference model has been studied recently by a number of authors; for example, see Jain et al [16], Hamdaoui and Ramanathan [15], and Gupta, Musacchio and Walrand [11]. A special case of this model, where two links are considered to be interfering if and only if they are incident (in the network graph) to a common node, has been studied earlier by Hajek [12], Hajek and Sasaki [13], and Kodialam and Nadagopal [17]. While we will study the more general model, it still does not incorporate parameters such as the actual signal-to-interference-and-noise ratio at the receiver node in its decision making process. In our work below we will also revisit the special case just mentioned and obtain some new results in the distributed setting.

The notion of a conflict or interference graph was introduced earlier, outside the context of wireless networks, by Chaitin [4] for solving the register allocation problem. A register is a high-speed memory location on a computer’s CPU and the register set is usually of a very limited size and much faster compared to ordinary memory. During the register allocation phase of compiling, a decision needs to be made as to which values to keep in registers and which to keep in memory at each point in the object code. One approach is for the compiler to construct a conflict graph, where the vertices of this graph correspond to values, and there is an edge between two vertices if and only if the two values are simultaneously live at some point in the code. A valid coloring of a graph is an assignment of one color to each vertex so that adjacent nodes are assigned different colors. If it is not possible to color the conflict graph using colors, where is the number of registers available for use, then one of the nodes (variables) is moved to memory and deleted from the conflict graph, and a valid coloring is again attempted. Many commercial and research compilers use a coloring approach on the conflict graph during the register allocation phase.

In the context of wireless networks, the notion of coloring the conflict graph has been used to study the spectrum allocation problem in cellular communications [14]. In this setting, the vertices of the conflict graph correspond to base stations. Two vertices are adjacent iff the corresponding base stations are close enough to interfere while using the same frequency. Each base station represents its area of service (such as a hexagonal region in the plane) and makes a demand for a certain number of frequencies which is proportional to the traffic demand for that region. The frequency assignment problem is to assign a set of frequencies (colors) to each base station, so that the demand for each base station is met, so that interfering base stations are assigned nonoverlapping sets, and so that the minimum number of frequencies is used. This problem is essentially equivalent to a problem sometimes referred to as the weighted vertex coloring problem.

In the admission control problem studied in our work, the vertices of the conflict graph represent the links in the communication network. The quality-of-service metric is specified in terms of the bandwidth desired by each link. This gives rise to one demand value for each vertex of the conflict graph, and this value could possibly be non-integral. The admission control problem is then to determine whether these demands can be satisfied using a specified amount of resource (total available bandwidth). This problem is different from the classical weighted vertex coloring problem in some ways. First, fractional solutions to the coloring problem are also admissible, as indicated by the linear programming formulation given below. Second, our emphasis is on decisions that can be made in a decentralized manner, i.e. using only localized information. Finally, the conflict graph sometimes has additional structure derived from the structure of the links in the network or the interference model. The admission control problem studied here is essentially that of obtaining, using only localized information, an upper bound on the amount of resource required to satisfy a demand pattern. The scheduling problem, which we do not study here, concerns how these resources are actually allocated or managed.

This paper is organized as follows. The admission control problem studied here is formulated precisely in Section 1.1. Decentralized solutions to this problem are then studied: Section 2 is on the row constraints, Section 3 is on the degree and mixed conditions, and Section 4 is on the clique constraints. These results provide sufficient conditions and distributed algorithms for admission control. The main new results here concern the worst-case performance of these algorithms. Finally, in Section 5 the results obtained thus far are applied to some specific examples such as unit disk networks and networks with primary interference constraints. In order to keep this paper as self-contained and accessible as possible, we recall along the way some of the known results and their proofs from the literature.

1.1 Model and problem formulation

We first state the flow control problem formally. Then, in order to avoid repeating trivialities throughout the paper, we will present an equivalent reformulation of the problem that ignores many of the constants and variables and involves just the essential details. We will work only with this reformulation in the rest of this paper.

Let be a network graph, where . Each link has a maximum transmission capacity of b/s, and there is a demand to use that link at some rate b/s. The total available bandwidth of the shared wireless medium is b/s. The conflict graph specifies which pairs of links interfere with each other: two links are adjacent in the conflict graph iff they interfere with each other when they are simultaneously active. The main problem we study is to determine, using only localized information, whether the set of demands can be satisfied. More precisely, an independent set of a graph is a subset of elements that are pairwise nonadjacent. If the set of links that are simultaneously active is an independent set, then these links cause no interference with each other and can have (a part of) their demands satisfied during the same time slot. Let denote the set of all independent sets of . [Note that this set can grow exponentially with the size of the graph. For example, a graph on vertices consisting of disjoint triangles has maximal independent sets.] A schedule is a map . The schedule assigns to each independent set a time duration , which specifies the fraction of time that the links in are active. A schedule is said to satisfy a set of demands if, for each , , and if the duration of the schedule is at most 1. A schedule is said to be optimal if it satisfies the demand of all the links and has minimum duration.

A schedule , as defined above, can be implemented as follows. We construct a periodic schedule with period seconds, say, where we can choose arbitrarily. Then divide the time axis into frames of duration , and further subdivide each frame into subintervals of length . The schedule is implemented by letting the set of links be active during the subinterval of duration . During this subinterval each link transmits at its maximum transmission rate . It is seen that if the schedule satisfies the demand vector, then so does this implementation. We shall assume throughout that the entries of vectors and are all rational numbers. Since the linear program defining the duration of an optimal schedule has a rational optimal solution, it can be assumed that the schedule is a map into the rationals. Let be the least common multiple of the denominators of . Then, divide the time axis into frames of duration , and further subdivide each frame into subintervals of equal length. Suppose ; then in each frame, we let the links be active for time slots; we let each link transmit at its maximum transmission rate during each time slot when it is active. Such an implementation meets the demands satisfied by the schedule .

Reformulation. Suppose we are given a network graph and a conflict graph that specifies which pairs of links interfere with each other. Let denote the amount of time when link demands to be active. A link demand vector is said to be feasible within time duration if there exists a schedule of duration at most that satisfies the demands. We will often assume, for simplicity of exposition, that . Note that a schedule is a map that assigns to each independent set of the conflict graph a time duration . A link is then active for total duration . The admission control problem is to determine whether there exists a schedule of duration at most that satisfies the link demand vector . The scheduling problem is to realize such a schedule. We are interested in solutions that can be implemented using only localized information and with low processing cost. For simplicity of exposition, we shall assume that is connected; if this is not the case we can work with each connected component separately and the results here still apply.

In a communication network with interference constraints, the problem of determining the minimum duration of a schedule is essentially that of computing the fractional chromatic number of a weighted graph. This problem, which corresponds to the problem solved by an optimal, centralized algorithm, is NP-hard in the general case [10]. The admission control mechanisms studied here provide upper bounds on the fractional chromatic number; furthermore, these upper bounds have the advantage that they can be computed efficiently and implemented in distributed systems.

Notation. It will be convenient to use the following notation [2]. Let be a simple, undirected graph. For , denotes the neighbors of . denotes the maximum number of vertices of that are pairwise nonadjacent. For , denotes the induced subgraph whose vertex-set is and whose edge-set is those edges of that have both endpoints in . For any and any , define .

1.2 Prior work and our contributions

The take off point for our work is the prior work of [11] and [15]. Their work proves that certain distributed algorithms provide sufficient conditions for admission control. They call these conditions the row constraints [11] (or rate condition [15]), the degree condition [15], mixed condition [15], and scaled clique constraints [11]. Furthermore, for unit disk networks, the scaled clique constraints are shown to be a factor of 2.1 away from optimal [11].

The main results of this paper are along the following lines: the exact worst-case performance of these distributed admission control mechanisms is characterized and it is thereby shown that these mechanisms can be arbitrarily far away from optimal; we then show that for some well known classes of networks and interference models, these distributed algorithms are actually within a bounded factor away from optimal. The classes of networks and interference models we study include unit disk networks and networks with primary interference constraints.

More specifically, we introduce the notion of the induced star number of a graph and show that it determines the exact worse-case performance of the row constraints. This implies, for example, that for unit disk networks there is a simple and efficient distributed admission control and distributed scheduling mechanism which is at most a factor of 5 away from optimal, and for networks with primary interference constraints this mechanism is at most a factor of 2 away from optimal. The performance of two other sufficient conditions - namely the degree condition and mixed condition - is also studied. Finally, the results obtained thus far are applied to some specific classes of networks and interference models. For example, it is seen that the scaled clique constraints are at most a factor of 1.25 away from optimal for networks with primary interference constraints. It is also seen that the different sufficient conditions studied here are in general incomparable (each condition is neither stronger nor weaker than the others).

Our results quantify the performance of some practically-efficient distributed mechanisms used for flow admission control. The induced star number of a network is seen to be a crucial parameter that effects the performance of such distributed systems, and hence needs to be as close to unity as possible when designing such networks. It is also seen that the row constraints can consume up to twice the amount of resources compared to the mixed condition for the same set of demands. Quantitative performance results of this nature assessing the performance of the various distributed algorithms are also obtained.

2 Row constraints

We now present a sufficient condition for flow admission control that can be implemented in a distributed manner. Given a conflict graph , link demand vector and , a sufficient condition for feasibility is given by the following result (see [15, Thm. 1], [11, Thm. 1]):

Proposition 1.

If for each , then the demand vector is feasible within duration .

Proof: Pick any ordering of the links . Assign link the time interval , and initialize . For each , assume has already been assigned time interval . Here, need not be one continuous time interval; it can be a union of disjoint intervals, but its overall length is . Since , it follows that . Hence, it is possible to assign to some subset which is nonoverlapping with the intervals already assigned to the neighbors of and which has duration . We can repeat this procedure for the remaining links , in turn.

These constraints are called the row constraints in [11] and the rate condition in [15]. Let be the 0-1 valued adjacency matrix of where iff or and are interfering. Let denote the vector whose every entry is 1. Then the sufficient condition above is equivalent to the condition on the rows of , hence the name row constraints. Since these constraints depend on the flow rate (the demand value) of the neighboring links, it is called the rate condition in [15], where the authors also study a degree condition which depends only on the number of neighboring links rather than on their actual demand values. We chose to use the phrase row constraints (rather than the rate condition) because all these results also apply in a general context where the variables need not refer to only rate values, or even just time durations, but to any demand for resources by competing entities, where the competition is modeled by a conflict graph.

The proof of Proposition 1 also gives a very efficient algorithm for checking feasibility. It provides both a distributed admission control mechanism as well as a distributed scheduling algorithm: when a link that is currently inactive makes a demand to be active for duration , the admission control mechanism can be implemented efficiently by just checking the condition above for and its neighbors. The information required by a link to check this condition is just its demand and the demand of its neighbors. Furthermore, the distributed scheduling algorithm that meets the demand for link needs to know only the time intervals already assigned to the neighbors of in order to determine the time interval for .

It is known (cf. [15, Thm. 5]) that this sufficient condition is also necessary if the conflict graph is the disjoint union of complete graphs. Indeed, if a given set of links are pairwise interfering, then the duration of an optimal schedule for those links is equal to the sum of their individual demands. It turns out that the converse is also true: the sufficient condition above becomes necessary only if the conflict graph is the disjoint union of complete graphs. To prove this, suppose that the conflict graph is connected but not complete. Then there exist three links and such that and interfere, and interfere, but and do not interfere (otherwise, the relation of two links being interfering would be an equivalence relation and the conflict graph would be complete). The link demand vector is feasible within duration [0,1] but does not satisfy the row constraint for . This proves the converse. We shall show that this observation - that the row constraints above are also a necessary condition if and only if the conflict graph is the disjoint union of complete graphs - can be deduced from a more general result (cf. Theorem 3).

2.1 Row constraint polytope and induced star number

In order to quantify how far the row constraints are from optimal, it will be useful to introduce its associated polytope. Given a conflict graph, let denote its independent set polytope. This polytope is defined as the convex hull of the characteristic vectors of the independent sets of the graph. Note that is exactly equal to the set of all link demand vectors which are feasible within one unit of time. For the given conflict graph, let denote the set of all link demand vectors that satisfy the row constraints for ; that is,

 Prow:={τ≥0:τ(ℓi)+τ(Γ(ℓi))≤1 ∀i}.

Since the row constraints are sufficient, . Also, note that . Define the scaling factor

 βrow:=inf{β≥1:PI⊆βrowProw}.

Equivalently,

 βrow=supτ∈PImaxi{τ(ℓi)+τ(Γ(ℓi))}.

So , and is the smallest scaling factor which converts the sufficient condition into a necessary one.

It has been pointed out (cf. [11]) that the row constraints can be arbitrarily far away from optimal. For example, suppose the network consists of links , where interferes with each of and there is no interference between the remaining links. Then the conflict graph is a star graph. The link demand vector is feasible within one unit of time, but the row constraint for has value which can be made arbitrarily close to as approaches 0. This shows that . We prove next that the opposite inequality also holds, i.e. the row constraints can be a factor away from the optimal schedule time for some demand vector only if the conflict graph contains a star graph on vertices as an induced subgraph.

Definition 2.

The induced star number of a graph is defined by

 σ(H):=maxv∈V(H)α(H[Γ(v)]).% \vrule width 4.0pt depth 2.0pt height 6.0pt

Hence, the induced star number of a graph is the number of leaf vertices in the maximum sized star of the graph. Note that equals 0 or 1 iff is the disjoint union of complete graphs. The induced star number of a graph determines exactly how close the row constraints are to optimal in the worst case:

Theorem 3.

Let be a conflict graph. The exact worst-case performance of the row constraints is given by .

Proof: It has already been pointed out above that . To prove the opposite inequality, suppose and let be any link with demand . It suffices to show that . Since links in must be scheduled at time intervals disjoint from those of and since , there exists a schedule satisfying the demands of just the links which has duration at most . The maximum number of pairwise non-interfering links in is , and so at most links can be simultaneously active during any time interval of this period of duration . Hence, . So .

It was mentioned that the row constraints are also necessary conditions if and only if the conflict graph is the disjoint union of complete graphs. This observation follows as a special case of Theorem 3: the row constraints which are sufficient conditions are also necessary iff , which is the case iff , which is the case iff each component of is a complete graph. While the induced star number of a graph can be arbitrarily large, for special classes of networks studied in the literature this quantity is bounded by a fixed constant. This happens to be in the case for unit disk graphs and for networks with primary interference constraints. We examine these special cases in Section 5.

2.2 A strengthening of the row constraints

We saw above that the row constraints are a sufficient condition for feasibility of a link demand vector, and the performance of this distributed algorithm is determined by the induced star number . We now show that a slight improvement to can be obtained.

Recall that the row constraint corresponding to link is that the sum total of the demand and the demands of all its interfering neighbors not exceed the available resource . It is easy to see that all the links in the network, except for any one designated link, say , can ignore the demand of up to one of its interfering neighbors.

Proposition 4.

Given a network and its conflict graph , pick any designated link . A sufficient condition for to be feasible within duration is that

 τ(ℓi)+τ(Γ(ℓi)) ≤T, i=1 τ(ℓi)+{τ(Γ(ℓi))−minℓj∈Γ(ℓi)τ(ℓj)} ≤T, i=2,…,m.

Note that this sufficient condition is equivalent to the row constraints when is complete. This is because the inequality for becomes , which implies that the set of inequalities for are also satisfied. If is not complete, then it is possible to do away with the notion of a designated link (i.e. every link can ignore the demand of up to one of its neighbors), provided the conflict graph is not of one exceptional type - the odd cycle. The proof is by reducing to the previous case by showing that the scheduling mechanism can always find a designated link for which the conditions above are essentially satisfied.

Proposition 5.

Suppose is not complete. Then the set of constraints

 τ(ℓi)+{τ(Γ(ℓi))−minℓj∈Γ(ℓi)τ(ℓj)} ≤ T,   i=1,…,m

is a sufficient condition for to be feasible within duration if is not an odd cycle. Furthermore, the smallest scaling factor that converts this sufficient condition into a necessary one is equal to exactly or , depending on the structure of .

Proof: Suppose the inequalities in the assertion are satisfied. Let be the minimum number of vertices of whose removal disconnects into more than one connected component. We consider three cases, depending on the value of (this proof method is from [3]):

: Let be a cut-vertex of , so that the removal of produces connected components . Since each is connected to and by the inequalities of the condition, the demand of each link in can be satisfied, using the scheduling mechanism given in an earlier proof and using as the designated link. It remains to show that the demand of can also be satisfied. Indeed, note that has neighbors and in and , respectively, where and are nonadjacent. Hence, the time slots of the schedule in can be permuted so that the time interval assigned to is a subset of that assigned to (or, in case , the time slots can be permuted so that the time interval assigned to is a subset of that assigned to ). Now can be scheduled as well since two of its neighbors have been assigned overlapping intervals.

: Since is not complete, there exist , and such that is adjacent to both and while and are nonadjacent. Since , is connected. First assign and the time intervals and , respectively. By the inequalities of the condition, we can now schedule the remaining links in some order, with as the designated link of the connected graph . Finally, can also be scheduled since two of its neighbors have been assigned overlapping intervals.

: Let denote the maximum degree of a vertex of . If , then the graph is either an odd cycle or an even cycle. If is an even cycle, say , then a feasible schedule is obtained by assigning to each link of odd index the time interval and to each link of even index the time interval . If is an odd cycle, say , note that the demand vector satisfies the inequalities of the condition but is not feasible within duration . So now assume . There exist and such that is a cut-vertex of . So suppose the removal of this cut-vertex decomposes into connected components . Now has neighbors and in and , respectively. Note that and are nonadjacent and is connected. As before, we can assign time intervals and to and , respectively, then schedule all the remaining links with as the designated link, and finally schedule as well.

This proves the sufficiency of the condition.

We now determine the performance of this distributed algorithm, where denotes the smallest scaling factor which converts this sufficient condition into a necessary one, i.e., replacing by in the inequalities above produces a set of necessary conditions. Let denote . Now contains the star graph on some nodes as an induced subgraph, with as the center node. For these nodes, the demand is feasible within one unit of time, and the left-hand-side of the inequality for given in the sufficient condition evaluates to , which can be made arbitrarily close to . Hence, . Also, since this sufficient condition is stronger than the row constraints, . So we have that .

Let denote the vertices of that can induce a maximum sized star of with some of their neighbors, i.e. ; here denotes the maximum size of an independent set of . (It suffices to consider just these vertices because the remaining vertices have a smaller value of , and hence the left hand side of the inequalities in the condition evaluate to at most .) If the degree of every vertex in is exactly , then , as we just showed. However, suppose the degree of some vertex in is at least , where are the neighbors of and the first of these neighbors form an independent set. Then the demand vector for given by is feasible within one unit of time, but the left-hand-side of the inequality for given in the condition can be made arbitrarily close to . Hence in this case. It follows that equals or , according as whether the degree of any vertex in exceeds or doesn’t exceed .

3 Degree and mixed conditions

It was shown that the row constraints provided a simple, distributed sufficient condition for feasibility of a given demand vector. In this condition, there was exactly one constraint associated with each link, namely, the sum total of the demand of the link and demands of its neighbors not exceed the available resource. We now describe an even simpler condition. We call this the degree condition since it requires knowing, for each link, the demand of that link and just the number (not actual demands) of links interfering with it.

Suppose link interferes with exactly other links, i.e. in the conflict graph has degree . Then, the following result provides another sufficient condition for admission control [15]:

Proposition 6.

A given demand vector is feasible within duration if for each .

Proof: Suppose the inequalities in the assertion are satisfied. Order the links so that . Assign the time interval . Assume links have already been assigned time intervals satisfying their demands. It follows from and the chosen ordering of the links that . Hence, it is possible to assign to a subset of of length which is disjoint from the intervals already assigned to its neighbors. It follows by induction that the demand vector is feasible.

The degree condition and the row constraints are seen to be equivalent when all links have the same demand value. Also, note that while the admission control mechanism specified by the degree condition can be implemented in a distributed manner, the scheduling mechanism given in the proof above requires ordering all the links according to their demand values, which is global information. We will see more examples of such conditions where the admission control mechanism is distributed but the scheduling mechanism realizing a feasible schedule is not distributed.

The degree condition is quite loose (far away from optimal) in that it admits a given demand vector only if for each (this is because, since is connected, we have ), and more generally, only if for each . In particular, when the demand values of the various links are asymmetric - in the sense that some links have small demand values, while others have demand values close to the maximum possible - then the demand vector will not be admitted by the degree condition even if it really is feasible.

The performance of the degree condition is determined, not surprisingly, by the maximum degree of a vertex in the conflict graph. More precisely, define

 Pdegree:={τ≥0:τ(ℓi)(d(ℓi)+1)≤1, ∀i}.

Then by Proposition 6. Define

 βdegree:=inf{β≥1:PI⊆βPdegree}.

Let denote the maximum degree of a vertex in .

Lemma 7.

For any conflict graph , the exact worst-case performance of the degree condition is given by .

Proof: Let be a link having exactly neighbors in the conflict graph. For these links, the demand vector is feasible within one unit of time, and can be made arbitrarily close to . Hence, .

Now suppose is feasible within one unit of time, and let be any link. Then, , and so . Hence, .

This implies that the degree condition is also necessary (and hence optimal) iff is the empty graph, i.e. iff no two links interfere with each other. It is possible to combine the row constraints and degree constraints to get a sufficient condition which is strictly stronger, as shown in [15]:

Proposition 8.

A link demand vector is feasible within duration if

 min{ τ(ℓi)+τ(Γ(ℓi)) , τ(ℓi)(d(ℓi)+1) }≤T, ∀ℓi∈L.

Proof: Suppose the inequalities in the condition are satisfied. Order the links so that . Assign the time interval . Suppose have already been assigned time intervals satisfying their demands. By the inequality in the condition for , either or ; in either case, can also be scheduled, as was shown in the earlier proofs where one of these conditions is satisfied.

As introduced in [15], we call this sufficient condition the mixed condition. Consider the conflict graph on the three links , where is adjacent to each of and , and and are nonadjacent. Let . Note that the link demand vector satisfies the row constraints but not the degree constraints, and the demand satisfies the degree constraints but not the row constraints. Hence these two conditions are incomparable. Also, the demand satisfies neither the row nor the degree constraints but does satisfy the mixed condition. Finally, the demand is feasible but does not satisfy the mixed condition. (This last example proves that the mixed condition is optimal (and hence, also necessary) iff is the disjoint union of complete graphs, i.e. iff , an observation which we can also deduce from a more general result; cf. Theorem 9.) These examples show that, in general,

 Prow,Pdegree  (Prow ∪ Pdegree)  Pmixed  PI,

where

 Pmixed:={τ:min{ τ(ℓi)+τ(Γ(ℓi)) , τ(ℓi)(d(ℓi)+1) }≤1, ∀ℓi∈L}.

Let denote the smallest scaling factor that converts the sufficient mixed condition into a necessary one; hence, given the conflict graph and its independent set polytope , we have that and

 βmixed=supτ∈PI maxℓi∈L min{ τ(ℓi)+τ(Γ(ℓi)) , τ(ℓi)(d(ℓi)+1) }.
Theorem 9.

The worst-case performance of the mixed condition is bounded as

 1+σ(GC)2 ≤ βmixed ≤ σ(GC),

where denotes the induced star number of . Moreover, the lower and upper bounds are tight; the star graphs realize the lower bound, and there exist graph sequences for which approaches the upper bound arbitrarily closely.

Proof: Since the mixed condition is satisfied whenever the row constraints are satisfied, , and so the upper bound follows from Theorem 3.

To prove the lower bound, suppose is a star graph on vertices , where is the center vertex and the leaf vertices form an independent set. We show that for this and its independent set polytope ,

 supτ∈PI maxℓi∈L min{ τ(ℓi)+τ(Γ(ℓi)) , τ(ℓi)(d(ℓi)+1) }=1+η2.

It can be assumed that , for if then is complete and so by the optimality of the mixed condition in this case. Set . In order to maximize

 maxℓi∈L min{ τ(ℓi)+τ(Γ(ℓi)) , τ(ℓi)(d(ℓi)+1) }

over , can be set to equal its maximum possible value of , for . The mixed condition evaluated at is bounded as

 min{ τ(ℓi)+τ(Γ(ℓi)) , τ(ℓi)(d(ℓi)+1) }≤τ(ℓi)+τ(Γ(ℓi))=(1−δ)+δ=1.

But trivially. Hence, to determine the worst-case performance of the mixed condition, it suffices to evaluate the mixed condition at just :

 βmixed = supδ∈[0,1] min{ τ(x)+τ(Γ(x)) , τ(x)(d(x)+1) } = supδ∈[0,1] min{δ+η(1−δ) , δ(1+η)}.

Note that iff . It can be seen that the optimum is attained at , giving for this star graph on vertices.

This proves the lower bound in the assertion, and it has also been shown that the class of star graphs realize this lower bound.

We now construct a sequence of graphs for which approaches the upper bound in the limit. We do this by determining the exact value of for a wider class of graphs which includes the class of star graphs as a special case. The property that needs to satisfy is that it has some that is adjacent to every other element of and such that the removal of disconnects into a disjoint union of complete graphs, i.e., has degree and .

We now claim the following: Suppose is such that is adjacent to all other members of and the removal of produces disjoint complete graphs on vertex sets . Then

 βmixed=η(1+∑|Li|)η+∑|Li|.

To prove this claim, suppose , , and satisfy the conditions of the claim. Recall that

 βmixed=supτ∈PI maxℓi∈L min{ τ(ℓi)+τ(Γ(ℓi)) , τ(ℓi)(d(ℓi)+1) }.

Let . In order to maximize

 maxℓi∈L min{ τ(ℓi)+τ(Γ(ℓi)) , τ(ℓi)(d(ℓi)+1) }

over , assign demands to the elements of arbitrarily so that . For this , the row constraint at each evaluates to exactly 1 and the mixed condition at each evaluates to at most 1. Since , is determined by the value of the mixed condition at just :

 βmixed = supδ∈[0,1] min{ τ(x)+τ(Γ(x)) , τ(x)(d(x)+1) } = supδ∈[0,1] min{ δ+(1−δ)η , δ(1+∑|Li|) }.

It can be verified that iff and that attains its optimal value of when . This proves the claim.

Note that the case of , makes a star graph, whereas letting approach infinity gives that approaches the upper bound of . This proves that the upper bound in the assertion is tight.

One general class of graphs that includes the star graphs, the even and odd cycles, the complete graphs and bipartite graphs are those that satisfy the following property: for each vertex in the graph , the neighbors of induce a disjoint union of complete graphs. For this general class of graphs there is a simple expression for the exact value of :

Theorem 10.

Suppose satisfies . Let denote the number of neighbors of and let denote the number of connected components induced by the neighbors of . Then

 βmixed=maxℓ∈Lηℓ(1+d(ℓ))ηℓ+d(ℓ).

This result implies that if is a star graph, a bipartite graph, or a cycle graph, then the exact worst-case performance of the mixed condition is . This value is about a half of or for some families of graphs; hence, in terms of the amount of resources requested, it is seen that the mixed condition can be an improvement over the row constraints and the degree condition by up to a factor of 1/2.

The proof of Theorem 10 is similar to the proof of the claim in the previous proof and so the details are omitted. The simplest example of a graph that does not satisfy the conditions of Theorem 10 is the graph (one edge removed from the complete graph on 4 vertices). For this graph, Theorem 9 immediately gives that the graph invariant is bounded as . A straightforward but lengthy computation yields the exact value of

4 Clique constraints

A necessary condition for a given link demand vector to be feasible can be obtained as follows. Suppose there exists a schedule of duration 1 satisfying demand . Then if is a clique in the conflict graph, the time intervals assigned to the distinct links in must be disjoint, hence . Thus, a necessary condition for to be feasible within duration is that for every maximal clique in the conflict graph. These constraints are called clique constraints [11]. As before, we can associate a polytope with this necessary condition; define

 Pclique:={τ:τ(K)≤1,∀K}⊇PI,

where runs over all the cliques (or equivalently, over just all the maximal cliques) of the conflict graph.

A perfect graph is one for which every induced subgraph has its chromatic number equal to its clique number. It is known that for a graph if and only if the graph is perfect.

Using the notion of the imperfection ratio of graphs, bounds on the suboptimality of clique constraints were obtained [11] for the case of unit disk graphs. More precisely, given a conflict graph and demand vector , let denote the minimum duration of a schedule satisfying (the optimal value of this linear program is also the smallest such that , and is often referred to as the fractional chromatic number of ). Let denote the maximum value of over all cliques in the conflict graph; so . The imperfection ratio of a graph is defined as

 imp(GC):=supτ≠0{T∗(τ)/Tclique(τ)}.

This quantity has been studied in [9]; it is finite and is achieved for any given graph. In the definition above, for a given demand vector , the numerator specifies the exact amount of resource required to satisfy the demand, as determined by an optimal, centralized algorithm. The denominator specifies a lower bound on the resource required to satisfy the demand, as determined by a particular distributed algorithm (the clique constraints). Their ratio is the factor by which the distributed algorithm is away from optimal for the given demand vector. The imperfection ratio, which maximizes this ratio over all demand patterns, is then the worst-case performance of the distributed algorithm. This argument is made more precise in the next proof.

The following general result is implicit in [11] (where the authors focus on unit disk graphs) and in [9]:

Proposition 11.

The largest scaling factor which converts the necessary clique constraints into a sufficient condition is ; i.e. the worst-case performance of the clique constraints is given by

 sup{β≤1:βPclique⊆PI}=1imp(GC);

and

 1imp(GC)Pclique⊆PI⊆Pclique.

Proof: Suppose . Then for all cliques . So . So .

Now suppose . It suffices to show that . Since , there exists a such that . Define . Then , but .

The problem of determining the imperfection ratio of various families of graphs has been studied in [9].

5 Examples

In this section we apply the results obtained so far to some special classes of networks and interference models. In Section 5.1, we examine a model of interference called primary interference constraints, which has been well-studied in the literature in the centralized setting; we examine we examine this problem in the distributed setting. In Section 5.2 we look at the unit disk model, which is quite popular and widely used by researchers in the sensor networks community to model the topology of a network.

5.1 Primary interference model

Given a network , suppose two links are considered to be interfering iff they share one or more endvertices in common. We refer to this kind of interference as primary interference. This interference model arises, for example, from the assumption that each node can communicate to only one other node at any given time. This interference model is perhaps the most well studied in the literature; for example, see [12], [13], [17]. The conflict graph for such a network is called a line graph. Note that in this kind of interference model, if link interferes with each of and , and doesn’t interfere with , then must interfere with exactly one of or . So we obtain a well-known result that if the conflict graph is a line graph then . It follows that for such networks the row constraints will be at most a factor 2 away from optimal.

More specifically, for this interference model, the row constraints on the conflict graph can be reformulated on the network graph as follows. Suppose link is incident between nodes and . Given link demand vector , let denote the sum of the demands of all links incident in to node . Then the row constraint in the conflict graph is equivalent to the constraint in the network graph. This equivalence yields the following sufficient condition:

Corollary 12.

Let be a network graph, and suppose two links interfere with each other if and only if they are incident to a common node. Then is feasible within duration if for each , . This sufficient condition is a factor of at most 2 away from optimal.

Remark. In the case of primary interference constraints, upper bounds on the minimum scheduling time can be obtained immediately by using upper bounds on the chromatic index of multigraphs; for example, see some of the results in [12], [17]. We note here a reverse implication: the row constraints above (and its proof) can be used to obtain an upperbound on the chromatic index of multigraphs. More precisely, let be a simple undirected graph. Let denote the number of parallel edges between vertices and . This defines a multigraph . A proper edge-coloring is an assignment of colors to edges so that any two edges that share one or more endvertices in common are assigned different colors. The chromatic index of a multigraph is the minimum number of colors required to properly color its edges. Let denote the number of edges incident to . Then, Corollary 12 implies that: the chromatic index of the multigraph is bounded from above by

Another distributed algorithm that can be used in networks having primary interference is given by the clique constraints. Trivially, a necessary condition for