ABSTRACT
Utility-based Sensor Selection ∗
Sensor networks consist of many small sensing devices that monitor
an environment and communicate using wireless links. The
lifetime of these networks is severely curtailed by the limited battery
power of the sensors. One line of research in sensor network
lifetime management has examined sensor selection techniques, in
which applications judiciously choose which sensors’ data should
be retrieved and are worth the expended energy. In the past, many
ad-hoc approaches for sensor selection have been proposed. In this
paper, we argue that sensor selection should be based upon a tradeoff
between application-perceived benefit and energy consumption
of the selected sensor set.
We propose a framework wherein the application can specify the
utility of measuring data (nearly) concurrently at each set of sensors.
The goal is then to select a sequence of sets to measure whose
total utility is maximized, while not exceeding the available energy.
Alternatively, we may look for the most cost-effective sensor
set, maximizing the product of utility and system lifetime.
This approach is very generic, and permits us to model many applications
of sensor networks. We proceed to study two important
classes of utility functions: submodular and supermodular functions.
We show that the optimum solution for submodular functions
can be found in polynomial time, while optimizing the costeffectiveness
of supermodular functions is NP-hard. For a practically
important subclass of supermodular functions, we present an
LP-based solution if nodes can send for different amounts of time,
and show that we can achieve an O(logn) approximation ratio if
each node has to send for the same amount of time.
Finally, we study scenarios in which the quality of measurements
is naturally expressed in terms of distances from targets. We show
that the utility-based approach is analogous to a penalty-based approach
in those scenarios, and present preliminary results on some
practically important special cases.
∗ This material is based upon work supported by the National Science
Foundation under Grants No. 0520235 and 0121778. Any
opinions, findings and conclusions or recommendations expressed
in this material are those of the author(s) and do not necessarily
reflect the views of the National Science Foundation (NSF).
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IPSN’06, April 19–21, 2006, Nashville, Tennessee, USA.
Copyright 2006 ACM 1-59593-334-4/06/0004 ...$5.00.
Fang Bian David Kempe Ramesh Govindan
Department of Computer Science
University of Southern California
Los Angeles, CA 90089
{bian, dkempe, ramesh}@usc.edu
Categories and Subject Descriptors: C.3 [Special-purpose and
Application-based Systems]: Real-time and Embedded systems,
F.2.1 [Analysis of Algorithms and Problem Complexity]: Optimization.
General Terms: Algorithms, Theory.
Keywords: Sensor networks, Utility, Sensor Selection, Complexity,
Sub-modular, Super-modular.
1. INTRODUCTION
Sensor networks consist of many small sensing devices that monitor
an environment and communicate using wireless links. The
limited energy of sensor nodes curtails the lifetime of sensor network
deployments. Therefore, a large body of literature has examined
methods for extending sensor network lifetime by carefully
managing communication and computation resources.
One thread has focused on techniques to maximize sensor network
lifetime by using routing or topology control [1, 2, 3, 4, 5, 6,
7, 8]. Somewhat orthogonal to this thread is the approach of asking
which and how much data can be collected over the lifetime of a
network. Work on this question is usually centered around one of
two paradigms: (1) Maximize the total amount of data collected, or
(2) Collect data from all sensors for as long as possible. The first
of these assumes that all data is equally interesting or important to
the application. In particular, optimizing for the total amount of
data will give undue preference to data measured close to a base
station, as that data can be extracted at relatively low cost. On the
other hand, the second objective is based on the assumption that
data collection is only worthwhile if data from all sensors is being
collected, and as soon as one node’s energy is depleted, the network
may as well not collect any data.
Clearly, both approaches are oversimplifications of reality. Which
data is useful depends heavily upon the specific application and its
needs. In many cases, individual sensor readings will be of little
use; the appeal of sensor networking lies in the ability to aggregate
and correlate sensor readings from different locations. In other scenarios,
readings from a group of proximate sensors may be considered
redundant, and it would suffice to obtain a reading from one
of them.
In this paper, we argue that the algorithms for deciding which
data to retrieve should be sufficiently generic to let the application
specify how useful particular measurements are. We call the usefulness
of a concurrently measured sensor set S its utility u(S). Given
a network topology and application-specified utility function u(·),
it is then the algorithm’s decision how to trade off the utility and
energy consumption of sensor sets in an optimal way, and maximize
the total utility extracted from the network until the network
ceases to function. Such a utility-based sensor selection approach
has also been proposed recently by Byers and Nasser [9].
Ideally, one would like to be able to find an (approximately) optimal
sequence of sets and associated communication scheme to
measure for arbitrary monotone utility functions. This goal seems
very ambitious: as we show, the problem of selecting an optimal
sequence of sets is NP-hard in many settings. In this paper, we
explore three natural and practically important classes of utility
functions in more detail. Specifically, we focus on submodular
functions (with returns for additional sensors diminishing for larger
sets), supermodular functions (with returns increasing for larger
sets), and a general framework of geometric covering objectives.
In the latter case specifically, we show that the utility-based approach
is analogous to a penalty-based approach that characterizes
the penalty of a sensor set as its collective distance from the targets
to be measured.
We show that the optimum sequence of sets for submodular functions
can be found in polynomial time, while optimizing the costeffectiveness
of supermodular functions is NP-hard. For a practically
important subclass of supermodular functions, we present an
LP-based solution if nodes can send for different amounts of time,
and show that we can achieve an O(logn) approximation ratio if
each node has to send for the same amount of time. Finally, for geometric
covering objectives, we show that finding the best sensor
set is NP-hard, and unless P=NP, the optimum solution cannot be
approximated to within better than a multiplicative factor of 1.822.
The paper is structured as follows. Section 2 formally describes
the utility-based sensor selection problem. Sections 3 and 4 then
examine the feasibility of solving several variants of the utilitybased
sensor selection problem for submodular and supermodular
functions. Section 5 discusses geometric coverage objectives. Finally,
Section 6 discusses related work and Section 7 summarizes
our contributions.
2. NETWORK AND PROBLEM SETUP
2.1 Network Model
Formally, our network model can be described as follows. The
network is considered as a directed graph G = (V,E) with n = |V |
nodes including a special root node r ∈ V . Each non-root node v
has a finite initial energy Ev. If node v is to sense data, it incurs
a sensing cost σv. To transmit data along the link e = (v,w) ∈ E,
node v incurs a transmission cost of τe per unit of data. The total of
sensing cost and transmission cost may not exceed a node’s energy.
When the two are equal, we say that the node’s energy is depleted.
This network model is consistent with current practice in sensor
network deployment. Most existing deployments, including the
James Reserve habitat monitoring network [10], the Great Duck Island
network [11, 12], and the Extreme Scaling network [13] are
tiered: they consist of a large number of small battery-powered
motes, sending data (perhaps after some local processing) to a smaller
number of well endowed upper-tier 32-bit embedded nodes (e.g.,
Stargates). The small-form-factor motes are most often battery
equipped, and thus constrain the system lifetime, while the uppertier
nodes are much more powerful, and rarely impose additional
constraints. For our purposes, this means that once data has reached
the upper-tier, it can be extracted from the network. This allows us
to consider all upper-tier nodes as one base station or root, for the
purposes of theoretical treatment.
This network model is also consistent with at least one proposed
sensor network architecture [14]. That architecture proposes to
place application-specific functionality in the upper-tier nodes. Applications
can generically task collections of motes. Each mote
senses the environment, processes the sensed data as specified in a
task, and returns the data to an upper-tier node. Indeed, our problem
formulation is motivated by this architecture: applications may use
our utility-based sensor selection framework to determine which
set(s) of sensors to task.
2.2 Utility-Based Sensor Selection
We are now ready to characterize the optimization problem of
utility-based sensor selection. In addition to the network with costs
and initial energies for nodes, we are given a monotone utility function
u : 2 V → R + . The meaning of this function is that if all nodes of
S are measured (nearly) concurrently, then the application derives
a benefit of u(S) from this measurement. The goal is to calculate a
sequence of measurements to give the maximum total benefit until
the sensor nodes’ energy is depleted.
Utilities naturally capture application requirements in many practical
sensor network scenarios. For example, in a beam-forming
application, a non-collinear set of sensors has a higher utility than
one that is collinear. In a surveillance application, a set of (approximately)
evenly spaced sensors around the periphery of the sensed
area has a higher utility than a clustered set of sensors in one corner
of the sensed area. In a structural monitoring application, a set of
sensors in which none are on the nodes (nulls in the structural mode
shapes) has a higher utility than a set where some are.
More formally, a feasible solution consists of a collection S =
{S1,...,Sk} of subsets of V to be measured, with a non-negative
amount of data ti to be transmitted from each set Si. In addition, the
solution must specify a flow f of value ∑i:v∈Si ti from each node v
to the root node r, such that the total cost of flows and sensing does
not exceed any node’s energy. (This flow captures the delivery of
all relevant data from node v to the root.) The goal is then to find a
feasible solution of maximum total utility.
We can state this as a linear program (with exponentially many
variables) as follows. For each set S ⊆ V , we have a variable tS
specifying the amount of data sent from S to the root. (Here, we
denote by δ + (v) and δ − (v) the set of outgoing and incoming edges
of v, respectively.)
Maximize ∑S⊆V u(S) ·tS
subject to ∑e∈δ + (v) fe − ∑e∈δ − (v) fe = ∑S:v∈S tS ∀v �= r
∑e∈δ + (v) τe fe + ∑S:v∈S σvtS ≤ Ev ∀v �= r.
The first constraint characterizes flow conservation: at every node,
the total amount of flow leaving exceeds the amount entering exactly
by the amount of data that the node itself is sending. The
second constraint simply states that the total of transmission and
sensing cost for no node exceeds the available energy.
Notice that the LP formulation does not contain a “time” component.
Indeed, the order in which the sets transmit their data does
not affect the total utility derived from a sequence. It is also not difficult
to see that the flow f can be decomposed into different flows
f S for each sending set S, while keeping the total amount ∑S⊆V f S
transmitted along any link e the same as fe. Hence, we are justified
in considering the simplified formulation above.
We term the problem as formalized above the Utility-based Sensor
Selection Problem (USSP). Notice that the formulation is very
similar to that proposed by Byers and Nasser [9], but the focus of
our work is significantly different from theirs (Section 6). Several
natural variants of the problem may be of practical importance:
• In the Integral Utility-based Sensor Selection Problem (IUSSP),
all data transmission amounts tS are required to be integral.
This variant is important in applications which require data
collection in discrete data units (for example, [15] describes
a scenario in which a structure’s response needs to be sam-
pled for no less than 1 second at high enough frequency in
order to have enough samples to compute a FFT).
• In the Utility-based One Set Sensor Selection Problem (UOSSP),
the amount of data is allowed to be positive for at most one
set, i.e., we are interested in collecting data only from one
set. This case is of importance in determining the set with
the best utility-to-energy tradeoff.
• Naturally, the previous two variants can be combined, by requiring
that only a single set transmit data, and do so for an
integral amount of time. We then obtain the Integral Utilitybased
One Set Sensor Selection Problem (IUOSSP).
Notice that even the LP for the basic USSP problem is unlikely
to be solvable explicitly for arbitrary utility functions, due to its
exponential size. Indeed, even specifying a utility function completely
would require giving a value for each set S, and thus space
exponential in the number n of sensors. These two observations
motivate studying restricted classes of utility functions of practical
importance. In the following sections, we investigate more closely
submodular and supermodular functions, as well as utility functions
applicable to geometric coverage settings.
3. SUBMODULAR UTILITY FUNCTIONS
In game theory, submodular functions are frequently studied as a
natural restriction of utility functions [16]. Recall that a function u :
2 V → R is submodular if u(S1 ∪ S2) + u(S1 ∩ S2) ≤ u(S1) + u(S2)
for all sets S1,S2. The practical importance of submodular functions
is more evident from an equivalent characterization: u is submodular
if and only if u(S1 + v)−u(S1) ≤ u(S2 + v)−u(S2) for all
sets S2 ⊆ S1 and elements v /∈ S2. This inequality captures the idea
of “diminishing returns”: the benefit of adding the element v to a
set is non-increasing as the set that v is added to increases. In sensor
networks, utility functions will be submodular if the data measured
by different sensors is (partially) redundant; it will cease to be so if
multiple measurements yield superlinear benefit.
For example, when sensors are deployed for measuring the average
temperature, one natural notion of utility is the expected reduction
in variance of the estimate of average temperature. Under
some natural assumptions, this function is submodular. In order to
maximize the system lifetime utility, one will then be trading off
accuracy in the estimate against taking measurements for a longer
time.
In this section, we show that USSP can be solved in polynomial
time for arbitrary submodular functions. The key observation is
that, without loss of generality, an optimal solution never measures
two sensors at the same time. This observation then allows us to
reduce the size of the linear program to polynomial, and solve it
explicitly.
LEMMA 1. Without loss of generality, the optimum solution to
USSP only retrieves data from singleton sets, i.e., tS = 0 for all
non-singleton sets S.
Proof. Let ((tS),( fe)) be an optimal solution for USSP, and assume
that the set S = {v1,v2,...,vk} (with k ≥ 2) transmits tS > 0
amount of data. Consider a solution which sets t ′ S := 0, and instead
increases t ′ {vi} := t {vi} +tS for i = 1,...,k. Each node v in the new
solution still needs to transmit the same total amount of data as before,
so f is still a valid flow, and the new ((t ′ S ),( fe)) is a feasible
solution.
From the diminishing returns property, we can inductively derive
that u(S) ≤ ∑ k i=1 u({vi}), and as the tS ′ values for all other sets S′
remain unchanged, the total utility of the solution changes by exactly
tS · (∑ k i=1 u({vi}) − u(S)) ≥ 0. In particular, the new solution
is at least as good as the original one. By continuing in this way,
we eventually arrive at a feasible solution in which only singleton
sets have non-zero amounts of data transmitted, and which has at
least the same total utility.
Based on the observation in the lemma, we can remove all variables
for non-singleton sets from the LP for USSP, arriving at the
following modified LP:
Maximize ∑v∈V u({v}) ·tv
subject to ∑e∈δ + (v) fe − ∑e∈δ − (v) fe = tv ∀v �= r
∑e∈δ + (v) τe fe + σvtv ≤ Ev ∀v �= r.
This new LP has polynomial size, and can thus be solved explicitly
in polynomial time. Notice also that it is enough for the
application to specify the utilities of singleton sets. Thus, our observation
allows us to circumvent the problem of needing exponential
space to describe the utility function completely — a complete
description is not necessary to find the optimum solution.
4. SUPERMODULAR UTILITY FUNCTIONS
Supermodularity in a sense captures the opposite of submodularity:
the benefit of individual nodes is larger as they are added to an
already larger set; more generally, the benefit from combining two
(disjoint) sets is at least as large as the sum of the individual benefits.
Formally, we say that a function u : 2 V → R is supermodular
if u(S1 ∪ S2) + u(S1 ∩ S2) ≥ u(S1) + u(S2) for all sets S1,S2 ⊆ V .
In sensor network applications, supermodular functions will occur
when the application requires the combination of diverse data measured
(nearly) simultaneously.
For example, in an aquatic monitoring application, sensors of
different types (temperature, light, chlorophyll, ...) may be deployed
in an expanse of water. An application may be interested
in a correlated measurement of all light sensors, a correlated measurement
of all sensors at depth 10ft, and a measurement of all
sensors that had measured a certain phenomenon the previous day.
Naturally, there may be overlap between these sets: for instance, a
light sensor may be at a depth of 10ft, and have measured the phenomenon
the previous day. If the application is such that missing
even one of the relevant sensors makes the measurement (nearly)
useless, for instance because correlations are to be analyzed in detail,
then the utility derived from a particular set S is simply the
number of categories (light, 10ft, phenomenon) that are completely
contained in S. This function is supermodular - in fact, it is a special
case of the category of set-weighted functions studied in more
detail below.
Notice that a function can be submodular and supermodular at
the same time; this happens exactly when the function is of the
form u(S) = ∑v∈S u({v}), i.e., the utility of any set is simply the
sum of utilities of the individual elements in the set. Though one
can solve the same LP used for the sub-modular case here, notice
that the resulting optimum singletons can be combined freely to get
another optimum solution.
The property of supermodularity allows us to infer an interesting
property of the optimal solution: without loss of generality, the
sending sets are nested. In other words, if S1,S2 are sets sending
data, then either S1 ⊆ S2, or S2 ⊆ S1.
LEMMA 2. Without loss of generality, the optimum solution for
USSP with a supermodular utility function consists of nested sets.
1
r
ˆv (E ˆv = k)
1 1 1 1
v1 v2 v3 v4
Figure 1: SETCOVER graph with n = 4
Proof. Let ((tS),( fe)) be an optimal solution for USSP, and assume
that there are sets S1,S2 such that neither S1 ⊆ S2 nor S2 ⊆ S1,
yet tS1 > 0 and tS2 > 0. Without loss of generality, assume that tS1 >
tS2 . Consider a solution which instead sets t′ S2 = 0, t′ = tS1 − tS2 ,
S1
and t ′ S1∪S2 = tS1∪S2 +tS2 , as well as t′ = tS1∩S2 +tS2 . In the new
S1∩S2
solution, each node still needs to transmit the same amount of data,
so ((t ′ S ),( fe)) is still feasible.
Since each set other than S1,S2,S1 ∪S2, and S1 ∩S2 still transmits
the same amount of data, the change in the total system utility is
u(S1 ∪ S2) ·tS2 + u(S1 ∩ S2) ·tS2 − u(S1) ·tS2 − u(S2) ·tS2
= tS2 · (u(S1 ∪ S2) + u(S1 ∩ S2) − u(S1) − u(S2))
≥ 0,
by the supermodularity of u. This reduces the number of pairs
(S1,S2) of non-nested sets by one, as it can be easily verified that
whenever S1 ∪ S2 is not nested with some set S ′ , then at least one
of S1,S2 is not nested with S ′ , either. Thus, we can continue this
process inductively, and arrive at a nested solution of at least the
same total utility.
While this lemma shows that the optimum solution gathers data
from at most n sets, it unluckily does not restrict the class of sets
that these could be drawn from, and thus, does not restrict the size
of the LP. Hence, it does not suggest a polynomial-time algorithm.
Indeed, we conjecture that USSP is NP-complete for supermodular
functions. While we currently have no proof of this conjecture, we
can show that the version requiring integral data amounts to be sent
is not only NP-hard, but cannot be approximated unless P=NP.
THEOREM 3. IUSSP with supermodular utility functions is NPhard.
Unless P=NP, it cannot be approximated to within any multiplicative
factor.
Proof. We will prove that it is NP-hard to decide if the system admits
a solution with positive total utility. That proves both the NPhardness
and approximation hardness result. We reduce from the
decision version of SETCOVER: Given a universe U = {1,...,n},
and a collection of subsets Si ⊆ U for i = 1,...,m, is there a collection
of k sets Si j covering the entire universe, i.e., such that
�kj=1 Si j = U.
Given an instance of SETCOVER, we construct an instance of
IUSSP with a supermodular utility function. The graph has one
node vi for each set Si, as well as a bottleneck node ˆv and the root
r. There is a directed link from each vi to ˆv, and from ˆv to r. All
links e have transmission cost τe = 1, and all sensing costs are 0.
The initial energy of each vi is 1, while the initial energy of ˆv is k.
The resulting graph for four sets is shown in Figure 1.
To complete the reduction, we need to specify the utility function.
For any node set T ⊆ {v1,...,vm}, we define the utility u(T ) :=
2 |T | if �
i:vi∈T Si = U, and u(T ) := 0 otherwise. This reduction can
be computed in polynomial time. The utility function, although
defining the value for 2m subsets, can be specified in polynomial
space (and time) by giving the collection of sets Si. It is also not
difficult to verify that u is in fact supermodular, by considering the
three cases that neither T1 nor T2 defines a set cover, that exactly
one of them does, and that both of them do.
We now claim that a positive total utility can be achieved in this
setting if and only if the SETCOVER instance had a solution of size
at most k. If {Si j } is a set cover of size at most k, then it is easy
to see that the node set {vi j } leads to a feasible solution, of utility
2k > 0. Conversely, if {vi j } is a node set contributing total utility
to an IUSSP solution, we first observe that by feasibility, it has size
at most k. Furthermore, since the utility is positive, {Si j } must be
a SETCOVER by the definition of u. Thus, we have established the
claim, proving both hardness results.
Notice that the same hardness results carry over to the case when
we are seeking a single set to send an integral amount of data, i.e.,
the integral version of UOSSP.
4.1 Set-weighted functions
While Theorem 3 shows a very strong hardness result for IUSSP
with supermodular functions, the functions it uses in the reduction
are arguably not very natural in the context of sensor networks. In
particular, one would assume that most utility functions specified
by applications have a more explicit compact representation. Here,
we focus on a further restriction on supermodular functions.
We term the class of functions set-weighted utility functions. A
set-weighted utility function is characterized by a collection P of
sets, and for each set P ∈ P a non-negative weight wP. The utility
of a set S is then defined as u(S) = ∑P⊆S,P∈P wP. If the collection
P is small enough (for instance, polynomial in n), then it provides
a natural and compact representation of u. Also notice that any
function u defined in this way is always supermodular.
For set-weighted utility functions, we can in fact find the optimum
solution to USSP by solving a linear program. The insight for
this linear program is based on Lemma 2: the optimum solution for
any supermodular utility function without loss of generality consists
of a nested collection of sets. In particular, we can think of
this collection as being ordered by decreasing size. Then, any solution
can be fully characterized by specifying how much data each
node sends, or, equivalently, how long each node sends. Suppose
that up to some time t, all nodes in S still send data (or, the nodes
in S each send t or more units of data). Then, these nodes up to
time t contribute utility t · u(S) = t · ∑P⊆S,P∈P wP. Based on this
observation, and changing the order of summation in the objective
function, we can rewrite the linear program for USSP as follows:
Maximize ∑P∈P wP ·tP
subject to tP ≤ sv ∀P ∈ P,v ∈ P
∑e∈δ + (v) fe − ∑e∈δ − (v) fe = sv ∀v �= r
∑e∈δ + (v) τe fe + σvsv ≤ Ev ∀v �= r.
Notice that we are introducing variables sv for the amount of data
sent by each node v. By expressing the utility function in terms of
contributions by subsets, we managed to eliminate the terms for
sets other than the P ∈ P. By solving this polynomial-sized LP,
we can find an optimal solution to USSP in polynomial time.
4.2 Set-weighted single set selection
In the context of understanding the solutions to the USSP problem,
an important question is which single set is most cost-effective,
i.e., gives the largest utility for its energy consumption. In other
words, which set S maximizes the product of the set’s utility u(S)
and the lifetime of the system if only S sends data? This is the problem
termed UOSSP above. Here, we show that even for the special
case of set-weighted supermodular functions, the UOSSP problem
is NP-hard.
THEOREM 4. UOSSP with set-weighted supermodular functions
is NP-hard.
Proof. We will give a reduction from the densest k-subgraph problem
[17, 18]: Given a graph G = (V,E) and a number k, as well as
a density requirement ρ, is there a subset S ⊆ V of size at most k
containing at least ρ ·k edges. (Here, we say that S contains an edge
e if it contains both of its endpoints.)
Given an instance of the densest k-subgraph problem, we construct
an instance of UOSSP with a set-weighted utility function as
follows: The graph G ′ = (V ′ ,E ′ ) consists of a sink node r, two bottleneck
nodes ˆv1, ˆv2, and a node v for each v of the given graph G.
Each such node v is connected to the bottleneck node ˆv2, and both
bottleneck nodes are connected to the sink r. All sensing costs are
0, and all transmission costs are 1. All nodes have initial energy of
1, with the exception of node ˆv2, which has initial energy of k.
The collection P contains each pair (u,v) of G ′ such that (u,v) ∈
E is an edge of the given graph, and assigns them a weight of 1.
In addition, the node ˆv1 is in P as a singleton set, having very
large weight 1 + |E| · k. We claim that the given graph contains a
k-subgraph of density at least ρ if and only if there is a node set to
sense that gives total utility at least 1 + k · (|E| + ρ).
For the first direction, observe that if the set S contains at most k
nodes and has density ρ, then sending one unit of data each from
nodes ˆv1 and from all nodes v ∈ S gives the desired total utility.
Conversely, we first observe that the maximum utility that can be
obtained without including ˆv1 is at most k|E|, as the nodes can send
at most k units of data, and the utility of any such set of nodes is
at most |E|. Thus, the optimum solution will always send one unit
of data from ˆv1, and is thus also forced to send at most one unit
of data from each other node v. But then, it will always send data
from exactly k nodes other than ˆv1 (else, some of the energy of ˆv2
would go unused). Since the total utility was assumed to be at least
1 + k · (|E| + ρ), the utility of the set S of nodes other than ˆv1 must
be at least ρ · k. This in turn means that the set S contains at least
ρ · k edges in G. This proves the correctness of the reduction, and
thus the NP-hardness of the problem.
While the UOSSP problem for set-weighted supermodular functions
is NP-hard to solve optimally, it can be approximated to within
a factor of O(logn) within polynomial time, using an LP-rounding
based approach. The algorithm first solves the corresponding linear
program for USSP (which is also the natural linear program for
UOSSP) in polynomial time, and then considers all candidate node
sets Cx of the form Cx = {v ∈ V | sv ≥ x}. Among all such sets Cx,
it chooses the one maximizing the product of u(Cx) and the system
lifetime if all of Cx sends. Notice that the algorithm only needs to
look at such sets Cx for values x = sv for some v. Thus, at most n
different sets need to be considered. For each set, calculating the
system lifetime is done in turn by solving the corresponding flow
LP.
The proof of the O(logn) approximation guarantee is based on
the following useful lemma.
LEMMA 5. If x1 ≥ x2 ≥ x3 ≥ ... ≥ xk ≥ 0 are k real numbers
with associated non-negative rational weights w1,w2,...,wk, then
maxn i=1 (xi · ∑ i j=1 w j) ≥ ∑ki=1 wixi , where Hk = Hk
∑ k i=1 1/i denotes the
kth Harmonic number.
Proof. To prove the claim, we first consider the case when all wi =
1 (thus, ∑ i j=1 w j = i). Let i∗ be the index maximizing xi · i. Thus,
we have that xi ≤ (i∗ · xi∗)/i, and
∑ k i=1 xi
Hk
≤ i ∗ · xi ∗ · ∑k i=1 1/i
Hk
= i ∗ · xi ∗ = maxi(i · xi).
If the wi are not all equal to 1, then we first multiply all of them
by the common denominator to make them integers — notice that
multiplying all weights by the same constant does not affect the
validity of the inequality to be proved. Then, if the number xi has
weight wi ∈ N, we replace it by wi copies of the number xi, each
with weight 1. Notice that this leaves both the left-hand side and
right-hand side in the inequality max k i=1 (xi · ∑ i j=1 w j) ≥ ∑k i=1 wixi
Hk
unchanged; for the right-hand side, this follows because wixi is re-
placed by ∑ wi
j=1 xi, and for the left-hand side, it can be seen that the
maximum after the replacement is attained at an index i ′ such that
xi ′ is the wi th copy of the number xi.
Using this lemma, we can state and prove the approximation
guarantee for the LP-rounding algorithm.
THEOREM 6. The above approximation algorithm achieves an
O(logn) approximation ratio.
Proof. First, the utility of a fractional solution that can use at most
one set is certainly upper-bounded by the solution that is allowed
to use multiple sets, i.e., the value of the LP-solution of USSP is an
upper bound on the optimum UOSSP value.
If the algorithm selects a set Sx, then the lifetime of Sx is at least
x (since the fractional solution was able to send at least x units of
data for each node in Sx). Thus, the algorithm obtains total utility
at least
x · u(Sx) = x · ∑P⊆Sx,P∈P wP.
If we sort the sets P ∈ P by non-increasing tP values, numbering
them P1,...,Pk, then we can rewrite and lower-bound the algorithm’s
total utility as maxk i=1 tPi · ∑i wPj j=1 .
On the other hand, the optimum fractional solution obtains total
utility ∑ k tPi
wPi i=1 . By the above lemma, the ratio between these two
quantities is bounded by Hk = O(logk), which in turn is O(logn)
whenever the size of the collection P is polynomial in n.
If we want to improve the approximation guarantee beyond the
factor O(logn) proved above, the approach will have to be based on
a different upper bound from the one given by the USSP LP. (Notice
that the USSP LP is also the natural LP relaxation for UOSSP.) For
the family of examples depicted in Figure 2 exhibits an O(logn) integrality
gap in the LP. The graph has n nodes in addition to the root.
Each node is connected to the root through an edge of transmission
cost 1. All the other n sensor nodes have sensing cost 0, unit initial
energy 1, and weights 1,1/2,1/3,...,1/n. The fractional solution
sends 1/i units of data from each node i, for a total utility of Hn.
On the other hand, the UOSSP solution can only chose one set, all
of whose nodes have to send the same amount of data. Since any
set of i nodes can send at most 1/i units (one of them must be a
node with energy at most 1/i), the maximum utility for UOSSP is
1, proving an integrality gap of Hn = Ω(logn).
4.3 Greedy Algorithms
The O(logn) LP-rounding algorithm given above implicitly shows
that selecting the best single set gives utility within O(logn) of selecting
the best sequence. For many types of problems (such as
maximizing the value of a submodular monotone function f (S) on
sets S), simple greedy algorithms based on adding or removing one
r
1 1 1
E : 1 E : 1/2 E : 1/3 E : 1/n
Figure 2: O(logn) Integrality Gap Example
element at a time lead to optimal solutions or good approximations.
Here, we show that neither the greedy addition nor removal algorithm
will give a good approximation for supermodular functions.
Specifically, we consider the following two algorithms. In the
description, we let tS be the maximum amount of data that the set S
can send if no other nodes are sending data.
Addition Algorithm: Start with the empty set S0 = /0. In each
iteration i (until all the nodes are selected), always add the node
vi such that the total utility tSi−1∪{vi} · u(Si−1 ∪ {vi}) is maximized.
(That is, set Si := Si−1 ∪ {vi}.) Output the best of all the Si.
Removal Algorithm: Start with all nodes S ′ 0 = V . In each iteration
i (until no nodes are left), always remove the node v ′ i such that
the total utility tS ′
i−1 \{v ′ i } · u(S ′ i−1 \ {v′ i }) is maximized. (That is, set
S ′ i := S′ i−1 \ {v′ i }.) Output the best of all the S′ i .
Even if the two algorithms are combined (i.e., the better of their
solutions is output), the approximation guarantee can be as bad as
Ω(n). The example has a network consisting of the sink r, a bottleneck
node ˆv, and n other nodes vi. The bottleneck ˆv is connected
to all other nodes with edges of transmission cost 1, and it has an
energy of 1. No other edges exist, and the energy of each vi is also
1. Then, it is easy to see that the amount of data that can be sent
by a set S is 1/|S|, and hence, the optimum solution will be the one
maximizing u(S)
|S| .
The supermodular utility function u is defined as follows: We
partition the nodes v1,...,vn into S1 := {v1,v2} and S2 := {v3,...,vn}.
Their utilities are u(S1) = 1 2 − εn for some very small constant ε,
and u(S2) = 1 2 +εn. The utility of the entire set is u({v1,...,vn}) =
1.
Each node in S1 has individual utility ε/2, and each node in S2
has utility ε. Any set S ⊇ S1 has utility u(S) = u(S1) + ε · |S ∩ S2|,
and any set S ⊇ S2 has u(S) = u(S2) + ε/2 · |S ∩ S1|. Finally, all
other sets S have utility u(S) = ε · |S ∩ S2| + ε/2 · |S ∩ S1|. It is not
difficult to verify that this function u is indeed supermodular.
The greedy addition algorithm will add all nodes from S2 first,
never encountering the set S1 as a possible solution. Similarly, the
removal algorithm will remove the nodes from S1 first. Hence, the
output of any solution based on the greedy algorithm will be the
set of all nodes, giving a total utility of 1/n. On the other hand,
the set S1 can send half a unit of data, and thus achieves utility
1
2 (1/2 − nε) ≈ 1 4 , which is better than the greedy solution by a
factor of Ω(n).
5. GEOMETRIC PENALTY FUNCTIONS
In the previous two sections, we investigated the special cases of
submodular and supermodular utility functions. One of our main
goals was to find a compact representation for utility, as arbitrary
utility functions require specifying up to 2 n different function values.
1
Another approach to obtain compactly represented utility functions
is based on the observation that the real-world application of
sensor networks frequently involves monitoring geometric entities
such as structures or habitats. In those cases, the quality of a solution
can often be characterized in terms of the average or maximum
distance of “interesting” points on the geometric entity from
the measuring sensors.
Specifically, let A ⊆ R2 denote an area to be monitored, and
w : A → R + a weight function measuring how “important” it is to
measure any given point x ∈ A. Notice that A may be discrete or
continuous. We let dx,y be a distance metric measuring how effectively
a node at point x can measure the point y; most frequently, we
will use the Euclidean distance dx,y = � (x1 − y1) 2 + (x2 − y2) 2 ,
but our framework also permits using different metrics, or distance
measures that are not truly metrics (such as the Euclidean distance
with the additional provision that the dx,y = ∞ if x and y are too
far apart). Given a set S of sensors, we use dS,x to denote the distance
of the point x from the sensor set, i.e., dS,x = miny∈S dy,x.
Based on this notion of distance, we can define the weighted maximum
and average distance penalty as ˆp(S) = maxx∈A dS,x, and
p(S) = �
x∈A wx · dS,xdx. (If A is a discrete set, the integral should
be replaced by a sum.)
Based on this penalty, we can now define the utility of a set
to be inversely proportional to its penalty, i.e., u(S) := 1/ ˆp(S) or
u(S) := 1/p(S). Alternatively, we can phrase the problem as a
penalty minimization instead of a utility maximization. In particular,
in many monitoring applications, the goal will be to chose
one best set of sensors that will monitor the environment. Thus,
the goal would be to choose a set S such that ˆp(S)/tS or p(S)/tS is
minimized subject to the constraint that the set S can indeed send
tS units of data.
Unfortunately, the problem of choosing the best single set is NPhard
for both of these objectives, at least if the amount of data sent
by all nodes must be integral.
THEOREM 7. If tS is required to be integral, then finding the set
S minimizing ˆp(S)/tS or p(S)/tS is NP-hard. In fact, unless P=NP,
ˆp(S)/tS cannot be approximated to within better than 1.822.
Proof. We give reductions from the Euclidean k-median and Euclidean
k-center problems, respectively. The former was shown to
be NP-hard by Papadimitriou [19], and the latter was shown to be
hard to approximate to within 1.822 by Feder and Greene [20]. In
both cases, we are given a finite set of points A ⊆ R 2 , and are asked
to find a subset S ⊆ A of size at most k. For the k-median problem,
the objective is to minimize ∑x∈A dS,x, and for the k-center problem,
to minimize maxx∈A dS,x.
For the reduction, in both cases, we let A be the area to be covered
by sensors. There is a sensor node vx with energy 1 located at
each point x ∈ A, and all the vx are connected to a bottleneck node
ˆv with edges of sending cost 1. The bottleneck in turn is connected
to the root r via an edge of sending cost 1, and has energy k.
To prove that this is a correct (and approximation-preserving, in
the case of k-center) reduction, first notice that if S is a solution for
k-center (or k-median), then having each node vx with x ∈ S send
one unit is feasible, and gives exactly the same objective value.
Conversely, if S is the set of nodes sending data in a solution to the
sensor selection problem, then by integrality and the energy constraints
of 1 on nodes vx, each node in S sends exactly one unit of
data. But then, the objective function value of the set S as a solution
to k-center or k-median is exactly its penalty in the corresponding
sensor selection problem. This completes the proof.
This result does not rule out the possibility that sending fractional
amounts of data may make the problem tractable, or the ex-
istence of good approximation algorithms for the objective p(S)/tS.
Indeed, as shown by Arora et al. [21], there is a Polynomial-Time
Approximation Scheme (PTAS) for the Euclidean k-median problem.
6. RELATED WORK
Energy-conservation in sensor networks has received a tremendous
amount of attention in the literature. For example, there is
extensive work on overhearing avoidance and node duty-cycling at
the MAC layer [22, 23, 24, 25]. As well, several pieces of work
have focused on topology control [26, 27, 28, 29] to conserve energy
in dense deployments. Our utility-based sensor selection is
largely complementary, in that applications, rather than the system,
indicate which set of sensors should be active. Also complementary,
for a similar reason, is work on increasing network lifetime
using energy-aware routing [1, 2, 3, 4, 5, 6, 7, 8].
Utility functions have significant prior history in networking,
having been used to model architectural questions relating to network
QoS [30] and network pricing [31]. In the sensor networks literature,
however, we have found relatively few applications thereof.
Perhaps closest to our work is that of Byers and Nasser [9], who
first proposed utility-based sensor selection. However, they have
analytically examined only the special class of utility functions
that depend exclusively on the size of the set. More recently, Isler
and Bajcsy [32] used utility functions to model the sensor selection
problem for target localization. By contrast to both these pieces of
work, we examine a broader class of utility functions. Finally, the
work of Mainland et al. [33] is tangentially relevant to ours. They
proposed a price-based resource management scheme for sensor
networks. Their focus is more on the mechanistic aspects of resource
management, whereas ours is algorithmic.
7. CONCLUSIONS
In this paper, we have examined a utility-based sensor selection
approach that enables sensor network applications to express
utilities associated with retrieving data from sets of sensors. We
study the feasibility of determining, subject to network lifetime
constraints, the sequence of sets whose data has the maximum total
utility.
We explore fractional and integral variants of the utility-based
sensor selection problem, as well as single-set variants thereof, on
three important classes of utility functions. We find that many variants
are NP-hard, and some hard to approximate. On the other
hand, submodular functions can be optimized efficiently, and an
important subclass of supermodular functions admits a fractional
solution via solving an LP, and an O(logn)-approximation when
nodes are constrained to send the same amount of data. Finally, we
show that geometric utilities can be cast into a penalty framework
for which we are able to prove preliminary hardness results.
This paper should be considered a first step towards a much more
thorough exploration of utility-based sensor set selection. The utilitybased
framework is a very natural way of expressing the true goal
of a senor network application, and being able to select (approximately)
optimal schedules for practical classes of utility functions
is crucial in making the best use of a deployment. The geometric
penalty framework appears to lead to more tractable classes than,
e.g., supermodular functions, and we would not be surprised if efficient
approximations are possible. Similarly, it would be desirable
to derive matching upper and lower bounds for supermodular
functions, and to characterize more generally the classes of utility
functions that would be useful for sensor network applications.
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