## 1 Introduction

A long-standing open problem is to develop simple, distributed scheduling algorithms for wireless networks that are provably efficient. Distributed mechanisms for admission control and scheduling have lower communication overhead and lesser complexity than optimal, centralized algorithms. A distributed, greedy scheduling algorithm for wireless networks which has been well-studied in the literature is maximal scheduling; it is suboptimal, and it is known that its worst-case performance is characterized by the interference degree of the conflict graph [2] [5] [6] .

Modeling interference using hypergraphs instead of graphs helps capture certain complexities and increase system capacity (cf. [13][12] [15]). In the present work, we extend the previous results in the literature on performance analysis of the maximal scheduling algorithm to the more general case of hypergraphs. We define the so-called interference degree of a hypergraph, which characterizes the worst-case performance of maximal scheduling.

## 2 System Model

Let be a wireless network, where is a set of nodes and is a set of wireless links. Due to wireless interference, links in the same vicinity cannot be simultaneously active. The interference is modeled using a (conflict) hypergraph , where is the ground set of the hypergraph and is a collection of subsets of such that each hyperedge is a minimal subset of links that cannot be simultaneously active due to interference.

The admission control problem is now formally stated. Let

be a link demand vector, where the quality-of-service requirement for each link is specified by the fraction

of each unit of time that link demands to be active. An independent set of the hypergraph is a subset that does not contain any hyperedge [1]. Let denote the set of all independent sets of . A schedule is a map that assigns a time duration to each independent set of the hypergraph. The total duration of the schedule is . The schedule satisfies demand if , for all . A link demand vector is said to be*feasible*if there exists a schedule of duration at most satisfying demand . Given a conflict hypergraph and link demand vector , the admission control problem is to determine whether is feasible. The present focus is on distributed mechanisms for admission control, wherein feasibility is determined using only localized information. Also, the admission control and scheduling problems have been well-studied for the conflict graph model (cf. [8] [9] [2] [4]), and the present work extends these results to hypergraph models.

An equivalent formulation of the admission control problem is as follows. Given the conflict hypergraph , with , let be the set of all independent sets of . Define the link-independent set incidence matrix by if and only if . Let by a link demand vector. The fractional chromatic number of the weighted hypergraph , denoted by

, is defined to be the optimal value of the linear program: minimize

subject to . A link demand vector is said to be*feasible*if . Let denote the convex hull of the characteristic vectors of the independent sets of . Then, is exactly the set of all link demand vectors that are feasible.

The notation used is standard. In the sequel, denotes the set . Given a conflict hypergraph , denotes the set of neighbors of link and is defined as the set of other links such that and belong to the same hyperedge:

The set of edges of that contain is denoted .

## 3 Distributed scheduling algorithms

In this section, sufficient conditions for a link demand vector to be feasible are given. The sufficient conditions for admission control given in this section can be extended to provide a feasible schedule when the demand vector is feasible. Thus, the results below give distributed algorithms for both admission control and scheduling. The first sufficient condition is an extension of the greedy coloring algorithm for graphs to the case of hypergraphs. A special case of this sufficient condition, obtained when the hypergraph is a graph, is the row constraints of [10] [7]. Theorem 3 and Corollary 4 are essentially from [11], but the proofs given here are based on our system model.

###### Lemma 1.

Let by a conflict hypergraph and let be a link demand vector. A sufficient condition for to be feasible is that

*Proof:*
By the given inequality for , it is possible to schedule , i.e. it is possible to assign to link a subset of total length . Suppose have already been scheduled. It will be shown that can also be assigned a subset of such that not all links in a hyperedge are simultaneously active (except possibly at endpoints of subintervals).

Let denote the set of hyperedges such that contains and all other links in have previously been scheduled, i.e. satisfies and . For , define the common time slots of all links in that have already been scheduled:

Because the set of links in a hyperedge cannot be simultaneously active, it is necessary that be disjoint from , for each . Also, . It follows from the given inequality for that . Hence, can also be assigned a subset such that .

Given a hypergraph with link set , let denote the set of all real, symmetric matrices such that (1) , (2) for all , if , and (3) for all and .

###### Lemma 2.

Let be a conflict hypergraph, let , and let be a link demand vector. Fix . Suppose satisfies , for all . Let be the collection of hyperedges of which contain and such that every link in has index at most . For , define . Then,

*Proof:*
The proof is by induction on . If , then

where we have used the facts for , and for . Now fix , where , and assume the assertion holds for all smaller values of . It suffices to prove

Define the modified time intervals and demands

Then, the left-hand side can be rewritten as

Also, . The right-hand side can be rewritten as

By the inductive hypothesis,

It remains to show

Because for , one obtains

###### Theorem 3.

[11, Theorem 1] Let be a conflict hypergraph and let be a link demand vector. A sufficient condition for to be feasible is that there exists some such that .

*Proof:*
Let be a link demand vector and suppose there exists satisfying

It needs to be shown that is feasible, i.e. that each link can be assigned a subset of total length such that not all links in any hyperedge are assigned the same subinterval (except possibly for endpoints of subintervals). Initially, when none of the links have been scheduled, , for all . By the inequality for given in the assertion, , whence link can be assigned the time interval . Suppose links have already been assigned time intervals , respectively. It suffices to show that can also be scheduled.

In order to ensure that the time interval assigned to link does not conflict with those assigned already to its neighbors, consider the set of hyperedges . In other words, it suffices to consider those hyperedges which contain the current link of interest and whose remaining links have already been scheduled; time intervals assigned to links with can be ignored at this time because for .

Define the common time slots , for . To ensure that not all links in a hyperedge are simultaneously active, it is necessary and sufficient that be disjoint from (except at endpoints of subintervals) for all . Hence, to show that the demand can be satisfied, it suffices to show that . By the inequality for given in the assertion, it suffices to prove

But this follows from Lemma 2.

A special case of an element of is the matrix defined as follows. For , define

###### Corollary 4.

[11] Let be a conflict hypergraph and let be a link demand vector. Define as above. Then, a sufficient condition for to be feasible is

*Proof:* It can be verified that the matrix belongs to . By Theorem 3, is feasible.

## 4 Worst-case Performance

Corollary 4 gave a sufficient condition for admission control. This is equivalent to giving an upper bound on the fractional chromatic number . Define

It follows from Corollary 4 that . The worst-case performance of a sufficient condition is defined to be the largest factor by which the upper bound is away from optimal, and is defined by the hypergraph invariant

Thus, . In this section, an analysis of the worst-case performance is carried out for the above sufficient condition.

###### Definition 5.

Given a hypergraph , define the following quantities for :

Define the interference degree of a hypergraph to be

*Remarks:* In [11, Theorem 2], it is claimed that the worst-case performance of Corollary 4 is essentially characterized by .
It seems to be claimed in [11, Theorem 2] that if a link demand vector is feasible, then will satisfy the sufficient condition of Corollary 4; however, there seems to be an error in their proof, and a counterexample is given below to show that the worst-case performance can be a factor of more than away from optimal.

In the special case where the conflict hypergraph is a conflict graph, when a link is scheduled, none of its neighbors can be scheduled during the same time slot. In this special case, if , and so is the largest size of an independent set in . The graph invariant is the interference degree or induced star number of the conflict graph (cf. [2] [5] [6]). Also, in the graph case, , for all . Thus, reduces to the induced star number of the conflict graph in this special case.

However, when analyzing the worst-case performance, the situation for hypergraphs is in general different from the situation of graphs mentioned in the previous paragraph. In a hypergraph, when focusing on a particular link

, two possible situations arise when the value of the resource estimate

is maximized. During a time slot, it is possible that link is not scheduled because all of its neighbors in some hyperedge containing it are already scheduled. In this case, the resource estimate is at most . However, it is possible in the hypergraphs situation that a link can be scheduled at the same time as some of its neighbors - for example, in the example hypergraph below, link can be scheduled at the same time as links because is an independent set of the hypergraph. In such cases, the maximum contribution to the resource estimate during one unit of time can be as large as . Thus, can’t be ignored for hypergraph models; it appears that the proof of [11, Theorem 2] has overlooked the term in the formula for the worst-case performance.###### Theorem 6.

There exist conflict hypergraphs and link demand vectors such that the sufficient condition of Corollary 4 can be away from optimal by a factor larger than .

*Proof:*
Let be the conflict hypergraph , where , . Here, links are labeled as instead of for simplicity of exposition. Consider the link demand vector . Then is feasible because there exists a schedule of duration at most satisfying : the two independent sets and can each be active for duration units, and this gives a schedule satisfying . The upper bound , whereas it can be verified that .

The demand vector given in the previous example is not uniform on the subset . However, it appears that the hypergraph has some kind of symmetry whereby the links in can be scheduled in a round-robin manner, and similarly for the links in , so that the demand vector satisfied by such a schedule has a uniform demand pattern on the subset and still attains the same value for the upper bound . Indeed, the automorphism group of the hypergraph can be used to show that the worst-case performance is attained by a uniform demand pattern.

An automorphism of a hypergraph is a permutation of that maps the set of hyperedges to itself. The automorphism group of the hypergraph, denoted by , is the set of all automorphisms of . In other words, , where is the full symmetric group acting on . Thus, if , where , , then fixes the point and is the automorphism group of the partition of a -element set into two -subsets, and hence is isomorphic to (cf. [3, p. 46]). Also, acts transitively on the set .

The exact value of the worst-case performance is computed next (see also Corollary 9 for another proof).

###### Lemma 7.

Let be the conflict hypergraph defined by , . Then, the worst-case performance of the sufficient condition of Corollary 4 is .

*Proof:*
Let . Then, , and the objective is to compute . If and link belongs to a different hyperedge than link , then , and so the value can also be attained for some and a suitable . Thus, .

Let be a demand vector and let . Let denote the demand vector obtained by permuting the components of according to permutation . Then, because an automorphism preserves the sizes of hyperedges. It follows that . The convex combination

has a uniform demand pattern on because acts transitively on this subset (cf. [14, p. 5]). Also, is feasible if is and . It follows that is achieved for some that is uniform on .

The hypergraph has maximal independent sets: are subsets of the form and . Consider a schedule that assigns a duration to each independent set () and duration to . Every demand vector uniform on is satisfied by a schedule of this form. The demand pattern satisfied by this schedule is and (). Thus, . The duration of this schedule is . Thus, is the optimal value of the linear program: maximize subject to , . Evaluating the objection function at the three vertices of the feasibility polytope, one obtains .

###### Theorem 8.

Let be a conflict hypergraph. The worst-case performance of the sufficient condition of Corollary 4 is given by the interference degree of the hypergraph, i.e.

*Proof:* First, it will be shown that . Let . Let and be such that is a feasible demand vector and . Let be a schedule satisfying , and suppose assigns duration to independent set . The contribution to due to demands satisfied during this -th time slot is at most because the contribution is at most if link is not active during this time slot, and the contribution is at most if link is active during this time slot. Summing over the contributions to during the entire duration , one obtains that . This proves that .

To prove the opposite inequality, suppose that the values of and attaining are known; denote these values by and , respectively. Choose the demand pattern to be the characteristic vector of or , according as whether or is larger. For this demand vector , . Hence, .

A -star of a hypergraph is a collection of edges satisfying the following condition: there exists some such that for all . A -star is a generalization of the star subgraph found in graphs and satisfies the property that any two edges have exactly one vertex in common, and this vertex which is common to all the edges is the center of the star. If itself is a -star, then it can be verified that the value of is the number of edges in the hypergraph, and the value of is the second parameter in the result below.

###### Corollary 9.

Let be a -star containing exactly edges of size . Then,

## 5 Concluding Remarks

It is important to characterize the worst-case performance of distributed admission control and scheduling algorithms because they can overestimate the network resources required to satisfy a given set of demands by up to this factor. The interference degree of a hypergraph was defined and it was shown that in the worst case, the performance of the maximal scheduling algorithm is away from that of an optimal, centralized algorithm by a factor equal to the interference degree of the hypergraph.

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