Wireless Capacity in General Metrics

Wireless Capacity with Oblivious Power in General Metrics

Magnús M. Halldórsson School of Computer Science
Reykjavik University
Reykjavik 101, Iceland
 and  Pradipta Mitra School of Computer Science
Reykjavik University
Reykjavik 101, Iceland

The capacity of a wireless network is the maximum possible amount of simultaneous communication, taking interference into account. Formally, we treat the following problem. Given is a set of links, each a sender-receiver pair located in a metric space, and an assignment of power to the senders. We seek a maximum subset of links that are feasible in the SINR model: namely, the signal received on each link should be larger than the sum of the interferences from the other links. We give a constant-factor approximation that holds for any length-monotone, sub-linear power assignment and any distance metric.

We use this to give essentially tight characterizations of capacity maximization under power control using oblivious power assignments. Specifically, we show that the mean power assignment is optimal for capacity maximization of bi-directional links, and give a tight -approximation of scheduling bi-directional links with power control using oblivious power. For uni-directional links we give a nearly optimal -approximation to the power control problem using mean power, where is the ratio of longest and shortest links. Combined, these results clarify significantly the centralized complexity of wireless communication problems.

1. Introduction.

How much communication can occur simultaneously in any given wireless network? This basic question addressed here of the capacity of a wireless network has attracted tremendous attention, led by the highly-cited work on non-constructive average-case analysis by Gupta and Kumar [8]. Less is known about algorithmic results on arbitrary instances.

In the problems we consider, we are given a set of links in a metric space. In the uni-directional model, the link represents a directed communication request from a sender to a receiver , each a point in the space. In the bi-directional model, communication is two way, thus both points in the link act as sender and receiver. In either model, a power is associated with each link (which may part of the input, or something the algorithm must find out). When communicating during the same time, two links and interfere with each other. We adopt the physical model or “SINR model” of interference, precisely defined in the Section 2. It is known to capture reality more faithfully than the graph-based models most common in the theory literature, as shown theoretically as well as experimentally [7, 13, 16].

Within this overall framework, a large number of problems can be defined. Even limited to the specific scope of this paper, at least the following dimensions in the problem space can be considered.

First, one can consider fixed versus arbitrary power. In the fixed version, the power that a link must use is a part of the input, whereas for the arbitrary power (or, “power control”) case, choosing the appropriate powers to optimize capacity is a part of the problem. For ease of implementation in distributed settings, research on both fixed and arbitrary power has focused on oblivious power assignments, where the power of a link is a (usually simple) function of the length of the link. We study how well oblivious power assignments can handle the power control problem and provide tight characterization (upper and lower bounds) for certain cases.

Second, there is the capacity problem (also known as the single slot scheduling problem), of finding the maximum number of links that can communicate simultaneously vs. the scheduling problem, which is to partition the links into the smallest number of slots such that each slot can be scheduled at once. We focus mainly on the capacity problem, but study the relationship between capacity and scheduling, specially in the context of the oblivious vs. arbitrary power dichotomy.

Finally, there is the issue of the metric space. The common assumption of the 2-dimensional Euclidean plane is natural, but for wireless transmission it is clearly a grand simplification. Antenna directionality, walls and obstructions, environmental conditions, and terrains all complicate and distort signal transmissions, questioning the geometric assumption. In this work, we are interested in exploring what can be said in the case of completely arbitrary metrics. This relates also to the issue of the path loss exponent (defined in Section 2), a constant that defines how quickly the signal falls off from its source. Most approximation results have depended on the assumption that [17, 14, 2, 6, 10], while it is known that can actually be equal to or smaller than 2 in real networks (see [18], for example).

The aim of our work is to clarify the relationships between the different problem variants, as well as obtaining improved bounds for the fundamental problems.

1.1. Our Contributions.

For general metrics, we provide a -approximation algorithm for the capacity problem for all length-monotone, sub-linear power assignments (these terms are defined in Section 2). In particular, this implies constant factor approximation for uniform, mean-power and linear-power assignments. The results apply for both uni- and bi-directional links. Previously, an -approximation was known only for uniform power in the plane with [10].

We use the above results to study the power control problem through the use of oblivious power, specifically, mean power. For bi-directional links, we achieve a -approximation algorithm improving the previous -approximation for metrics with bounded Assouad dimension (or doubling metrics)[9], where is the number of links. For uni-directional links, we improve and generalize the -approximation of [9] for doubling metrics to a -approximation, where is the ratio between longest to the shortest linklength. This nearly matches the lower bound of for oblivious power assignments [9].

Algorithmic results for capacity imply approximation algorithms for the corresponding scheduling problems with an extra multiplicative factor. In all cases, these improve and/or generalize existing bounds. Among these, the -approximation algorithm for scheduling bi-directional links with power control deserves special mention. It improves on previous poly-logarithmic bounds [12, 3], and we additionally show it to be best possible for any algorithm using oblivious power assignment. This is done by providing an example of links in a metric space that can be scheduled using some power assignment in one slot, yet require slots using oblivious power. Observe that in this construction all links are of equal length, for which uniform power is known to result in a constant-factor approximation in the 2-dimensional setting (with ) [9]. Hence, this also implies that when scheduling, general metrics are strictly harder than doubling metrics.

Another feature of our results is that they work for all positive values of the path loss exponent , generalizing a host of results that required . Finally, noting that the approximation factors we achieve are exponential in (which is a constant for typical values of ), we prove an inapproximability result showing that in general, exponential dependence on the path loss exponent is necessary.

The techniques used can be viewed as a step in the evolution of SINR analysis. The key addition is to treat both the interference caused as well as interference received. We treat the pairwise affectances as a matrix, and build on sparsity properties. In addition, we extend the key technical lemma – the red-blue lemma – of [6, 10] from the plane to arbitrary distance metrics.

1.2. Related Work.

Early work on scheduling in the SINR model was focused on heuristics and/or non-algorithmic average-case analysis (e.g. [8]). In contrast, the body of algorithmic work is mostly on graph-based models that ultimately abstract away the nature of wireless communication. The inefficiency of graph-based protocols in the SINR model is well documented and has been shown theoretically as well as experimentally [13, 16].

In seminal work, Moscibroda and Wattenhofer [15] propose the study of the scheduling complexity of arbitrary set of wireless links. Early work on approximation algorithms [17, 14, 2] produced approximation factors that grew with structural properties of the network.

The first constant factor approximation algorithm was obtained for capacity problem for uniform power in [6] (see also [10]) in with . Fanghänel, Kesselheim and Vöcking [4] gave an algorithm that uses at most slots for the scheduling problem with linear power assignment , that holds in general distance metrics. Recently, Kesselheim and Vöcking [12] gave a distributed algorithm that gives a -approximation for the scheduling problem with any fixed length-monotone and sub-linear power assignment. Our capacity algorithm yields an improved -approximate centralized algorithm.

The situation with power control has been more difficult. Moscibroda, Oswald and Wattenhofer [14] showed that using either uniform or linear power assignment results in only a -approximation of the capacity and scheduling problems. In terms of , their construction gives a -lower bound. Avin, Lotker and Pignolet [1] show non-constructively that the ratio between the optimal capacity with power control and optimal capacity with uniform power is in , for any . Fanghänel et al. [3] showed that any oblivious power scheduling results in a -lower bound. In terms of , this was shown to be in [9]. A -approximation algorithm was given in [9], that holds in any doubling metric where is strictly greater than the doubling coefficient (generalizing the Euclidean setup). This was shown to lead to -approximation in general distance metrics [5].

Much of the difficulty of the power control scenario has to do with the asymmetry of the communication links. Fanghänel et al. [3] introduced a bi-directional version, where both endpoints of each link are senders and receivers (using the same power). They give an algorithm for arbitrary metric that approximates scheduling within a factor. This was improved to a -factor for doubling metrics in [9].

In a very recent breakthrough, Kesselheim obtained a -approximation algorithm for the capacity problem with power control for doubling metrics and for general metrics [11]. In contrast, in the power control part of our paper, we are interested in what can achieved through locally obtainable oblivious assignments, whereas the power assignment in [11] is necessarily non-oblivious.

2. Model and Preliminaries.

Given is a set of links, where each link represents a communication request from a sender to a receiver . The distance between two points and is denoted . The asymmetric distance from link to link is the distance from ’s sender to ’s receiver, denoted . The length of link is denoted simply . Let denote the ratio between the maximum and minimum length of a link. We assume that each link has a unit-traffic demand, and model the case of non-unit traffic demands by replicating the links.

Let denote the power assigned to link , or, in other words, transmits with power . We assume the path loss radio propagation model for the reception of signals, where the signal received at point from a sender at point with power is where the constant denotes the path loss exponent. We adopt the physical model (or SINR model) of interference, in which a node successfully receives a message from a sender if and only if the following condition holds:


where is a universal constant denoting the ambient noise, denotes the minimum SINR (signal-to-interference-noise-ratio) required for a message to be successfully received, and is the set of concurrently scheduled links in the same slot. We say that is SINR-feasible (or simply feasible) if (1) is satisfied for each link in .

Let -Capacity denote the problem of finding a maximum SINR-feasible set of links under power assignment . Let PC-Capacity denote the problem of finding a maximum cardinality set of links and a power assignment that makes these links feasible. Let denote the optimal capacity of a linkset under any power assignment.

The above defines the physical model for uni-directional links. In the bi-directional setting, the asymmetry between senders and receivers disappear. The SINR constraint (1) changes only in that the definition of distance between link changes to

With this new definition of inter-link distances, all other definitions and conditions remain unchanged. For all problems we will consider, when the problem name is prefixed with Bi-, we assume bi-directional communication.

A power assignment is length-monotone if whenever and sub-linear if whenever .

There are two specific power assignments we will be using throughout the paper. The first is , or the uniform power assignment, where every link transmits with the same power. The second is , or the mean power (or “square-root”) assignment, where is proportional to . It is easy to verify that both and are length-monotone and sub-linear.

Affectance.  We will use the notion of affectance, introduced in [6] and refined in [10], which has a number of technical advantages. The affectance of link caused by another link , with a given power assignment , is the interference of on relative to the power received, or

where is a constant depending only on the length and power of the link . We will drop when it is clear from context. Let . For a set of links and a link , let and . Using this notation, Eqn. 1 can be rewritten as , and this is the form we will use. This transforms the relatively complicated Eqn. 1 into an inequality involving a simple sum that can be manipulated more easily.

We will frequently consider the affectances arranged as a matrix , where . Note that feasibility of a set is equivalent to , where is the appropriate affectance matrix.

We will require two simple matrix-algebraic results. Before we state them, we need a few definitions. For a matrix (vector ), let () denote the sum of entries in the matrix (vector). We say that a matrix is -approximately-symmetric if, for any two indices and , for some . For a matrix and vectors and , we will use the notation to mean for all .

The proof of the following two claims can be found in the appendix (Section A.1).

Claim 1.

Let be a matrix that contains only non-negative entries. Assume for some . Then, for any , there are at least rows (columns) of for which .

Lemma 2.

Let be a number and be a positive -dimensional vector. Let be a non-negative -approximately-symmetric -by- matrix such that . Then, .

Signal-strength and robustness.  A -signal set is one where the affectance of any link is at most . A set is SINR-feasible iff it is a 1-signal set. Let be the maximum number of links in a -signal set given a power assignment (and let be the corresponding maximum for bi-directional links). We drop , and when understood from context.

The following results are essential.

Proposition 2.1 ([3],Prop. 3).

Let be set of links and be a power assignment for . Then, for any , . This is constructive.

Lemma 3 ([9]).

Let be links in a -signal set. Then, .

3. Fixed Power in General Metrics.

In this section, we will deal with the -Capacity problem for fixed power assignments. The most widely studied fixed power assignments are oblivious, where the power assigned to a link depends only on the length of the link. Our results are not restricted to oblivious power, but we will see that they do apply for popular oblivious assignments considered in the literature, such as and linear power (where ).

We use the simple greedy algorithm C described in Fig. 1 for -Capacity. Algorithm C is closely related to Algorithm A(c) in [10], differing primarily in that it considers also the outgoing affectances from the given link . Also, the last line was added to ensure feasibility in general metrics; as shown in [10], it is not needed in when .

Algorithm C(L,P)
Suppose the links
are in non-decreasing order of length.

for to do
then add to

Figure 1. Algorithm for the capacity problem

3.1. Constant Factor Approximation for -Capacity.

We will show the following.

Theorem 4.

Algorithm C approximates -Capacity within a constant factor for any length-monotone, sub-linear .

The feasibility of the solution output by C is evident by virtue of the last line of the algorithm. We proceed to show that it gives constant-factor approximation. We first compare the final output to the set .

Lemma 5.

Let and be the sets created by the algorithm C on a given link set. Then, .


Observe that by the algorithm construction, the sum is upper bounded by . Then the Lemma follows from Claim 1 by plugging in . ∎

The main task is to relate the size of the set to the optimal solution. For this, we extend the “blue dominant centers” lemma of [6] (and “blue-shadowed lemma” of [10]) to arbitrary metric spaces and length-monotone sub-linear power assignments.

For a set and link , let and .

Lemma 6 (Red-Blue Lemma).

Let and be disjoint sets of links, referred to as red and blue links in a length-monotone, sub-linear power metric. If and is a -signal set, then there is a blue link such that .

This lemma is a consequence of the following two symmetric lemmas.

Lemma 7.

Let and be disjoint sets of links, referred to as red and blue links in a length-monotone power metric. If and is a -signal set, then there is a set of size at least such that for all , .

Lemma 8.

Let and be disjoint sets of links, referred to as red and blue links in a sub-linear power metric. If and is a -signal set, then there is a set of size at least such that for all , .

We prove Lemma 7 below and relegate proof of the very similar Lemma 8 to the appendix.

Proof of Lemma 7.

For each link , we will assign a set of “guards” . For different , the ’s will be disjoint and each will be of size at most . We claim that each link that remains in after removing all the guard sets satisfies the lemma.

Here’s how we choose the guards. We process the links in in an arbitrary order. Initially set . For each link , add to the link with the sender nearest to (if one exists); also add the link with the receiver nearest to (if one exists; possibly, ); finally, remove the links in from and repeat the loop.

Figure 2. Example configuration of guards and points. Guard (on left) is the nearest blue sender to red point (in center) among those of links larger than .

Since , , by construction. Consider any link . In what follows, we will show that the affectance of any link on is comparable to the affectance of one of the guards of on . Once we recall that the guards are blue, this implies that the overall affectance on from is not much worse than that from .

Since , is non-empty and contains guards and (potentially ). Consider any . Let denote and observe that since , .

We claim that . Before proving the claim, let us see how this leads to the proof of the Lemma. If the claim is true, then by the triangular inequality, Since and the power metric is length-monotone, . Thus, Summing over all links in ,

To prove the claim that , let us suppose otherwise for contradiction. Then, by the triangular inequality, Since was chosen into , its receiver is at least as close to as , that is , and its sender is also at least far as , or . So, . Now, and But since is a -signal set, this is a contradiction by Lemma 3. Hence, any link in satisfies . ∎

Lemma 9.

Let be the set found by the algorithm and . Then, .


By Claim 1, there is a set of size at least such that for all , . By definition, .

We claim that . Suppose otherwise. Then, . Applying Lemma 6 with and , we find that there is a link in that satisfies . The operation of the algorithm is then such that the algorithm would have added the link to , which is a contradiction. ∎

The proof of Thm. 4 is now straightforward.


We can bound . The first inequality is by Lemma 5, the second by Lemma 9, and the last equality is a consequence of Proposition 2.1. ∎

We note that our algorithm applies equally in the bi-directional setting.

3.2. The Necessity of Exponential Dependence on .

In all results achieved in prior work as well as this work, the approximation factor depends exponentially on . Here we show a simple approximation preserving reduction from the maximum independent set problem to prove that this dependence is necessary in general. However, this does not rule out better dependence on for special cases, say for small values of on the plane.

Theorem 10.

For and any fixed , there can be no polynomial time algorithm that approximates -Capacity in general metrics to within a factor better than , unless .


By reduction from the maximum independent set problem in graphs. Given graph , form a link for each vertex , with . If , set , while if , set . It is easy to show that the triangle inequality holds. Set assume and . Any set of vertices is an independent set if and only if the corresponding set of links are SINR-feasible. To see this, first consider a set of vertices that contains at least one edge . Then . So, is not feasible. On the other hand, if is an independent set, then for all ,

The reduction is clearly approximation preserving. Since independent set is -inapproximable unless P=NP (see [19]) for any fixed , the capacity problem is hard to approximate within a factor of . ∎

4. Approximating Power Control Using Oblivious Power.

We will prove the following results.

Theorem 11.

There is a -approximation algorithm for PC-Capacity in general metrics that uses the mean power assignment .

Theorem 12.

There is a -approximation algorithm for Bi-PC-Capacity in general metrics that uses the mean power assignment .

Thus for bi-directional links, we find that mean power is essentially the best possible. For uni-directional links, oblivious power is known to be sub-optimal, but we achieve nearly tight bounds in light of lower bounds in [9].

For simplicity we assume . We shall also assume that the noise is negligible, which is acceptable in the power control case since we can scale the powers to make the noise arbitrarily small. The following simple lemma makes this rigorous.

For any oblivious power assignment and any number define the assignment as . That is, is a simple linear scaling of .

Lemma 13.

Assume a given link set and ambient noise . Then for any oblivious power assignment , there exists a number such that the assignment has the following property: If a set is feasible using assuming zero noise, then there is a set of size which is feasible assuming ambient noise . Also, such an can be found efficiently.


We will be comparing the case where the noise is zero versus the case where the noise is . We will use , , etc. to denote the value of and affectances in the zero noise case, and the un-superscripted versions ( etc.) to denote the noisy case.

First we claim that we can choose such that for all . Recall that depends only on . Thus we can easily choose such that such that for all . Thus (since we have assumed ).

On the other hand, if the noise level is zero then . Consider a set that is feasible under assuming zero noise. Then for all . Now,

The statement of the Lemma now follows from Prop. 2.1. ∎

4.1. Uni-directional Capacity with Power Control via Oblivious Power.

In [9], Halldórsson achieved a approximation factor for uni-directional capacity with power control for doubling metrics via . Using tools developed in this paper, the same approximation can be achieved for general metrics in a straightforward manner. We additionally improve the bound to .

First we need an weaker result on a class of power assignments we call Lipschitz power assignments.

Definition 14.

A power assignment is Lipschitz if there is some constant such that , for any two links with .

We prove the following.

Theorem 15.

Let be any Lipschitz power assignment. Then, for any linkset , .

For Lipschitz power assignments, the above result is the best possible, given the matching lower bound for both uniform and linear power [14, 1].

Before proving Thm. 11, we need a weaker result for Lipschitz power assignments.

Theorem 16.

A set of links is said to contain nearly equilength links if for all , . Let be any Lipschitz power assignment. Then, for any linkset of nearly equilength links, .


First note that for nearly equilength links, the affectance for any Lipschitz assignment varies from that of uniform power by no more than a constant factor. Thus by the signal strengthening property (Proposition 2.1) we safely assume .

Let be an optimal -signal subset of . Let be a power assignment for which is a -signal set. Let be the affectance of link on under uniform power. Then, is the affectance under . Viewed as a matrix where , feasibility of using is equivalent to where .

We claim that for all . To show this assume and we will prove the inequality in both directions. Since , we get . By the triangular inequality and .

Then, . Hence, is -nearly symmetric. By Lemma 2, the set of links with affectance at most is of size at least . By signal strengthening (Prop. 2.1), applied both to and , there is a 1-signal subset in of size at least , while . ∎

Thm. 16 can be used to achieve Thm. 15 (a generalization of a result from [9] to arbitrary distance metrics) in a straightforward manner.

Proof of Thm. 15.

Given an arbitrary linkset, we can divide it into nearly-equilength groups. Let these groups be for . Then by Thm. 16, . Let, . Now,

We now combine the use of mean power with Thm. 15 to get an algorithm with better approximation ratio in terms of the dependence on . To to do this, we modify ideas from [9]. A key idea in [9] was to partition the input links into length based classes. We adopt this idea but employ a more efficient partitioning of the input links and combine it with Thm. 15 for our result.

For this we will need the following lemma, a slight variation of Lemma 4.2 in [9].

Lemma 17 ([9]).

Let be a set of links that are SINR-feasible using some power assignment, and let be a link that is shorter than the links in by a factor of at least . Suppose . Then .

Partitioning the input into levels.  For an input set , assume the shortest link has length . Define non-overlapping sets, for all , where . Let us call these sets levels. We partition into and , where and . (and ) is a union of levels such that intra-level distances vary by no more than while inter-level distances vary by at least . In what follows, we will present an -approximation algorithm that takes as input such a set. Since, , we can claim the same approximation factor for any input set with additional multiplicative factor of .

Algorithm and analysis.  We will adopt the convention that the levels are sorted in increasing order, thus if , links in are longer than links in .


for do
Divide such that lengths
of links within varies by a
factor of at most
for each do

for in increasing order do
add to if both:
1. , and
2. contains no link s.t.


Figure 3. Algorithm for PC-capacity
Lemma 18.

Consider the set selected by . Then .


We partition both and into levels: where and where . Also let . We aim to prove or equivalently .

Now let us define for as the set of links in that fail condition 2 due to links selected from level . In other words,

Let . By Lemma 17,


for some constant .

With this in mind let us investigate what PC does for a single . We find that PC runs on the algorithm suggested by Thm. 15. This is done by partitioning the links in to nearly equilength sets , running algorithm C separately on each and choosing the largest feasible set found. The only difference is due to condition 2, in other words, the approximation factor achieved is not in relation to but rather . Since for any , by construction, we can claim that

for some constant . Using Eqn. 2, this implies that

which yields that , as desired. ∎

To complete the proof of Thm. 11, all we need is to relate to . For , the average affectance from links at the same level is bounded by construction. Additionally, also by the design of the algorithm, one can observe that the total affectance from links at other levels is at most . Now arguing that is within a constant factor of takes the same route we have already seen via Lemma  1 and Proposition  2.1.

4.2. Bi-Directional Capacity with Power Control.

Let denote the optimal bi-directional capacity with respect to power assignment and the optimal bi-directional capacity with respect to any power assignment.

We show that is optimal for bi-directional capacity modulo constant factors.

Theorem 19.

For any linkset , .


Let be a power assignment with which the optimal capacity is attained, and let be the feasible subset in of cardinality . Let be a function defined by . Recall that in the bi-directional setting, , and affectance under is symmetric and given by . Observe that

Since is feasible w.r.t. , it holds that for each link in , , or . Thus, if where is the affectance matrix for , we have that , where is the vector of the values for all . Since is symmetric, , by Lemma 2, and by Claim 1 there is a subset of with at least links such that for each link . Namely, for each link ,

Hence, is a -signal set under . Finally, by signal-strengthening, there is a 1-signal subset in of size . ∎

In light of the fact that C(L) is a constant factor algorithm for by Thm. 4 (which works for bi-directional links as well), Thm. 12 is a simple consequence of Thm. 19 .

5. Scheduling vs. Capacity.

Our capacity results immediately imply results for the corresponding scheduling problems, within a logarithmic factor. We state these below. Recall that the scheduling problem is to the capacity problem as what the graph coloring problem is to the independent set problem.

Corollary 20.

There is an -approximation algorithm for link scheduling under any fixed length-monotone, sub-linear power assignment . Further, for scheduling with power control, by using square-root power, we get both - and -approximation for uni-directional case and -approximation for bi-directional case. All results apply in general distance metrics and for all fixed .

The relationship between capacity and scheduling has remained an open question for some time. For instance, while a constant factor approximation for capacity with uniform power (in doubling metrics) was obtained in [6], a corresponding result for scheduling has been elusive, with a faulty claim in [10]. We show here that scheduling can indeed be more difficult by a logarithmic factor.

5.1. A Lower Bound.

In this section we provide bounds on how well oblivious power assignments can do in a general metric space. Lower bounds of this type have been proved previously for uni-directional links on the line [3, 9], giving and lower bounds. However, these arguments don’t say anything about bi-directional scheduling.

We show a construction of equilength links that are feasible with some power assignment, but using oblivious power, one needs slots to schedule the links. The reader may recall that for the capacity problem, in contrast, -approximation is possible for the bi-directional case via oblivious power (Thm. 19). Since the links have equal lengths, the only possible oblivious assignment is . The construction works for both uni-directional and bi-directional cases.

We start with a single link and define the other links in the link set in relation to . For some suitably large , define sets of links for , such that , and for all , , where is a fixed constant to be defined later. Now, , and the total number of links . It is easy to see that


Furthermore, let and . All other distances are defined by transitivity. Let and .

Theorem 21.

There exists a power-assignment such that is SINR-feasible using , whereas cannot be partitioned into SINR-feasible sets using uniform power .

The following two technical lemmas imply the two claims in Thm. 21. Let the matrix be defined by entries and , and the vector be defined by .

Lemma 22.

There exists a power assignment such that .

Lemma 23.

Take any partition of into sets ( ). Then there exists a link in some such that , thus proving that is not feasible usin