Scheduling Resource-Bounded Monitoring Devices for Event Detection and Isolation in Networks
Waseem Abbas, Aron Laszka, Yevgeniy Vorobeychik, Xenofon Koutsoukos
aa r X i v : . [ c s . N I] A ug JOURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, JUNE 2016 1
Scheduling Resource-Bounded Monitoring Devicesfor Event Detection and Isolation in Networks
Waseem Abbas, Aron Laszka, Yevgeniy Vorobeychik, and Xenofon Koutsoukos
Abstract —In networked systems, monitoring devices such assensors are typically deployed to monitor various target locations.Targets are the points in the physical space at which eventsof some interest, such as random faults or attacks, can occur.Most often, these devices have limited energy supplies, andthey can operate for a limited duration. As a result, energy-efficient monitoring of various target locations through a setof monitoring devices with limited energy supplies is a crucialproblem in networked systems. In this paper, we study optimalscheduling of monitoring devices to maximize network coveragefor detecting and isolating events on targets for a given networklifetime. The monitoring devices considered could remain activeonly for a fraction of the overall network lifetime. We formulatethe problem of scheduling of monitoring devices as a graphlabeling problem, which unlike other existing solutions, allowsus to directly utilize the underlying network structure to explorethe trade-off between coverage and network lifetime. In thisdirection, first we propose a greedy heuristic to solve the graphlabeling problem, and then provide a game-theoretic solution toachieve optimal graph labeling. Moreover, the proposed setup canbe used to simultaneously solve the scheduling and placementof monitoring devices, which yields improved performance ascompared to separately solving the placement and schedulingproblems. Finally, we illustrate our results on various networks,including real-world water distribution networks.
Index Terms —scheduling, networked systems, network cover-age, graph labeling, potential games, dominating sets.
I. I
NTRODUCTION
Detection and isolation of unwanted events such as faults,failures, and malicious intrusions is a fundamental concern in avariety of practical networks. For example, leakage detectionin water distribution networks can reduce physical damageas well as financial losses [1]. For this purpose, monitoringdevices, such as sensors, are typically deployed strategicallythroughout the network. Spatially distributed systems overlarge areas may often be monitored only by battery-powereddevices, as wired deployment can be prohibitively expensiveor impossible. If the power supply provided by batteries isinsufficient for continuous monitoring during the intendedlifetime of a system, batteries must be replaced regularly. Sincethe cost of battery replacement for a large number of devicescan be very expensive, one of the primary design concerns for
W. Abbas is with the Institute for Software Integrated Systems, VanderbiltUniversity, Nashville, TN, 37212 USA (email: [email protected])A. Laszka is with the Department of Electrical Engineering and ComputerScience at the University of California, Berkeley, CA 94720, USA (email:[email protected]).Y. Vorobeychik, and X. Koutsoukos are with the Department of Elec-trical Engineering and Computer Science, Vanderbilt University, and alsowith the Institute for Software Integrated Systems, Vanderbilt University,Nashville, TN 37212 (emails: [email protected], [email protected]). such systems is increasing the time until the batteries of thesensors are depleted. At the same time, it is desired to maintaina certain level of monitoring in terms of the number of targetscovered throughout the network lifetime. Here, targets are thepoints in the physical space at which events of interest canoccur. For instance, in water distribution networks, events canbe the pipe bursts, and so targets can be the water pipes, whichneed to be monitored through sensors such as battery operatedpressure sensors.One of the primary approaches for conserving battery poweris “sleep scheduling.” The idea is to have only a subsetof the sensors activated at any given time, and to turn off(i.e., “sleep”) the remaining ones, thereby conserving power.By activating different sets of devices one after another, theoverall lifetime of a system can be substantially increased.Previous works have mostly focused on finding schedules thatensure complete coverage, that is, guaranteeing that everytarget is monitored by some device at any given moment intime (e.g., [2], [3]). However, complete coverage is a verystrict requirement, which severely limits the sets of devicesthat may be asleep at the same time. In fact, coverage (i.e.,ratio of monitored targets to the total number of targets)is a submodular function of the set of active devices inmost models (e.g., [4], [5]), which roughly means that at-taining complete coverage is disproportionately expensive ascompared to achieving reasonably good coverage. Managingenergy resources of monitoring devices via their schedulingto achieve an appropriate coverage of targets is a significantissue in networks where extended network lifetime is a criticalrequirement.In this paper, we design efficient scheduling schemes fora set of monitoring devices with limited battery supplies toachieve maximum target coverage for a given network lifetime.Scheduling of such devices to achieve complete network cov-erage is a special case of this general formulation. We modelthe network as a graph, in which monitoring devices could bedeployed at a subset of nodes, and the targets could be nodesand/or edges. Each monitoring device has a limited activetime, and covers a subset of targets within its range duringits active time. For a given network lifetime, the objective isto determine the maximum possible coverage, both in terms ofthe detection and isolation of (events at) targets, and a scheduleof monitoring devices to obtain the optimal coverage.In this direction the main contributions of the paper are:(1) We show that the optimal scheduling of monitoring devicesis an APX-hard problem, that is, there is no polynomial-timeapproximation scheme (PTAS) for the problem unless P=NP.(2) We provide a graph-theoretic formulation of the scheduling
JOURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, JUNE 2016 problem by showing that it is equivalent to a unique graphlabeling problem, which allows us to directly exploit thenetwork structure to obtain optimal schedules.(3) To solve the graph labeling, and hence the schedulingproblem, we propose two solutions; first, a greedy heuristicthat runs in polynomial time, and gives near optimal solutionsfor many networks as we illustrate. However, in general,performance guarantees of the heuristic in terms of the op-timality of the solution remain unknown. Second, we presenta game-theoretic solution, in which we show that the labelingproblem can be posed as a potential game, for which efficientalgorithms, such as binary log-linear learning (BLLL), areknown that asymptotically give globally optimal solutions withan arbitrary high probability.(4) Moreover, we illustrate that the game-theoretic solution al-lows simultaneously optimizing the placement and schedulingof monitoring devices, which gives better results as comparedto separately solving the placement and scheduling. Note thatthe placement problem involves selecting optimal locationsto deploy a given set of monitoring devices to maximize thetarget coverage within networks.(5) We analyze the performance of the approach throughsimulations on various networks including real-world waterdistribution networks and random networks. For random net-works, we also provide analytical results to determine theperformance of random scheduling, which does not requireany information about the network structure.(6) Finally, we consider some practically relevant special casesof the problem, such as scheduling to maximize networklifetime while ensuring complete coverage of the targets withinthe network.The rest of the paper is organized as follows: In SectionII, we explain our system model and define the schedulingproblem. Section III addresses the issue of complexity of theproblem. In Section IV, we present a graph labeling basedformulation of the scheduling, and in Section V proposesolutions to the graph labeling problem. Section VII illustratessimulation results, and section VIII presents a particular caseof interest of the scheduling problem. In Section IX, weprovide an overview of related work, and conclude the paperin Section X.II. S
YSTEM M ODEL AND P ROBLEM F ORMULATION
In this section, first, we present the system model bydescribing all the major components involved, and then weformulate the problem of optimal scheduling of resourcebounded monitoring devices in networks. (a) Network Graph –
We model the network as an undi-rected graph , G ( V, E ) , in which V is the set of nodes, and E is the set of edges given by the unordered pairs of nodes. Twonodes are adjacent if there exists an edge between them. The neighborhood of a node v , denoted by N ( v ) , is the set of allnodes that are adjacent to v , i.e., N ( v ) = { u : ( u, v ) ∈ E } ,and the neighborhood of a subset of nodes S , denoted by N ( S ) , is S v ∈ S N ( v ) . The degree of a node v , representedby δ ( v ) , is simply δ ( v ) = | N ( v ) | . Moreover, a path is asequence of nodes such that any two consecutive nodes in the path are adjacent, and the number of edges includedin the path is the length of the path. Any two nodes aresaid to be connected if there exists a path between them.The distance between connected nodes u and v , denotedby d ( u, v ) , is the length of the shortest path between them.Similarly, the distance between node u and edge e = ( i, j ) is d ( u, e ) = max( d ( u, i ) , d ( u, j )) . The network graph abstractsinteractions among various nodes within the network. (b) Targets – They are a subset of nodes and/or edges, de-noted by Y ⊆ ( V ∪ E ) , that could be subjected to an abnormalactivity (or event ), such as pipe failure, and therefore, need tobe monitored by monitoring devices. (c) Monitoring Devices – These are the devices that aredeployed at a subset of nodes S ⊆ V in the network, andcan monitor the other nodes and/or links within the networkfor any unusual activity, for instance, link failure detectionsuch as pipe burst in water networks. We refer to any suchabnormal activity on a target as an event . A monitoring devicecan monitor all nodes and edges for events that lie within somepre-specified distance, referred to as the range , of the device.If u is the node at which a monitoring device with the range λ is deployed, then the device covers (monitors) all the nodesand edges in the set { v ⊆ V : d ( u, v ) ≤ λ } ∪ { e ⊆ E : d ( u, e ) ≤ λ } . In other words, a target is covered if and only if it lieswithin the range of some monitoring device. Each device isresource-bounded in terms of the available battery supply ,denoted by B , which means that a device can be active (orcan be operational) for only B time duration. Furthermore, amonitoring device has only two output states – event detectedat some target without knowing the exact location of the target,and no event detected. A. Network Performance Measures
We are interested in measuring the quality of monitoringof targets through a set of monitoring devices, both fromthe detection and isolation perspectives. In detection , theobjective is just to detect any abnormal activity on some targetirrespective of determining the exact location of it, whereas in isolation , the goal is to uniquely detect the target at which theabnormal activity occurs. Moreover, we refer to the overalllifetime of the network, i.e., duration for which monitoring oftargets for detection (isolation) is considered, as the networklifetime T . To simplify, we divide the time into time slots ofequal length. The battery supply B of a monitoring devicecould be represented by the number of time slots, say σ , inwhich the device could remain active. Moreover, the networklifetime T could be represented by the total number of timeslots, say k , for which the detection (isolation) of targets isconsidered. Note that T and B represent the actual duration of overall network lifetime and battery lifetime of individualmonitoring device respectively, whereas, k and σ , which arechosen to be positive integers, represent respectively the total number of time slots and the time slots for which each devicecould remain active. (a) Detection Measure – Let there be a total of m targets,and m i be the number of targets that are covered by the BBAS et al. : SCHEDULING RESOURCE BOUNDED MONITORING DEVICES 3
TABLE IL
IST OF S YMBOLS
Symbol Description G ( V, E ) network graph S set of monitoring devices ( S ⊆ V ) Y set of targets ( Y ⊆ ( V ∪ E )) λ range of monitoring device N ( v ) neighborhood of a node vN ( S ) neighborhood of a subset of nodes ST network lifetime in terms of actual time duration B duration for which a device can remain active k network lifetime in terms of the total number of time slots σ number of time slots in which a device can remain active D average detection measure (1) I average isolation measure (2) S i nodes at which devices are active in the i th time slot G ( V , X ) bi-partite graph representation of the network G ( V, E ) monitoring devices that are active in the i th time slot. Wedefine the average detection performance , denoted by D , as D = 1 k k X i =1 (cid:16) m i m (cid:17) . (1) (b) Isolation Measure – We observe that event at target i canbe distinguished from an event at target j if and only if thereexists a monitoring device that gives different outputs in caseof events at i and j . In other words, there exists a monitoringdevice at some node such that exactly one of the target (either i or j , but not both) is covered by the monitoring device. Ifsuch a monitoring device exists, we say that the target-pair i, j is covered . The event at target i can be uniquely detected(or can be distinguished from events at all other targets) if alltarget-pairs i, j ( j = i ) are covered. If m is the total numberof targets, then there is a total of ℓ = (cid:18) m (cid:19) target-pairs. Inthe j th time slot, let ℓ j be the number of target-pairs that arecovered by the active sensors. Then, we define the averageisolation performance , denoted by I , as I = 1 k k X j =1 (cid:18) ℓ j ℓ (cid:19) (2)where k is the total number of time slots. A list of symbolsused throughout the paper is given in Table I. B. Problem Formulation
Consider a network G ( V, E ) in which S ⊆ V is thesubset of nodes at which monitoring devices with ranges λ are deployed, and Y ⊆ ( V ∪ E ) are the set of targets. Eachmonitoring device could remain active in at most σ of the totalof k time slots due to battery supply constraints. In each timeslot i , let S i ⊆ S be the subset of nodes with active monitoringdevices. Thus, we get a schedule of (active) monitoring devicesas S , S , · · · , S k . The objective is to determine the maximum average detec-tion performance D (or average isolation performance I ) fora given network life time, represented by k time slots, underthe battery constraints of monitoring devices, represented by σ time slots, and also a schedule of monitoring devices thatachieves the maximum D (or I ). It is obvious that as k increases, the maximum values of D (or I ) decrease. So, in a way, our goal is to understand arelationship between k and D (or I ), and design a systematicscheme to obtain a schedule for activating monitoring deviceswith limited battery supplies to obtain the desired networkperformance. Note that the scheduling problem for a completecoverage of targets, in which the objective is to determine aschedule that ensures D = 1 throughout the network life is aspecial case of the above problem.III. P ROBLEM C OMPLEXITY
In this section, we show that the problem of finding aschedule that maximizes the average detection performance fora given network lifetime and battery supplies, as discussed inSection II-B, is APX-hard. APX-hardness implies that (unlessP=NP), there does not exist a polynomial-time algorithm thatcan solve the problem to within arbitrary multiplicative factorof the optimum.In our case, for a target τ , if Q τ represents the fraction ofthe total number of time slots in which an event on τ canbe detected (i.e., τ is covered), then the expected value ofdetecting an event on an arbitrary target, denoted by Q is Q = 1 | Y | X τ ∈ Y Q τ . (3)Note that Q and D have exactly same values for a givenschedule ( S , S , · · · , S k ) , and therefore, they both measurethe average detection performance of the schedule. We formu-late finding a schedule that maximizes detection performanceas the following optimization problem: Definition (Maximum Average Detection):
Given a graph G =( V, E ) , a set of monitoring devices S ⊆ V , a set of targets Y ⊆ ( V ∪ E ) , range of the monitoring device λ , a network lifetimerepresented by k time slots, a battery supply represented by σ time slots, find a schedule ( S , S , . . . , S k ) that maximizesthe average detection performance Q . Theorem 3.1:
The Maximum Average Detection Problemis APX-hard.We show APX-hardness by reducing a well-known APX-hard problem, the Maximum Cut Problem [6] to the detectionproblem. The Maximum Cut Problem is defined as follows:
Definition (Maximum Cut Problem):
Given a graph G =( V, E ) , find a disjoint partition V , V of V that maximizesthe number of edges | E ( V , V ) | between V and V . Proof (Theorem 3.1) –
We prove APX-hardness by showingthat there exists a PTAS-reduction from the Maximum CutProblem to the Maximum Average Detection Problem. First,we define a polynomial-time mapping from an instance of thecutting problem to an instance of the detection problem:
JOURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, JUNE 2016 • let the network of the Maximum Average DetectionProblem be the graph of the Maximum Cut Problem; • let the set of monitoring devices be S = V ; • let the set of targets be Y = E ; • let the range of the monitoring device be λ = 1 ; • let the network lifetime be k = 2 time slots; • and let the battery supply be σ = 1 time slot.Second, we define a polynomial-time mapping from asolution ( S , S ) of an instance of the detection problem (i.e.,a schedule) to a solution ( V , V ) of the corresponding instanceof the cutting problem (i.e., a cut): V := S and V := S . (4)Next, observe that if an edge is cut by ( V , V ) , then thecorresponding target is covered by both S and S , whichimplies Q τ = 1 . On the other hand, if an edge is not cut by ( V , V ) , then the corresponding target is covered in only onetime slot, which implies Q τ = . Consequently, for any pairof solutions ( S , S ) and ( V , V ) , we have Q ( S , S ) = 1 | E | X τ ∈ E ( V ,V ) X τ E ( V ,V ) (5) = 12 + 12 | E ( V , V ) || E | . (6)Using the same argument, we can also show that if aschedule ( S , S ) is an optimal solution to the detectionproblem, then the cut ( V = S , V = S ) is also an optimalsolution to the cutting problem, and vice versa. Therefore, ifa schedule ( S , S ) is at most (1 − ǫ ) times worse than theoptimal schedule, then the corresponding cut ( V , V ) is atmost (1 − ǫ ) times worse than the optimal cut. Consequently,there is a PTAS-reduction from the Maximum Cut Problem tothe Maximum Average Detection Problem.As a consequence, we cannot optimally solve the maximumaverage detection problem in a polynomial time. Hence, weneed efficient heuristics that can provide reasonably goodsolutions with acceptable time complexities. In this regard,it becomes crucial to maximally exploit the structure of theproblem in a systematic way. To achieve this objective, wefirst provide a graph-theoretic formulation of the schedulingproblem in the next section, and then provide efficient solutionto the problem using a game-theoretic setting in Section V.IV. A G RAPH -T HEORETIC F ORMULATION OF THE S CHEDULING P ROBLEM
In this section, using various graph-theoretic notions, weformulate the scheduling problem as a graph labeling problem.In the next section, a solution approach is presented to solvethe corresponding graph labeling, thus solving the the originalscheduling problem.Our approach is to first obtain a bi-partite graph , denoted by G ( V , E ) , from a given graph. This bi-partite graph illustratestargets and the monitoring devices with given ranges coveringthose targets. We then formulate the scheduling problem onthe original network G ( V, E ) as a graph labeling problem onthe bi-partite graph G ( V , E ) . A. Bi-partite Graphs in the Cases of Detection and Isolation1) Case 1 – Detection:
When scheduling of monitoringdevices is required with an objective to maximize the averagedetection score D , as described in Section II-A, the bi-partitegraph G ( V , E ) is simply obtained as follows: the vertex set V is the union X ∪ Y , where X = S ⊆ V is the set of nodescorresponding to the set of monitoring devices, and Y = Y isthe set of targets in the original network G . Moreover, each x ∈ X is adjacent to vertices in Y that are at most λ distanceaway from x in G . An example is shown in Figure 1.
2) Case 2 – Isolation:
If maximizing the average isolationmeasure I , as in Section II-A, is the objective of scheduling,then G ( V , E ) is obtained as follows: As in the case of detec-tion, the vertex set of the bi-partite graph is V = X ∪Y , where X = S ⊆ V corresponds to the set of monitoring devices. Toobtain Y , we define a node for every pair of targets in Y . Therewill be (cid:18) | Y | (cid:19) such nodes in Y . As for the edge set E of the bi-partite graph, let y ∈ Y corresponds to the (unordered) targetpair ( τ , τ ) ∈ Y . Then, each x ∈ X is adjacent to y ∈ Y in G if and only if exactly one of the targets τ or τ is within λ distance from (the monitoring device corresponding to) x inthe original network G . In other words, in the bi-partite graph G , there will be no edge between x and y that corresponds tothe target pair ( τ , τ ) , if and only if the monitoring device x covers both targets τ and τ in G , or does not cover any ofthe targets τ and τ . An example is illustrated in Figure 1. Example:
Consider a graph G ( V, E ) in Figure 1. Let S = { , , } ⊆ V be the set of monitoring devices and edges inthe set Y = { e , e , e , e } be the targets. Moreover, eachmonitoring device has the range λ = 2 . The bi-partite graphs G ( V , E ) for the scheduling of monitoring devices to maximizethe detection and isolation measures are shown in Figures 1(b)and 1(c) respectively. The vertex set of bi-partite graphs in bothcases is V = X ∪Y , where X = S . For the detection case, Y = Y , and for the isolation case, Y = { e , e , e , e , e , e } ,where e ij corresponds to the pair of edges ( e i , e j ) in Y . Notethat an edge between x ∈ X and e ij ∈ Y indicates that themonitoring device at x covers the target pair ( e i , e j ) , or inother words, can distinguish between events at e i and e j . e e e e e , e , e , e , e , e , (b) (c) e e e e e e (a) Fig. 1. (a) An example network graph G ( V, E ) . Bi-partite graph represen-tations for (b) detection and (c) isolation. BBAS et al. : SCHEDULING RESOURCE BOUNDED MONITORING DEVICES 5
B. A Graph Labeling Problem and its Equivalence to theScheduling Problem
After obtaining the bi-partite graph G ( V = X ∪ Y , E ) froma given network G ( V, E ) , we can re-write the detection andisolation scores as in (1) and (2) respectively in terms of G .Note that if S i ⊆ X is the subset of active monitoring devicesin the i th time slot, then for the detection (isolation), the set oftargets (target-pairs) covered by S i is simply the neighborhoodof set S i , i.e., N ( S i ) = S x ∈ S i N ( x ) . Here, N ( x ) is theneighborhood of node x as defined in Section II. Hence, for agiven schedule ( S , S , · · · , S k ) where k is the total number oftime slots, the average detection (isolation) measure is simply (1 /k ) k P i =1 | N ( S i ) | . Thus, given a bi-partite graph G ( X ∪ Y , E ) ,network life in terms of k time slots, and battery supplyconstraint in terms of σ time slots, the problem of findingan optimal schedule that maximizes the average detection(isolation) measure as described in Section II-B becomesequivalent to finding a set of k subsets { S , S , · · · , S k } ,where S i ⊆ X , such that max { S , ··· ,S k } k X j =1 | N ( S j ) | , (7)and each node x ∈ X is included in at most σ such subsets.The above problem can be cast as a graph labeling problemas described below. Graph Labeling Problem:
Let K = { , , · · · , k } be the setof labels, and L be the set of all σ -subsets of L . Note that |L| = (cid:18) kσ (cid:19) . Moreover, we define f : X −→ L (8)i.e., f is a set function that assigns s ∈ L to each vertex in X , or in other words assign a subset of σ labels from K toeach x ∈ X . Also, for y ∈ Y , we define F ( y ) as follows: F ( y ) , [ x ∈ N ( y ) f ( x ) . (9)Note that | F ( y ) | is simply the number of distinct labelsavailable in the neighborhood of y . The objective is to obtainan assignment of labels to the nodes in X (i.e., (8)) such thatObjective: max f X y ∈Y | F ( y ) | (10)Here, the objective is to assign σ labels to each node in X suchthat the sum of the number of distinct labels available in theneighborhood of y , ∀ y ∈ Y , is maximized. The schedulingproblem in (7) and Section II-B, is equivalent to the graphlabeling problem described above. Proposition 4.1:
The problem of obtaining an optimalschedule that maximizes the average detection (isolation)measures of a set of monitoring devices with limited batterysupplies that cover a set of targets (target-pairs) for a given The cardinality of each subset is σ , where σ is some positive integer. network lifetime, which is divided into k time slots, is equiv-alent to the graph labeling problem as defined in Equations(8)–(10). Proof –
In the graph labeling problem, let the subset oflabels assigned to the vertex x , i.e., f ( x ) ∈ L , correspondto the indices of time slots in which the monitoring devicecorresponding to x is active. Since x has at most σ distinctlabels by the definition of f , the monitoring device corre-sponding to node x can be active in at most σ time slots.Hence, the battery supply condition that requires a monitoringdevice to be active in at most σ time slots, is always satisfied.Moreover, F ( y ) indicates time slots in which the target (target-pair) y ∈ Y remains covered by some x ∈ X . Then, (1 /k ) P y ∈Y | F ( y ) | is simply the average detection (isolation)measure. The set of vertices that have label i correspond tothe monitoring devices active in the i th time slot, i.e., S i .Thus, finding a labeling (8) that maximizes (10) is basicallyfinding a schedule ( S , S , · · · , k ) that maximizes the averagedetection (isolation) measure.An illustration of the graph labeling for the schedulingproblem is given below. Example:
In Figure 2, instances of optimal labeling ofgraphs in Figures 1(b) and 1(c) are shown for K = { , , · · · , } and σ = 2 . Here |K| = 5 means that the givennetwork lifetime spans five time slots. Each node x has at mosttwo labels, which represents that owing to battery constraint,a node can be active in at most two of the time slots. Thenode labels indicate time slots in which they remain active,thus, giving us optimal schedules. Here, the optimal detectionscore is 0.75, which could be obtained with the schedule S = S = { } , S = { } , S = { , } , S = { } . Similarly,the optimal isolation score is 0.633, which could be obtainedwith the schedule S = { , } , S = { } , S = { } , S = { } , S = { } . e e e e e , e , e , e , e , e , (a) (b) , , , , , , Fig. 2. Graph labelings for K = { , , · · · , } and σ = 2 . Node labels, i.e., f ( x ) are shown in colors. V. S
OLUTIONS TO THE G RAPH L ABELING P ROBLEM
In this section, we provide two solution approaches to thegraph labeling problem. The first one is a simple greedyheuristic, whereas, in the second approach, we utilize game-theoretic concepts. The greedy heuristic runs in polynomialtime, and gives a near optimal solution for many practicalnetworks as illustrated in the next section. However, in general,
JOURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, JUNE 2016 the approximation ratio of the algorithm is not known. Onthe other hand, the game-theoretic solution returns a graphlabeling that is globally optimal with high probability.
A. Greedy Heuristic
The graph labeling problem closely resembles the set cov-ering problem, since we have to ‘cover’ the set of targetsusing a set of monitoring nodes, each of which could cover agiven subset of the targets. Since the straightforward greedyalgorithm is known to be an efficient approximation algorithmfor the set covering problem, we can expect it to performwell for the graph labeling problem also. Hence, we formulatea simple greedy heuristic for the graph labeling problem asfollows (Algorithm 1): For a given labeling set K and σ ,iteratively select a combination of a label in K and a sourcenode in X that maximizes the sum of number of distinct labelsavailable in the neighborhoods of all target nodes in Y . Notethat in each iteration, only a source node with less than σ labels could be selected. Algorithm 1
Greedy Heuristic Given: σ , K = { , , · · · , k } Initialization: X ′ ← X , f ( x ) ← ∅ , ∀ x ∈ X While |X ′ | 6 = ∅ do ( x, ℓ ) ← argmax x ∈X ′ ,ℓ ∈K P y ∈Y | f ( y ) | f ( x ) ← f ( x ) ∪ { ℓ } If | f ( x ) | = σ do X ′ ← X ′ \ { x } End If End While If n is the total number of source nodes, m be the numberof target nodes, and k be the total number of labels in thelabeling set, then greedy heuristic could be executed in atmost O ( σkn m ) time as there are O ( σn ) iterations and eachiteration could take O ( knm ) time. Greedy heuristic gives asimple strategy to solve the labeling problem, however, wedo not know the quality of the solution returned by it, that is,how far is the greedy solution from the optimal one. Therefore,we present a game-theoretic solution by posing the labelingproblem as a potential game, for which algorithms are knownthat maintain globally optimal solution with high probabilityas time goes to infinity, as discussed below. B. Game Theoretic Solution to the Graph Labeling Problem
Game theory concepts have been extensively employed tosolve locational optimization problems, such as maximizingcoverage on graphs (e.g., [7], [8]) and distributed control ofmultiagent systems (e.g., [9], [10]). In a particular approach,the idea is to determine a potential function that captures theoverall global objective. The players’ individual utility func-tions are then appropriately aligned with the global objectivesuch that the change in the utility of the player as a result ofunilateral change in strategy equals the change in the globalutility represented by the potential function. The players’strategies are then designed to ensure that local actions lead to the global objective. It turns out that this problem formulationand design can be realized using a class of non-cooperativegames known as potential games , which are now extensivelyused for various distributed control optimization problems.A finite strategic game Γ( P , A , U ) consists of a set of play-ers P = { , , · · · , n } , action space A = A × A × · · · × A n where A x is a finite action set of the player x ∈ P ,and a set of utility functions U = {U , U , · · · , U n } where U x : A → R is a utility function of the player x . If a = ( a , · · · , a x , · · · , a n ) ∈ A denotes the joint action profile,we let a − x denote the action of players other than the player x . Using this notation, we can also represent a as ( a x , a − x ) .A game is a potential game if there exists a potentialfunction , φ : A → R such that the change in the utility of theplayer x as a result of a unilateral deviation from an actionprofile ( a x , a − x ) to ( a ′ x , a − x ) is equal to the correspondingchange in the potential function. More precisely, for everyplayer x , a x , a ′ x ∈ A x , and a − x ∈ A − x , we get U x ( a x , a − x ) − U x ( a ′ x , a − x ) = φ ( a x , a − x ) − φ ( a ′ x , a − x ) (11)In the case of potential games, there exist algorithms, suchas log-linear learning (LLL) [11], [12] and binary log-linearlearning (BLLL) [13] that could be utilized to drive the playersto action profiles that maximize the potential function. Thesealgorithms embody the notion of convergence of such gamesto the most efficient Nash equilibrium, particularly in scenarioswhere utility functions are designed to ensure that the actionprofiles that maximize the global objective of the systemcoincide with the potential function maximizers [11], [13].More precisely, in potential games, LLL and BLLL algorithmsguarantee that only the joint action profiles that maximize thepotential function are stochastically stable [13]. The LLL andBLLL are in fact, nosiy best-response algorithms that inducea Markov chain over the action space with a unique limitingdistribution that depends on the noise parameter. As the nosieparameter reduces to zero, the limiting distribution has a largepart of its mass over the set of potential maximizers (see e.g.,[13], [14] for details).The basic idea behind these algorithms is to have noisy best response dynamics, in which the noise parameter allowsthe selection of suboptimal action occasionally by the players.The probability of selecting a suboptimal action is dependentof the pay-off difference between the optimal and suboptimalcases. Thus, formulating the graph labeling problem as apotential game would allow us to use the above mentionedlearning algorithms to find the most efficient solutions to thegraph labeling problem. Thus, our objective now is to designa potential game corresponding to the labeling problem ongraphs, and incorporate learning algorithms for the potentialgames to achieve the desired labeling.
1) A Potential Game for the Graph Labeling:
We designa potential game Γ( P , A , U ) to obtain a labeling of a graphthat achieves the objective in (10), thus solving the schedulingproblem. In our game, the set of players is the vertex set X in the vertex partition ( V = X ∪ Y ) of the bipartite graph G ,i.e., P = X . For each player x ∈ X , the action set A x is the BBAS et al. : SCHEDULING RESOURCE BOUNDED MONITORING DEVICES 7 set of all σ -subsets of the labeling set K = { , · · · , k } . Wealso need to have a potential function that captures the globalobjective. For this, we define S j as the set of vertices with thelabel j , i.e., S j = { x ∈ X : j ∈ f ( x ) } (12)A potential function is then defined as φ ( a ) , k X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ x ∈ S j N ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (13)Note that φ ( a ) is simply the total number of nodes in Y having a label j ∈ K in their neighborhoods, summed over allthe labels, which is equivalent to the P y ∈Y | F ( y ) | in (10). Thus, φ ( a ) indeed captures the global objective.Moreover, we define the utility function of the player x asthe total number of labels made available by a x to the nodesin N ( x ) that otherwise would not have been available to thenodes in N ( x ) . For instance, in Figure 2(a), node has labels { , } , which represents the action a . Moreover, for the twoneighbors of node , i.e., e and e , node is the only one withthe label ; and for the node e , node is the only one withthe label . Thus, U ( a , a − ) = 2 + 1 = 3 . More precisely,we define U x ( a x , a − x ) as U x ( a x , a − x ) , k X j =1 a xj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N ( x ) \ [ z ∈ S j \{ x } N ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (14)where, a xj = (cid:26) if j ∈ a x (= f ( x ))0 otherwise.Next, we show that with the potential function as definedin (13), and the utility function as in (14), the game designedabove is indeed a potential game. Theorem 5.1: Γ( P , A , U ) is a potential game if utilities aredefined as in (14). Proof – The potential function, as defined in (13) can bewritten as, φ ( a x , a − x ) = k X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ x ∈ S j N ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k X j =1 a xj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N ( x ) \ [ z ∈ S j \{ x } N ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ z ∈ S j \{ x } N ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k X j =1 a xj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N ( x ) \ [ z ∈ S j \{ x } N ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + k X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ k ∈ I x \{ i } N ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = U ( a x , a − x ) + k X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ z ∈ S j \{ x } N ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (15)Similarly, for a = ( a ′ x , a x ) , we get φ ( a ′ x , a − x ) = U ( a ′ x , a − x ) + k X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ z ∈ S j \{ x } N ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (16) Subtracting (16) from (15) gives us the desired result, i.e., φ ( a i , a − i ) − φ ( a ′ i , a − i ) = U ( a i , a − i ) − U ( a i , a − i ) Since our graph labeling problem can be formulated asa potential game, using the results in [13] we deduce that if players adhere to the binary log linear algorithm (statedbelow), then the objective in (10) is maximized . In other words,if σ unique labels from a total of k labels are assigned toeach node x ∈ X as per below algorithm, then the numberof distinct labels in the neighborhood of every node y ∈ Y islikely to converge to the maximum value. Algorithm 2
Binary Log-Linear Learning [13] Initialization:
Pick a small ǫ ∈ R + , an a ∈ A , and totalnumber of iterations. While i ≤ number of iterations do Pick a random node x ∈ X , and a random a ′ x ∈ A x . Compute P ǫ = ǫ Ux ( a ′ x,a − x ( t )) ǫ Ux ( a ′ x,a − x ( t )) + ǫ Ux ( ax,a − x ( t )) . Set a x ← a ′ x with probability P ǫ . i ← i + 1 End While
Note that initially the nodes are assigned σ -element subsetsof labels randomly. Afterwards, in each iteration, a node isselected at random, and a σ -subset of labels that improve theoverall labeling to attain the objective in (10), is selected witha certain probability.VI. S IMULTANEOUS P LACEMENT AND S CHEDULING OF M ONITORING D EVICES
So far, we have considered the optimal scheduling ofresource bounded monitoring devices, assuming that theirplacement is fixed, i.e., locations at which monitoring devicesare deployed are given. If S is the set of all such nodesat which monitoring devices could be deployed, then the placement problem is to select a subset X ⊆ S with thegiven cardinality such that the number of targets (pair-wisetargets) that are covered, i.e., lie within the range of at leastone such device, is maximized. Typically, to maximize thecoverage of targets for a given network lifetime, the placementproblem is first solved, followed by the determination ofefficient schedules for the monitoring devices.However, for a given network lifetime, and a fixed num-ber of resource bounded monitoring devices, simultaneouslyoptimizing their placement and scheduling maximizes theaverage detection (isolation) measure. For instance, considerthe network in Fig. 3, in which three monitoring deviceswith λ = 1 and σ = 2 are deployed to cover the max-imum number of nodes for k = 4 . Fixing the placementof devices at nodes { , , } , optimal schedule (for instance, S = S = { } , S = S = { , } ) gives D = 0 . , whereasthe maximum possible D under the conditions is . , whichcould be obtained by placing the devices at nodes { , , } andwith a schedule S = S = { , } , S = S = { } .The BLLL based algorithm to schedule a set of monitoringdevices with fixed locations, presented in Section V-B, can JOURNAL OF L A TEX CLASS FILES, VOL. XX, NO. X, JUNE 2016 S ,S S ,S S ,S S ,S S ,S S ,S (a) (b) Fig. 3. (a) Optimal schedule for a given placement. (b) Optimal placementand schedule of three monitoring devices with λ = 1 , σ = 2 for k = 4 . be modified to simultaneously optimize the placement aswell as scheduling of such devices to maximize the averagedetection (isolation) measure. This modification is presented asAlgorithm 3 below. Fixing the number of monitoring devices |X | , the objective is to select X ⊆ S , and assign at most σ labels to each node from a labeling set K = { , , · · · , k } sothat the average detection measure D (or the isolation measure I ) is maximized . The labeling of nodes selected in X will thengive the schedule.In this case, players P are the monitoring devices, for whichwe need to find the locations, i.e., the nodes at which they aredeployed, as well as schedules, i.e., time slots in which theybecome active. Using the same notations as in Section V-B,here, action of a player is the selection of ( x, a x ) ∈ ( S × A x ) ,where S is the set of all such nodes at which a monitoringdevice could be placed, and A x is the set of all possible σ -subsets of the labeling set K . Previously, the choice of x wasfixed for a monitoring device and the player’s action comprisedof only selecting a x . Similarly, utility of a player for the choiceof an action ( x, a x ) here is simply the number new labels thatbecome available in the neighborhood of node x as a result ofassigning labels in a x to x . In the search of a better solution, ineach iteration, a new action ( s, a s ) is selected with a certainprobability for a randomly selected player. It simply meansthat with a certain probability, either new labels are assignedto the node at which (randomly selected) player is located, ora new node as well as a new set of labels (selected at random)are chosen for the player. Algorithm 3
Simultaneous Placement and Scheduling Initialization:
Pick a small ǫ ∈ R + and the number ofiterations. Select randomly a subset of nodes X ⊆ S , andassign labels to nodes in X , i.e, select a ∈ A . While i ≤ number of iterations do Randomly select a node x ∈ X . Randomly select a node s ∈ ( S \ X ) ∪ { x } , and a s ∈ A s . Compute P ǫ = ǫ Us ( as,a − x ) ǫ Us ( as,a − x ) + ǫ Ux ( ax,a − x ) . With probability P ǫ , set X ← ( X \ { x } ) ∪ { s } , andselect a s for node s . i ← i + 1 End While
Simulation results for the above algorithm are illustratedin Section VII-C. Using various networks, it is shown thatsimultaneously selecting the locations for monitoring devicesas well as scheduling them using Algorithm 3, gives improved average detection as compared to the one obtained by solvingthe placement and scheduling separately.VII. N
UMERICAL R ESULTS
In this section, we present numerical results on the simplegreedy and BLLL based algorithms for the scheduling andplacement of monitoring devices on urban water distributionnetworks and random geometric networks as explained below.
A. Scheduling Monitoring Devices in in Water DistributionNetworks
Water distribution networks can be modeled as undirectedgraphs in which edges represent the pipes and nodes representthe junctions. To detect pipe bursts and leakages, pressuresensors are deployed at junctions, which could sense thepressure transient generated as a result of pipe burst within acertain distance (range) from the sensor. The distance thresholdbased model has been used in water networks in the contextof sensor placement problems, e.g., [15], [16]. The pressuresensors are battery operated devices with limited batterylifetime. Thus, top operate these sensors for an extendedperiod of time, they need to be scheduled. Here, we simulatescheduling algorithms, including simple greedy and BLLLbased algorithm for the efficient scheduling of monitoringdevices, which are pressure sensors in this case, to obtain highvalues of D in two different water distribution networks. Thedetails of these networks, referred to as the Water Network 1 and
Water Network 2 , are as follows:Water Network 1 [17], [18] has 126 nodes, 168 pipes, onereservoir, one pump, and two storage tanks. This benchmarkwater distribution network has been extensively studied in thecontext of sensor placement problems for water quality. Waternetwork 2 [19] is a grid system in Kentucky with 366 pipes,270 nodes, three tanks, and five pumps. The layouts of bothnetworks are illustrated in Figure 4. For both the networks,we consider that the sensors are deployed at the junctionsas source nodes (monitoring devices), i.e., X , and the set ofpipes, which are edges in the corresponding network graph, astargets, i.e., Y . Moreover, for each sensing device, we assume σ = 2 , and compute D for a network lifetime, given by k time slots, using greedy and BLLL algorithms. For each BLLLinstance, we perform 20,000 iterations by selecting ǫ to be . . The plots of D as a function of k for various ranges ofsensing devices (as defined in Section II) are given in Figure5. We can see that both greedy and BLLL gives approximatelysame results. However, BLLL has an advantage over thegreedy algorithm as it allows to simulatneously solve theplacement as well as scheduling problem (as discussed inSection VI), which gives improved D as compared to individ-ually solving placement and scheduling problems. Moreover,if BLLL is run for sufficiently large number of iterations,the algorithm converges to the optimal solution. Similar plotscan be obtained for the scheduling of monitoring devices tomaximize the average isolation measure I by first obtainingthe appropriate network representation as outlined in SectionIV-A. In Figure 6, the convergence of BLLL algorithm is BBAS et al. : SCHEDULING RESOURCE BOUNDED MONITORING DEVICES 9
Fig. 4. Schematics for water networks 1 and 2.
Water network 1 k D λ = 2 (Greedy) λ = 2 (BLLL) λ = 3 (Greedy) λ = 3 (BLLL) Water network 2 k D λ = 2 (Greedy) λ = 2 (BLLL) λ = 3 (Greedy) λ = 3 (BLLL) Random geometric graphs k D |X| = 0 . n (Greedy) |X| = 0 . n (BLLL) |X| = 0 . n (Greedy) |X| = 0 . n (BLLL) Fig. 5. Plots of D as a function of network lifetime k for scheduling on water networks and random geometric networks, assuming that each monitoringdevice has a battery lifetime of σ = 2 time slots. D Water network 1 k = 10k = 16 D Water network 2 k = 10k = 16 D Random geometric graph k = 10k = 16
Fig. 6. Plots of D as a function of (BLLL) iterations to illustrate the convergence of BLLL algorithm for the scheduling of monitoring devices with σ = 2 and k = 10 , . illustrated. For both water networks, D as a function ofiterations is shown for λ = 2 , σ = 2 , and k = 10 and . Weobserve the algorithm converges to near optimal value fast,within about 5000 iterations, and the improvements thereafter,are quite small. B. Scheduling Monitoring Devices in Random Geometric Net-works
Random geometric networks are a form of spatial networksin which nodes are deployed uniformly at random in a certainarea. An edge exists between two nodes if the Euclideandistance between them is at most r , which is often referred toas radius of the sensing footprint. Owing to a wide variety ofapplications in various domains, such as modeling of wireless A TEX CLASS FILES, VOL. XX, NO. X, JUNE 2016 sensor networks, these networks have been extensively studied.For our simulations, we consider a network with nodes,deployed uniformly at random over an area of × , and r = 2 . The set of targets here is the set of all nodes. Moreover,a certain fraction of nodes (either or ) are selected atrandom as source nodes, i.e., nodes with monitoring devices.A monitoring device has a battery lifetime of at most σ = 2 time slots, and can monitor targets that are at a Euclideandistance of at most from it. In Figure 5, D as functions of k are illustrated using greedy and BLLL algorithms. Each pointon the plots is an average of fifty randomly generated graphinstances. In Figure 6, the convergence of BLLL algorithm isshown for some instances of random geometric graphs with100 nodes, out of which 20 randomly selected nodes containmonitoring devices.
1) Random Scheduling in Random Networks:
Another spe-cial case of interest is related to the quality of random schedul-ing, i.e., given a total of k time slots, if each node remains ac-tive in σ time slots chosen randomly, then what is the averagedetection performance of such a random scheduling? Here, wediscuss this question for random networks, including randomgeometric networks and networks that could be modeled byErd˝os-R´enyi random graphs. Though random scheduling isinferior to the BLLL based scheduling in terms of the detection(or isolation) performance, it is useful in many scenariossince it neither requires any sort information regarding thenetwork structure, nor requires any coordination between themonitoring devices. The average detection measure of randomscheduling in random geometric networks is given below. Proposition 7.1:
Let G ( V, E ) be a random geometric graphin which each node contains a monitoring device that remainsactive in σ time slots that are randomly chosen from a totalof k time slots, which correspond to the overall lifetime ofthe network. If each node in a graph is also a target, then theaverage detection performance of this random scheduling is D ( G ) = 1 − ( k − σ ) k exp (cid:18) − σλπr k (cid:19) (17)where r is the radius of the sensing footprint of node, and λ is the number of nodes per unit area.A proof of the above theorem is given in the Appendix. Asabove, it can be shown that in the case of Erd˝os-R´enyi randomgraphs with n nodes, denoted by G n,p , in which any two nodesare adjacent with some probability p , this random schedulingscheme results in an average detection performance given by D ( G n,p ) = 1 − ( k − σ ) k exp (cid:18) − σk np (cid:19) (18)Note that in (18), it is assumed that all the nodes havemonitoring devices and all the nodes need to be covered. C. Simultaneous Placement and Scheduling of MonitoringDevices Using Algorithm 3
We illustrate the Algorithm 3 for the water network 1 andthe random geometric graph here. For the water network 1, In terms of the (graph) distances as defined in Section II, the range of eachmonitoring device is λ = 1 , as the Euclidean distance of at most 2 betweentwo nodes u and v implies d ( u, v ) = 1 . we set the number of monitoring devices to be , whereeach device has a range λ = 2 . The set of pipes (or edges inthe corresponding network graph) are the targets that need tobe covered by these devices. We simulate two scenarios; inthe first case we use Algorithm 3 to simultaneously select thenodes and schedules for the monitoring devices; in the secondscenario, we first solve the placement problem by selectingthe 25 nodes, say X ⊂ V , that maximize the number of edgesthat are at most distance 2 from some node in X , and thensolving the scheduling problem using Algorithm 2. We notehere that the placement problems, in this context, are typicallysolved using some variant of the minimum set cover problem,or the maximum coverage problem in case the number ofmonitoring devices is fixed (e.g., [4], [5], [20]). Since thenumber of devices is fixed here, and the targets to be coveredare edges, we use the maximum coverage problem to place (agiven number of) monitoring devices at nodes that maximizethe number of edges that are at most λ = 2 distance fromat least one of the selected nodes. Moreover, since maximumcoverage problem is NP-hard, we use a greedy heuristic, whichgives best approximation ratio, to solve it.The results are illustrated in Figure 7. It can be seen that Al-gorithm 3 (simultaneously solving placement and scheduling)is always giving higher average detection D . For the randomgeometric graph, we simulate instances consisting of 50 nodesdeployed at random in an area of × unit , out of which10 could contain monitoring devices capable of covering nodeswithin a Euclidean distance of 100 units. The targets hereare nodes, and the objective is to maximize the averagedetection for a given network lifetime. As with the waternetwork example, average detection is improved if placementand scheduling is solved simultaneously using Algorithm 3 ascompared to optimizing placement and scheduling separately.For all cases, the battery lifetime of each monitoring deviceis assumed to be σ = 2 time slots. In Figure 8, we illustratethe convergence of Algorithm 3 for the water network 1 andrandom geometric graph example. given the network lifetime k = 10 time slots. k D k D Fig. 7. Comparison of simultaneously optimizing scheduling and placementusing Algorithm 3 versus separately optimizing placement and scheduling interms of D as a function of k . VIII. S
PECIAL C ASE : S
CHEDULING TO M AXIMIZE N ETWORK L IFETIME W HILE E NSURING C OMPLETE C OVERAGE
An important special case of the scheduling problem isto control the activity of monitoring devices such that the
BBAS et al. : SCHEDULING RESOURCE BOUNDED MONITORING DEVICES 11 D Water network 1 D Random geometric graph
Fig. 8. Plots of D as a function of iterations of Algorithm 3 showing theconvergence of the algorithm. overall network lifetime is maximized while ensuring completecoverage , i.e., D = 1 . In a basic setting, we consider that allnodes in a graph need to be covered at all times, and each nodeis equipped with a monitoring device that can remain active inat most σ time slots. Then, the objective is to schedule thesemonitoring devices such that the number of time slots k , inwhich all of the nodes remain covered through a subset ofactive devices, is maximized.The problem is related to the notion of dominating sets ingraphs . Definition
A dominating set is a subset of vertices in a graph S i ⊆ V , such that for every u ∈ V , either u ∈ S i , or thereexists some v ∈ S i such that v ∈ N ( u ) .In other words, considering the targets to be the set of nodes(i.e., Y = V ), the ranges of monitoring devices λ to be 1, thenetwork is guaranteed to be completely covered whenever theset of nodes with active monitoring devices form a dominatingset in the network graph. Moreover, in the case of targets beingedges (i.e., Y = E ), a dominating set of active monitoringdevices with ranges λ = 2 is also sufficient for the completecoverage of targets within the network. Thus, to maximize theoverall network lifetime while ensuring complete coverage oftargets, the problem of finding distinct dominating sets in agraph is of great importance. The problem of finding distinctdominating sets under certain constraints has been of greatinterest owing to its wide variety of applications (e.g., [21],[22], [23], [24]). There are two approaches to maximize thenumber of distinct dominating sets under the constraint onthe number of times a node can appear in a dominating set –disjoint dominating sets, and non-disjoint dominating sets. A. Disjoint Dominating Set Based Approach
One way to approach this problem is to partition the vertexset such that each set in the partition is a dominating set, andall dominating sets are pair-wise disjoint. Such a partition isknown as the domatic partition , and the maximum number of(disjoint dominating) sets that can be obtained is known as the domatic number , denoted by γ . Since dominating sets are pair-wise disjoint in such a partition, each vertex belongs to onlyone of the dominating sets. Moreover, since each node can beactive for σ time slots, each dominating set can remain activefor σ time slots. If only one dominating set is active at anytime instant, which is sufficient for the complete coverage, then the lifetime of the network achievable through this approachis given by k = σγ (19)time slots, where γ is the domatic number of a graph. Thedomatic partition problem is known to be NP-hard [25]. Var-ious sensor scheduling schemes that utilize domatic partitionshave been proposed to maximize the network lifetime whileensuring complete coverage (e.g., [24], [26], [27]). B. Non-Disjoint Dominating Set Based Approach
Another way to approach the network lifetime maximizationwhile maintaining complete coverage is by using the non-disjoint dominating sets of active nodes. Using this approach,it is possible to obtain a better lifetime as compared tothe disjoint dominating sets based approach [2], [28]. Asan illustration, consider the network in Fig. 9, which has adomatic number . We assume that each node can be activefor two time slots, i.e., σ = 2 , then using disjoint dominatingsets approach, we get the network lifetime of k = 4 time slots.However, it is possible to obtain five distinct dominating setssuch that each node appears in at most two such sets, as shownin Fig. 9(b), thus, yielding a network lifetime of k = 5 timeslots. , , , , , , , , , Fig. 9. (a) Two disjoint dominating sets are shown. (b) Five non-disjointdominating sets, indicated by the nodes with the same labels, are shown.Each node belongs to two distinct dominating sets.
The problem of finding the maximum number of dominatingsets under the constraints on the number of times a node canbe included in a dominating set is related to the notion of ( k, σ ) -configurations [29], [30] as defined below. Definition ( ( k, σ ) -Configurations in Graphs) Let σ , k be twopositive integers, and K = { , · · · , k } be the set of labels, then ( k, σ ) -configuration of a graph is the assignment of σ distinctlabels from the set K to each node in the graph such that forevery i ∈ K and every node in v , the label i is assigned to v or one of its neighbors.An example of (5 , -configuration is shown in Fig. 9(b).Note that the set of nodes corresponding to a particular labelin K constitute a dominating set. So, if a graph has an ( k, σ ) -configuration, it is possible to have k distinct (possibly non-disjoint) dominating sets such that each node can be includedin at most σ such dominating sets. Thus, for a given σ , themaximum value of k , say k ∗ , for which ( k ∗ , σ ) -configurationexists, is of particular interest as it provides a scheduling A TEX CLASS FILES, VOL. XX, NO. X, JUNE 2016 scheme based on the non-disjoint dominating sets to maximizenetwork lifetime while ensuring complete coverage.Obviously, for σ = 1 , the maximum k for which ( k, -configuration of a graph G exists, is equal to the domaticnumber γ of G . Thus, given a ( γ, -configuration of G withthe labeling set { ℓ , ℓ , · · · , ℓ γ } , a ( k, σ ) -configuration couldbe obtained for some σ > and k = σγ by simply replacingeach label ℓ i by a set of labels { ℓ i, , · · · , ℓ i,σ } . Thus, fora given σ , if k ∗ is the maximum value for which ( k ∗ , σ ) -configuration of a graph exists, then k ∗ ≥ σγ. (20)Consequently, the non-disjoint dominating sets approach isalways at least as good as disjoint dominating sets approach,though it often performs better. An interesting question here isunder what conditions or specific instances k ∗ > σγ ? In thisregard, first we note that every connected graph has γ ≥ , andtherefore, for a given σ , k ∗ is always at least σ . However,there exists many graphs for which γ = 2 , but k ∗ > σ .For instance, many cubic graphs have a domatic number of2, e.g., the one shown in Figure 9. However, the followingtheorem asserts that all cubic graph have k ∗ ≥ σ for a given σ . Theorem 8.1: [30] Any cubic graph has an ( k, σ ) -configuration with k = ⌊ σ/ ⌋ , and such a configuration canbe found in polynomial time.Recently, it has been shown in [29] that the above result istrue even for a bigger class of graphs as stated in Theorem8.2. Here, K , is a star graph with one central node of degreesix, and six end nodes each with a degree one ( K , = ). Theorem 8.2: [29] Let G be a graph such that– G has a minimum degree at least two,– no subgraph of G is isomorphic to K , , and– G = { , , , , , , , } ;then G has an ( k, σ ) -configuration with k = ⌊ σ ⌋ .The above result is particularly useful as proximity graphs(e.g., random geometric graphs), which are often used tomodel the limited range communication in networks such aswireless sensor networks, are always K , -free. As pointedout in [29], a large number of graphs in this family have adomatic number of 2, thus, non-disjoint dominating set basedstrategy is strictly better than the disjoint dominating set basedstrategy in those cases for maximizing the network lifetimewhile ensuring complete coverage of targets.IX. R ELATED W ORK
One of the earliest efforts to conserve battery power throughscheduling sensor devices is the work of Slijepcevic andPotkonjak [31]. In [31], the authors consider the problem ofmaximizing lifetime while preserving complete coverage of anarea, which they formulate as the Set K-Cover Problem. Tosolve this problem, they introduce a heuristic for finding mu-tually exclusive sets of sensors such that each set completelycovers the monitored area. In a follow-up work, Abrams etal. introduce three approximation algorithms for a variation graphs in which each vertex has a degree three. of the Set K-Cover Problem [32]. Later, Deshpande et al.study several generalizations of the Set K-Cover Problem, anddevelop an approximation algorithm based on a reduction toMax K-Cut [33].Besides the Set K-Cover Problem, researchers have studiedvarious other formulations of the scheduling problem. Mosci-broda and Wattenhofer consider disjoint dominating-set basedclustering in sensor networks [26]. The authors study theproblem of maximizing the lifetime of a sensor network, andprovide approximation algorithms for multiple variations ofthe problem. Cardei et al. study schedules that consist of non-disjoint sets of sensors and continuously monitor all targets [2].They model the solution as the maximum set covers problem,and propose two heuristics based on linear programming anda greedy approach. Koushanfar et al. consider the problemof scheduling sensor devices such that the values of sleepingdevices can always be recovered from the measurements ofactive devices within a given error bound [34]. The authorsfirst introduce a polynomial-time isotonic regression for re-covering the values of sleeping devices, and building onthis regression, they then formulate the scheduling problemas domatic partitioning problem, which they solve using anILP solver.Our approach is most related to the work of Wang etal., who study the trade-off between maximizing lifetimeand minimizing “coverage breach,” that is, minimizing thetotal amount of time that each target is not covered by anysensors [35], [28]. The authors propose organizing the sensorsinto non-disjoint sets, and introduce an algorithm based onlinear programming as well as a greedy heuristic. In a follow-up work, Rossi et al. propose an exact approach based on acolumn-generation algorithm for solving the scheduling prob-lem, and they also derive a heuristic from their approach [36].However, graph-theoretic formulation proposed in this paperallows us to directly exploit the network structure to obtainoptimal schedule for a given network lifetime maximizingthe detection or identification of targets. Moreover, unlikeprevious solutions, our game theory based solution couldsimultaneously solve the placement as well as schedulingproblems, which gives improved performance as compared toseparately solving placement and scheduling.A few research efforts have considered simultaneous place-ment and scheduling. Krause et al. study simultaneous place-ment and scheduling of sensor devices for monitoring spatialphenomena, such as road traffic [37]. The authors assumethat for any set of active sensors, the “sensing quality” isgiven by a submodular function, and they aim to maximizethe worst-case sensing quality. T¨urko˘gulları et al. considerthe problem of maximizing lifetime through sink placement,scheduling, and determining sensor-to-sink flow paths, underenergy, coverage, and budget constraints [38]. To solve thisproblem, they propose a mixed-integer linear programmingmodel as well as a heuristic, which is more scalable but lacksperformance guarantees.A number of studies have focused on the placement of sen-sor nodes, without considering sleep scheduling. Younis andAkkaya have surveyed earlier literature on node placement, in-cluding the placement of sensor nodes [39]. Krause et al. con- BBAS et al. : SCHEDULING RESOURCE BOUNDED MONITORING DEVICES 13 sider the problem of deploying sensors for detecting maliciouscontaminations in large-scale water-distribution networks [5].Based on the submodularity of realistic objective functions, theauthors design scalable placement algorithms with provableperformance guarantees. Furthermore, they show that theirmethod can be extended to multicriteria optimization and ad-versarial objectives. Hart and Murray provide a survey of sen-sor placement strategies for water-distribution networks [40].Finally, besides scheduling, researchers have also studiedother similar approaches for conserving battery power. Forexample, Zhao et al. consider selective collaboration of sensorsin order to minimize communication and communication,which increases the longevity of networks of battery-poweredsensors [41]. The authors focus on the problem of tracking,and they study optimizing the information utility of data forgiven costs of communication and computation.X. C
ONCLUSIONS
We studied the problem of scheduling resource boundedmonitoring devices in networks to maximize the detectionand isolation of failure events for a given network lifetime.We showed that the scheduling problem is equivalent to agraph labeling problem, which allowed direct exploitation ofthe network structure to obtain optimal schedules. To solvethe graph labeling problem, we presented a game-theoreticsolution. We also showed that the detection (isolation) perfor-mance of monitoring devices deployed within the network wasbetter when the placement and scheduling problems for thesedevices were solved simultaneously as compared to the case inwhich the optimal placement of these devices was solved firstfollowed by the computation of optimal schedules. Our graphlabeling formulation and game-theoretic solution allowed usto simultaneously solve placement and scheduling problems.We demonstrated results for various networks including waterdistribution and random networks. The graph labeling problempresented here could be useful in solving resource allocationproblems in other domains such as multi-agent and multi-robotsystems. Moreover, the proposed approach could be effectivein characterizing and comparing network topologies in termsof the coverage performance, i.e., using resource-constraintmonitoring devices, which network structures could result inhigher detection and isolation performances?A
PPENDIX P ROOF OF T HEOREM u iscovered in an arbitrary time slot i . In this direction, we observethatPr (cid:0) u is not covered in the i th slot (cid:1) = Pr (cid:18) u is not activein the i th slot (cid:19) Y v ∈ N ( u ) Pr (cid:18) v is not activein the i th slot (cid:19) (21) Here,Pr (cid:18) u is not activein the i th slot (cid:19) = 1 − (cid:18) σ − k − (cid:19)(cid:18) kσ (cid:19) = k − σk . (22)The second term in (21) is the probability that none ofthe nodes in the neighborhood of node u is active in the i th time slot. The probability of having j neighbors in N ( u ) in arandom geometric graph is given by Poisson distribution, i.e., ( λπr ) j e − λπr j ! . Thus, Y v ∈ N ( u ) Pr (cid:0) v is not active in the i th slot (cid:1) = ∞ X j =0 (cid:0) λπr (cid:1) j e − λπr j ! (cid:18) k − σk (cid:19) j (23)Inserting (22) and (23) in (21), we getPr (cid:0) u is not covered in the i th slot (cid:1) = e − λπr ( k − σ ) k ∞ X j =0 j ! (cid:18) λπr ( k − σ ) k (cid:19) j = e − λπr ( k − σ ) k e λπr k − σ ) k = (cid:18) k − σk (cid:19) e − σλπr k (24). Thus, we get the desired result asPr (cid:0) u is covered in the i th slot (cid:1) = 1 − (cid:18) k − σk (cid:19) e − σλπr k R EFERENCES[1] M. Farley and S. Trow,
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