Maximizing Capacity in Arbitrary Wireless
Networks in the SINR Model: Complexity and
Game Theory
Matthew Andrews
Bell Labs, Murray Hill NJ
Email: andrews@research.bell-labs.com
Michael Dinitz
Carnegie Mellon University, Pittsburgh PA
Email: mdinitz@cs.cmu.edu
Abstract—In this paper we consider the problem of maximiz-
ing the number of supported connections in arbitrary wireless
networks where a transmission is supported if and only if the
signal-to-interference-plus-noise ratio at the receiver is greater
than some threshold. The aim is to choose transmission powers
for each connection so as to maximize the number of connections
for which this threshold is met.
We believe that analyzing this problem is important both in its
own right and also because it arises as a subproblem in many other
areas of wireless networking. We study both the complexity of the
problem and also present some game theoretic results regarding
capacity that is achieved by completely distributed algorithms.
We also feel that this problem is intriguing since it involves both
continuous aspects (i.e. choosing the transmission powers) as well
as discrete aspects (i.e. which connections should be supported).
Our results are:
• We show that maximizing the number of supported connec-
tions is NP-hard, even when there is no background noise.
This is in contrast to the problem of determining whether or
not a given set of connections is feasible since that problem
can be solved via linear programming.
• We present a number of approximation algorithms for the
problem. All of these approximation algorithms run in
polynomial time and have an approximation ratio that is
independent of the number of connections.
• We examine a completely distributed algorithm and analyze
it as a game in which a connection receives a positive payoff
if it is successful and a negative payoff if it is unsuccessful
while transmitting with nonzero power. We show that in this
game there is not necessarily a pure Nash equilibrium but
if such an equilibrium does exist the corresponding price of
anarchy is independent of the number of connections. We
also show that a mixed Nash equilibrium corresponds to a
probabilistic transmission strategy and in this case such an
equilibrium always exists and has a price of anarchy that is
independent of the number of connections.
This work was supported by NSF contract CCF-0728980 and
was performed while the second author was visiting Bell Labs in
Summer, 2008.
I. INTRODUCTION
In this paper we consider the problem of maximizing the
number of successful transmissions in the physical SINR
model. In our basic model we are given a set of transmitter-
receiver pairs located in the plane and each has an associated
SINR requirement. The aim is to satisfy as many of the
requirements as possible.
Maximizing the transmission capacity in wireless networks
has been studied in many contexts. Typically this work can
be partitioned along two axes. On one axis we have the two
models that are typically used to model channel conditions.
The simplest case is the unit-disk graph (UDG) model in
which transmissions interfere if and only if they are within
distance 1.
1
A more complex model is the SINR model in
which each transmission is given a power and we assume a
distance-dependent path loss. A transmission is deemed to be
successful if the signal-to-interference-plus-noise-ratio (SINR)
is more than some specified threshold.
On the other axis is the structure of the networks that are
being considered. One option is to look at random networks
under a certain distribution of node placements and transmitter-
receiver pairings. In this case the typical goal is to calculate the
expected capacity of the system and examine how it changes
as the density of the network increases. Another option is to
simply look at a worst-case topology. In this case it makes
no sense to consider some notion of average capacity since
that could depend greatly on what the topology looks like, and
so we are more interested in the complexity of calculating the
optimum capacity and in determining how close we can come to
optimality via efficient algorithms (both centralized algorithms
and distributed protocols).
For random networks, the problem of calculating transmis-
sion capacity has been examined in both the UDG model and
the SINR model by Gupta and Kumar [10]. They showed
that in both cases the average source-sink capacity scales as
Θ(
1
√
n log n
) where n is the number of nodes in the network.
2
For UDGs, the maximum cardinality set of transmissions
corresponds to a maximum independent set in the graph.
Maximum independent set in UDGs was shown to be NP-
1
We remark that there are many variants of the UDG model. For example in
the Tx model of [23] a transmission from u
2
to v
2
suffers interference from
a transmission from u
1
to v
1
if and only if the distance between u
2
to u
1
is
at most the distance from u
1
to v
1
.
2
A difference between Gupta-Kumar and our work (aside from the fact that
they look at random networks whereas we look at worst-case networks) is that
they consider the case of multihop transmissions. (This is of course necessary in
their setting since they consider random source-destination pairs which may not
be supportable with one-hop transmissions.) However, for simplicity we shall
focus on the case of single hop transmissions. (This is for example the case that
arises when we apply our results to the scheduling problems that we discuss
in Section I-E.) We plan to extend our results to multi-hope transmissions in
future work. We believe that this could be done using ideas similar to those used
by Kumar et al. [13] for solving multicommodity flow problems in wireless
networks.
hard in [4]. However, it is known due to arguments about the
geometry of disks that the simple greedy algorithm in which we
continually pick nodes and delete all their neighbors leads to
a 6-approximation for Maximum Independent Set in Unit Disk
Graphs [14]. Morever, for any ε > 0 more complex algorithms
can give a (1+ε)-approximation in polynomial time [12], [17],
[7].
There has been less work on the complexity of calculating
the maximum possible capacity in arbitrary networks under the
SINR model. (We discuss some previous work on this problem
in Section I-D.) We believe that addressing this question is
important for two reasons. First, although analyzing capacity
in random networks is important for determining what level
of transmissions will be possible in completely unstructured
networks, there are many situations where the network will
have some sort of structure and the transmission capacity may
be very different than what is possible in random networks.
In these cases we believe that knowing the complexity of
calculating the maximum number of transmissions is important.
Second, as is well-known the Unit Disk Graph model does
not capture many features of wireless networks. One reason
for this is that receivers will hear interference from all other
transmitters, even if they are far away. A more important reason
is that interference at a receiver is a cumulative effect of
multiple transmitters whereas in the Unit Disk Graph model
interference is simply a local binary property.
Before we can describe our results in detail we must present
our model. We begin with a basic model that allows us to
demonstrate most of our techniques and show what results
can be obtained in this model. We then define a number of
extensions to the model and describe how our results change
in these cases.
A. Basic model
We consider a set of n connections in the plane. Each
connection i has a transmitter t
i
and a receiver r
i
. We let
d(u, v) be the Euclidean distance between two points u and
v. We use d
i
to denote d(t
i
, r
i
) and refer to it as the distance
of connection i. Suppose that a node u is transmitting with
power p. Following [18] we assume that for some parameters
d
0
and α the received signal at another point v is given by
p · min{(d
0
/d(u, v))
α
, 1}. We refer to min{(d
0
/d(u, v))
α
, 1}
as the path loss between u and v and denote it by g(u, v). We
refer to α as the path loss exponent. We make the traditional
assumption that α > 2 since otherwise the total energy received
over the plane would be more than the total energy transmitted.
We assume that for any connection i the distance d(t
i
, r
i
) is
either 0 or else lies between d
min
and d
max
for some parameters
d
max
, d
min
≥ d
0
. The running times and the performance
guarantees of many of our algorithms will depend on the ratio
d
max
/d
min
. For simplicity in the remainder of the paper we
shall normalize distances so that d
min
= 1. Hence in the
following all of our formulas with a dependence on d
max
will
in fact depend on d
max
/d
min
in the unnormalized case.
Let p
i
be the power used by connection i (which can be
zero). We assume that there is a maximum power p
max
with
which any node can transmit. The signal received at receiver
r
i
is given by p
i
g(t
i
, r
i
) and the interference heard from the
other connections is
P
j6=i
p
j
g(t
j
, r
i
). We also assume that
there is some background noise level W and so the signal-
to-interference-plus-noise-ratio (SINR) is (p
i
g(t
i
, r
i
))/(W +
P
j6=i
p
j
g(t
j
, r
i
)). We assume that each connection is for
a single application type such as Voice-over-IP for which
there is a fixed signal-to-noise requirement that we denote
by τ. In other words, connection i is satisfied if and only if
(p
i
g(t
i
, r
i
))/(W +
P
j6=i
p
j
g(t
j
, r
i
)) ≥ τ.
Our aim in this paper is to maximize the number of satisfied
connections, i.e. we wish to choose the transmission power
levels p
i
so as to maximize,
|{i :
p
i
g(t
i
, r
i
)
W +
P
j6=i
p
j
g(t
j
, r
i
)
≥ τ}|.
We refer to this problem as MAX-CONNECTIONS and we
denote the maximum achievable value by OPT.
We remark that if OPT equals n, then the optimum powers
can be found using linear programming. This is because all we
need to do is find powers such that p
i
≥ 0 and,
p
i
g(t
i
, r
i
) ≥ τ(W +
X
j6=i
p
j
g(t
j
, r
i
)). (1)
If such powers exist then clearly any linear programming algo-
rithm will find them. Moreover, there are also many distributed
algorithms for finding these powers. See for example the work
of Yates on uplink power control [22]. However, we are mostly
concerned with situations where it is not possible to support all
connections. In this case linear programming will not work
since we need to make the discrete decision about which
connections to support before we make the continuous decision
about what power levels to use for the supported connections.
We can think of the problem as being one of maximizing the
number of satisfied linear inequalities of the form (1). In general
the problem of maximizing the number of satisfied inequalities
in a linear system cannot be approximated to within a factor
better than n
δ
for some δ > 0. (See Arora et al. [1].) However,
our problem has significant geometric structure. Our aim is to
exploit this structure to get better bounds than the bounds of
[1].
Throughout the paper we shall assume that α and τ are
constants. Hence expressions that utilize O(·) will sometimes
be hiding dependencies on α and τ . In the case when W 6= 0,
we will also make the assumption that d
max
is bounded
away from the absolute distance limit with no outside in-
terference, i.e. there is some constant such that d
max
≤
(1 − )d
0
(p
max
/τ)
1/α
.
B. Results from the basic model
• Our first result is a hardness result. In Section II we show
that MAX-CONNECTIONS is NP-hard and so we should
not expect to obtain a polynomial-time exact algorithm.
• Given that the problem is NP-hard, in Section III we
turn our attention to approximation algorithms for MAX-
CONNECTIONS. Our first algorithm runs in polynomial
time and gives an O(log d
max
)-approximation. For the
case of zero background noise we describe a second algo-
rithm that gives an O(1)-approximation in time n
O(d
2
max
)
.
• The approximation algorithms presented in Section III
are centralized. Although this might be appropriate in a
situation where we are given a network configuration and
we wish to analyze the capacity, centralized algorithms are
unlikely to be useful if we wish to optimize capacity as
a network evolves. Distributed algorithms are much more
likely to be useful. In Section IV we consider the extreme
case of completely decentralized algorithms that do not
exchange any information but instead selfishly maximize
their payoffs in a game that we design in which a strategy
is a transmit power.
We first show that our game does not always have a pure
Nash equilibrium. On the other hand, we show that in
any mixed Nash equilibrium (of which there is always at
least one) the expected number of connections that are
supported is always within a O(d
2α
max
) factor of OPT (i.e.
the price of anarchy is O(d
2α
max
)). Thus if a pure Nash
does exist it is close to optimal.
C. Extended model
We now briefly describe some ways to extend our model
together with the results that we can obtain when these new
features are introduced.
• The first extension is to assume that each connection i has
a weight w
i
and the goal is to maximize the weighted total
of supported connections. In this model we can slightly
modify the proof from the basic model to obtain a similar
O(log d
max
)-approximation algorithm.
• Another extension is to assume that there are multiple
carriers in the system that do not interfere. Each connec-
tion must be assigned to a separate carrier. These carriers
might be different channels in an 802.11 system or they
might be different frequency bands in an OFDM system
such as 3GPP’s Long-Term Evolution (LTE) standard.
In this case we have three decisions to make, namely
which connections should be supported, which powers
should they be assigned, and which channels should they
be assigned. We remark that the third problem can be
thought of as providing a frequency reuse pattern for the
connections. In this model all of the results from the basic
model continue to hold other than losing another constant
factor independent of the number of carriers.
D. Related work
As already mentioned, Gupta and Kumar [10] looked at the
problem of maximizing the number of satisfied connections in
random multihop networks in both the SINR model and the
UDG model. For arbitrary networks under the UDG model,
(1+ε)-approximations were obtained in [12], [17], [7]. In terms
of completely distributed algorithms, game theoretic results
have been obtained in a number of different contexts. In [11],
Huang et al. looked at the problem of maximizing an aggregate
network utility in a situation where each connection’s utility is a
concave function of the connection rate and nodes are allowed
to share pricing information. They show that this distributed
algorithm will converge to a local optimum. In [19], Saraydar
et al. look at a game-theoretic algorithm for choosing powers on
the uplink of a single cell wireless system. In [20], Stolyar and
Viswanathan study fractional frequency reuse algorithms for
joint channel assignment and power control in cellular OFDM
systems and provide a game theoretic algorithm that always
leads to a stable solution. In [2], Bahl et al. provide distributed
algorithms inspired by game theory for the problem of sizing
cells and assigning users to basestations. However, none of this
work and to the best of our knowledge no other work studies
the problem of comparing the quality of a stable solution with
the global optimum.
Previous papers that consider the complexity of capacity
maximization in the SINR model include [3], [5], [9], [15],
[16]. Of these, Goussevskaia et al. [9] is the most closely
related to our work. The authors show NP-hardness and provide
O(log d
max
) approximation algorithms for a similar objective
to ours. They also consider a related objective of minimizing
the number of “rounds” required to serve all connections.
However, a key distinction between our work and [9] is that [9]
assumes that all transmission powers are fixed. In other words
it only addresses the combinatorial aspects of the problem
(deciding which connections should be scheduled at a given
time) and does not consider that the continuous aspects (which
transmission powers should be used). In contrast, our results in
Sections II and III assume that selection of transmission powers
is part of the problem.
The results of [9] were extended to the multihop case by
Chafekar et al. in [5]. The papers [15], [16] showed that we
can schedule all connections in a number of rounds that is only
an O(log
2
n) factor than a lower bound based on an intrinsic
measure of interference at the receivers. In [3], Borbash and
Ephremides present a linear programming formulation of the
problem (but which may not always have polynomial size).
E. Remarks
We remark that throughout this work we focus on a situation
where we are simply trying to maximize the capacity of a set
of conections transmitting at a given time. We do not explicitly
address scheduling issues such as timesharing between different
sets of connections.
However, we note that in many scheduling algorithms, the
scheduling decision involves finding a set of feasible trans-
missions that maximizes some notion of weighted capacity.
For example, the scheme of Kumar et al. [13] for realising
multicommodity flow solutions in multihop wireless networks
involves finding maximum feasible sets of transmissions. In
addition the well-known backpressure algorithm for stabilizing
queues lengths in wireless networks whenever possible (see
[21]) involves finding at each time-step a feasible transmission
set of maximum weight where the weight of a transmission
is derived from the difference between a queue length at the
transmitting node and a corresponding queue length at the
receiver. In Section V-B we mention how our techniques may
be extended to weighted problems. Hence our analysis provides
algorithms for solving these subproblems while at the same
time showing that finding exact solutions to the subproblems
is NP-hard.
Fig. 1. The graph that forms the basis of our NP-hardness reduction.
1.2
1
Fig. 2. The gadget.
II. NP-HARDNESS
In this section we show that the MAX-CONNECTIONS prob-
lem in arbitrary networks under the SINR model is NP-hard.
Our reduction follows the basic strategy of the NP-hardness
reduction for MIS in unit disk graphs. However, the reduction is
somewhat more complicated since we have to deal with the fact
that interference comes from arbitrary distances. The reduction
starts from the NP-hardness of MIS in planar cubic graphs.
Specifically it is known (see e.g. [4]) that MIS is NP-hard in
graphs where all nodes are on the edges of a grid with squares
of size M , edges are of size 1, all nodes have degree at most
3 and each degree 3 node is incident to linear arrays of size at
least M/4 (see Figure 1). Note that any maximum independent
set will include at most every other node along an edge of the
grid. The proof becomes somewhat complex since we need to
show that all power levels will lead to an infeasible solution
for any non-independent set.
A. Gadget
We now describe a gadget that will be used in the eventual
hardness proof. The purpose of the gadget is to represent a
degree-3 node in our grid. We consider three linear arrays of
nodes. (See Figure 2.) Each node serves as both the transmitter
and receiver for a single connection. The first linear array is at
positions (0, 1.2), (0, 2.2), (0, 3.2), . . .. The second linear array
is at positions (1.2, 0), (2.2, 0), (3.2, 0), . . .. The third linear
array is at positions (0, −1.2), (0, −2.2), (0, −3.2), . . .. Lastly
we have a single node at (0, 0). We let the path-loss exponent
α = 2.05, the signal-to-noise ratio threshold τ = 1.00001 and
the maximum power p
max
= 1. We also suppose that each
linear array has a at least ` nodes for some parameter `. The
first result about this gadget follows directly from the chosen
value of τ.
Lemma 1: There is no feasible solution that contains adja-
cent nodes from one of the linear arrays.
Proof: Consider two adjacent nodes from a linear array.
The distance between them equals 1. Consider the transmission
with the smallest power. The SINR for that transmission will
be at most 1. Hence the SNR constraint is not satisfied.
Fig. 3. The feasible configurations.
Fig. 4. The infeasible configurations.
Hence it remains to see what configurations are feasible that
only use alternating members of a linear array. The following
facts can be verified numerically.
Lemma 2: The following configurations are feasible for ar-
bitrarily large `, even when there is a background noise level
of ε = 0.01.
• (0, 0), (0, 2.2), (0, 4.2), . . . , (2.2, 0), (4.2, 0), . . . ,
(0, −2.2), (0, −4.2), . . . (See Figure 3 (left).)
• (0, 1.2), (0, 3.2), . . . (1.2, 0), (3.2, 0), . . . ,
(0, −1.2), (0, −3.2), . . .. (See Figure 3 (right).)
For sufficiently large ` the following configurations are not
feasible, even if there is no background noise level.
• (0, 0), (0, 1.2), (0, 3.2), . . . , (2.2, 0), (4.2, 0), . . . ,
(0, −2.2), (0, −4.2), . . . (See Figure 4 (left).)
• (0, 0), (0, 2.2), (0, 4.2), . . . , (1.2, 0), (3.2, 0), . . . ,
(0, −2.2), (0, −4.2), . . . (See Figure 4 (middle).)
• (0, 0), (0, 2.2), (0, 4.2), . . . , (2.2, 0), (4.2, 0), . . . ,
(0, −1.2), (0, −3.2), . . . (See Figure 4 (right).)
It is easy to see that by making M sufficiently large we can
guarantee that for any node a the interference caused to a by
nodes at distance at least M/4 from a is at most ε. Note that
M will depend only on ε. It is also easy to see from Lemma 2
that in a single linear array it is feasible for every other node
to transmit at p
max
= 1 even with background noise of 0.01.
We can use the above gadget to show NP-hardness in the
following manner. First we can make sure that in the grid
example where MIS is hard every node on the corners of the
grid have degree 3 and every other node has degree 1 or 2. (See
Figure 1.) We then place a copy of the gadget around every
degree-3 node so that the linear arrays correspond to degree 1
or 2 nodes.
For the first direction of the reduction we would like to show
that for any MIS in the original graph, the corresponding nodes
can transmit in our wireless instance. This is easy to see by
using Lemma 2, since close to the center of each gadget we
know that the interference from outside the gadget is at most
ε, so it is still feasible. The only non-obvious case is when
two gadgets meet at the center of a chain, but this is clearly
still feasible since at the center of the chain everything within
distance M/4 is just part of the chain, so is still feasible by
broadcasting at power 1 (which is consistent with the feasible
gadget solution).
Now we need to show that any maximum feasible solution
forms an independent set in the original graph. An important
observation is that we can without loss of generality assume
that in any maximum feasible solution every other node in a
linear array is transmitting. If not, then we could always add to
the number of nodes transmitting in the linear array by turning
off one of the degree 3 nodes. We can repeat this process until
every linear array has half its nodes transmitting. Lemma 2
then implies that we cannot have a degree 3 node transmitting
together with one of its neighbors, and Lemma 1 implies that
no other adjacent nodes are transmitting. This any maximum
feasible set also forms an independent set, completing the
reduction.
III. APPROXIMATION ALGORITHMS
Due to the NP-hardness of our problem we now turn our
attention to approximation algorithms. Ideally we would like
to adapt one of the polynomial time approximation schemes
(that give a (1 + ε)-approximation for any ε) to the SINR
model. However, we are unable to do that, mainly because the
analyses of these algorithms make critical use of the fact that
two transmissions only interfere if the transmitters are close
to each other. However, in the SINR model interference can
occur at arbitrary distances which makes it difficult to directly
adapt these analyses. However, in this section we show that if
d
max
is constant then we can obtain constant approximation
algorithms in polynomial time. More generally, we present an
O(log d
max
)-approximation that runs in polynomial time, and
for the case in which the background noise W = 0 we give an
O(1)-approximation that runs in time O(n
d
2
max
).
Before we present these algorithms we start with a density
lemma that we shall use both for these results and for our
game-theoretic results in Section IV. This lemma states that any
feasible solution can only have a limited number of receivers
in any fixed area.
Lemma 3: Consider a square S with side-length d
0
. In any
feasible solution the maximum number of connections with a
receiver in square S is 3
α
/τ.
Proof: Without loss of generality we assume that the
background noise is 0. Having a non-zero background noise can
only reduce the number of connections that can be supported.
Suppose that all nodes in the feasible solution transmit at
a power such that the received signal is a constant ¯p, i.e.
p
i
min{1, (d
0
/d(t
i
, r
i
))
α
} = ¯p. Let i and i
0
be two connections
such that both r
i
and r
i
0
lie in S.
The interference caused by connection i at receiver r
i
0
is p
i
min{1, (d
0
/d(t
i
, r
i
0
))
α
} ≥ p
i
min{1, (d
0
/(d(r
i
, r
i
0
) +
d(t
i
, r
i
)))
α
} Recall that we assume that either r
i
=
t
i
or d(t
i
, r
i
) ≥ d
0
. In addition, by the geome-
try of the square S we know that d(r
i
, r
i
0
) ≤ 2d
0
.
This implies that p
i
min{1, (d
0
/(d(r
i
, r
i
0
) + d(t
i
, r
i
)))
α
} ≥
1
3
α
p
i
min{1, (d
0
/d(t
i
, r
i
))
α
} ≥ ¯p/3
α
. Hence if there are more
than 3
α
/τ such connections the inteference experienced by all
of them would be enough to prevent the SINR constraint being
satisfied for all connections.
We now remove the condition that the received powers for
every connection are the same. However, in this case the SINR
value for some connection must be worse than it was when
the received signal powers were the same. This implies that
if there are more than 3
α
/τ connections, then for any set of
transmission powers there will be some connection whose SINR
constraint is not satisfied.
Corollary 4: Suppose now that square S has side-length d.
In any feasible solution the maximum number of connections
with a receiver in square S is 3
α
d
2
/τ(d
0
)
2
.
Proof: Divide square S up into subsquares of size d
0
and
then apply Lemma 3.
Lemma 5: Now consider a ball B of radius d. In any feasible
solution the maximum number of connections with a receiver
in ball B is 3
α
· 4d
2
/τ(d
0
)
2
.
Proof: Follows immediately from the fact that any circle
with radius d is contained in a square with side-length 2d.
The following extension of Lemma 3 will also be useful.
Lemma 6: Consider a square S with side-length d. In any
feasible solution the maximum number of connections such that
d(t
i
, r
i
) ≥ d and r
i
is in square S is 3
α
/τ.
Proof: The analysis is almost identical to that of Lemma 3
once we note that in this case d(r
i
, r
i
0
) ≤ 2d ≤ 2d(t
i
, r
i
) for
all i, i
0
.
In the next two theorems we present our approximation
algorithms for the MAX-CONNECTIONS problem in the SINR
model.
Theorem 7: There exists a polynomial time algorithm that
always finds a solution to MAX-CONNECTIONS that is within
a factor O(log d
max
) of optimal.
Proof: We divide all connections into classes based
on distance. Class F
j
contains all connections i such that
d
max
/2
j−1
≥ d(t
i
, r
i
) ≥ d
max
/2
j
. (We remark that a similar
decomposition was used in [9].) Note that in the optimal solu-
tion there must exist a j such that F
j
contains OPT/ log d
max
connections. In the following we will consider each j in turn
and obtain a constant approximation for the connections in
F
j
only. We focus on a j for which d
max
/2
j
≥ d
min
. The
connections for which d
t
i
,r
i
= 0 can be handled similarly.
We now divide the problem into squares of side d
max
/2
j
.
(See Figure 5.) We refer to these squares as j-squares. From
each j-square S, if there is at least one receiver in S then we
choose one arbitrarily, and restrict ourselves to the problem
on these connections. Note that Lemma 6 implies that each j-
square only contains at most 3
α
/τ receivers from the optimal
solution on F
j
, so as long as we can support at least a constant
fraction of our chosen connections we are still within a constant
of the optimal solution on F
j
.
We now restrict our attention to 1 out of every k
2
j-squares
in an evenly spaced pattern for some parameter k, i.e. squares
located at the same coordinates mod k. (See Figure 5). We can
partition the plane into k
2
such sets of squares. We show that
in each set we can support one connection in each square, so
by taking the best set we are only losing another k
2
factor.
Consider some j-square S, and consider the set I of j-
squares in the same pattern set that are offset from S by exactly
