Near-optimal Observation Selection using Submodular Functions

Andreas KrauseCarnegie Mellon University

Carlos GuestrinCarnegie Mellon University

Abstract

AI problems such as autonomous robotic exploration, auto-matic diagnosis and activity recognition have in common theneed for choosing among a set of informative but possibly ex-pensive observations. When monitoring spatial phenomenawith sensor networks or mobile robots, for example, we needto decide which locations to observe in order to most effec-tively decrease the uncertainty, at minimum cost. These prob-lems usually are NP-hard. Many observation selection objec-tives satisfy submodularity, an intuitive diminishing returnsproperty – adding a sensor to a small deployment helps morethan adding it to a large deployment. In this paper, we surveyrecent advances in systematically exploiting this submodu-larity property to efficiently achieve near-optimal observationselections, under complex constraints. We illustrate the ef-fectiveness of our approaches on problems of monitoring en-vironmental phenomena and water distribution networks.

IntroductionIn many artificial intelligence applications, we need to ef-fectively collect information in order to make best decisionsunder uncertainty. In this setting, we usually need to trade-off the informativeness of the observation and the cost ofacquiring the information. When monitoring spatial phe-nomena using sensor networks, for example, we can decidewhere to place sensors. Since we have a limited budget,we want to place the sensors only at the most informativelocations. Hence we want to select a set A ⊆ V of lo-cations, and want to maximize some objective F (A) mea-suring the informativeness of the selected locations, subjectto a constraint on the number of sensors we can place, i.e.,|A| ≤ k. If we collect information using mobile robots,or if the placed sensors need to communicate wirelessly,we have more complex constraints on how we can makethese observations. In the multi-robot case, for example,the chosen locations must lie on a collection of paths. Inthe wireless communication case, the locations need to beclose enough to enable efficient wireless communication.These optimization problems are generally NP-hard (Krause& Guestrin 2005b). Therefore, heuristic approaches havecommonly been applied, which cannot provide performanceguarantees. In our recent work (Krause & Guestrin 2005c;Guestrin, Krause, & Singh 2005; Krause et al. 2006; 2007;Singh et al. 2007), we have presented several efficient algo-rithms for approximately solving these optimization prob-lems. In contrast to the heuristic approaches, our algorithms

Copyright c© 2007, American Association for Artificial Intelli-gence (www.aaai.org). All rights reserved.

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Figure 1: (a) Monitoring Lake Fulmor [courtesy of NAMOS(http://robotics.usc.edu/∼namos) & CENS (http://cens.ucla.edu)];(b) Water network sensor placement in BWSN challenge.

have rigorous theoretical performance guarantees. In or-der to achieve these guarantees, we exploit a key propertyof many natural observation selection objectives: In mostproblems, adding an observation helps more, if we havemade few observations so far, and helps less if we alreadyhave made many observations. This intuitive diminishing re-turns property is formalized by the concept of submodular-ity, which will be introduced in the following. We will illus-trate the methodology on two important observation selec-tion problems we have considered in the past, environmentalmonitoring, and securing water distribution networks.

Applications and Selection ObjectivesEnvironmental MonitoringConsider, for example, the monitoring of algae biomass ina lake. High levels of pollutants, such as nitrates, can leadto the development of algal blooms. These nuisance algalblooms impair the beneficial use of aquatic systems. Mea-suring quantities, such as pollutants, nutrients, and oxygenlevels, can provide biologists with a fundamental character-ization of the ecological state of such a lake. Unfortunately,such sensors are expensive, and it is impractical to cover thelake with these devices. Hence, a set of robotic boats (as inFig. 1 (a)) have been used to move such sensors to variouslocations in the lake (Dhariwal et al. 2006). In order to makemost effective measurements, we want to move the robots,such that the few observed values help us predict the algaebiomass everywhere in the lake as well as possible.

Geometric objectives. A common approach for observa-tion selection has been to use a geometric approach. Withevery potential sensing location s ∈ V , one associates a ge-ometric shape Rs, a sensing region, which is usually takenas a disk (c.f., Bai et al. 2006), or a cone (for modeling cam-era viewing fields). Once an observation has been made,all points in the sensing region Rs are considered observed.The objective function is then FG(A) = |

⋃s∈A Rs|, where

|⋃

s∈A Rs| is the cardinality (or volume) of the covered set.

Probabilistic objectives. The assumption made bygeometric objectives is that every location is either fullyobserved or unobserved. Often, observations are noisy,and by combining several observations we can achievebetter coverage. To address this uncertainty, the spatialphenomena is often modeled probabilistically. In theabove example, we would discretize the lake into finitelymany locations V , associate a random variable Xs withevery location s ∈ V and describe a joint distributionP (XV) = P (Xs1 , . . . ,Xsn) over all random variables. Wecan then use observations at a set XA = xA to predict thephenomenon everywhere, by considering the conditionaldistributions P (XV\A | XA = xA). We can also use thisconditional distribution to quantify the uncertainty in theprediction. A good observation selection will have smalluncertainty in the prediction everywhere.

Several such objective functions have been considered inthe literature. In (Guestrin, Krause, & Singh 2005), we con-sider the mutual information between a set of chosen lo-cations and their complement. This criterion is defined asFMI(A) = I(XA;XV\A) = H(XV\A) − H(XV\A | XA).It hence measures the decrease in Shannon entropy in theunobserved locations (prior uncertainty H(XV\A)) achievedby making observations, leading to posterior uncertaintyH(XV\A | XA). As a model, we use Gaussian Processes(GPs), which have been frequently used to model spatialphenomena (c.f., Cressie 1991). The distribution P (XV) isthen a multivariate normal distribution and the mutual infor-mation can be computed in closed form.

Securing Water Distribution SystemsWater distribution networks (as in Fig. 1 (b)), which bringthe water to our taps, are complex dynamical systems, withimpact on our everyday lives. Accidental or malicious intro-duction of contaminants in such networks can have severeimpact on the population. Such intrusions could potentiallybe detected by a network of sensors placed in the water dis-tribution system. The high cost of the sensors makes opti-mal placement an important issue. We recently participatedin the Battle of Water Sensor Networks (BWSN), an interna-tional challenge for designing sensor networks in several re-alistic settings. A set of intrusion attacks S was considered;each attack refers to the hypothetical introduction of con-taminants at a particular node of the network, at a particulartime of day, and for a specified duration. Using EPANET 2.0(Rossman 1999), a simulator provided by the EPA, the im-pact of any given attack can be simulated, as well as, for anyconsidered sensor placement A ⊆ V , the expected time todetection and estimated affected population computed. Anoptimal sensor placement minimizes these objectives.Minimizing adverse effects. In the water distribution set-ting, we want to minimize adverse effects caused by a con-taminant introduction. For every possible contamination at-tack i ∈ S, we can compute the penalty πi(t) incurred whendetecting the intrusion at time t (where t = 0 at the start ofthe simulation). For example, πi can measure the estimatedpopulation affected (across the entire network) by attack i attime t. For a sensor s ∈ V and attack i ∈ S, we can use thesimulation to determine the time of detection, T (s, i). Nat-

urally, for a set of sensors, T (A, i) = mins∈A T (s, i). Wecan then define the penalty reduction for a sensor placementA ⊆ V as Ri(A) = πi(Tmax) − πi(T (A, i)). The finalobjective is then the expected penalty reduction, where theexpectation is taken w.r.t. a probability distribution P overthe attacks: FR(A) =

∑i P (i)Ri(A).

Submodularity of Observation SelectionIn the previous section, we presented a collection of practi-cal observation selection objectives, FG, FMI, and FR. Allthese objectives (and many other practical ones) have thefollowing key property in common: Adding an observationhelps more if we have made few observations so far andhelps less if we have made many observations. This dimin-ishing returns property is visualized in Fig. 2(c). The scoreobtained is a concave function of the number of observationsmade. This effect is formalized by the concept of submod-ularity (c.f., Nemhauser, Wolsey, & Fisher 1978). A real-valued function F , defined on subsets A ⊆ V of a finite setV is called submodular if for all A ⊆ B ⊆ V and for all s ∈V\B, it holds that F (A∪s)−F (A) ≥ F (B∪s)−F (B).Hence, many observation selection problems can be reducedto the problem of maximizing a submodular set functionsubject to some constraints (e.g., the number of sensorswe can place). Constrained maximization of submodularfunctions (and all objective functions discussed here) ingeneral is NP-hard. The following sections survey efficientapproximation algorithms with provable quality guarantees.

Optimizing Submodular FunctionsCardinality and Budget Constraints. We first considerthe problem where each location s ∈ V has a fixed positivecost c(s), and the cost of an observation selection A ⊆ V ,c(A) is defined as c(A) =

∑s∈A c(s). The problem then is

to solve the following optimization problem:A∗ = argmaxA F (A) subject to c(A) ≤ B, (1)

for some nonnegative budget B. The special case c(A) =|A| is called the unit cost case.

A natural heuristic which has been frequently used for theunit cost case is the greedy algorithm. This algorithm startswith the empty setA = ∅, and iteratively, in round j, it findsthe location sj = argmaxs F (A ∪ s) − F (A), i.e., thelocation which increases the objective the most, and addsit to the current set A. The algorithm stops after B sensorshave been chosen. Surprisingly, this straight-forward heuris-tic has strong theoretical guarantees:Theorem 1 (Nemhauser, Wolsey, & Fisher (1978)) If Fis a submodular, nondecreasing set function and F (∅) = 0,then the greedy algorithm is guaranteed to find a set A,such that F (A) ≥ (1− 1/e) max|A|=B F (A).Hence, the greedy solution achieves an objective value thatachieves at least a factor (1 − 1/e) (≈ 63%) of the optimalscore. In order to apply Theorem 1, we must verify that theobjective functions are indeed nondecreasing and F (∅) = 0.

1FMI is approximately nondecreasing (Guestrin, Krause, &Singh 2005), which preserves the approximation guarantee up toan arbitrarily small additive error.

A set function is called nondecreasing if for all A ⊆ B ⊆ Vit holds that F (A) ≤ F (B). All objective functionsdiscussed above are nondecreasing1, and satisfy F (∅) = 0.

The results extend to cases, where c(s) is arbitrary.Here, a slight modification of the algorithm, combiningthe greedy selection of the location with highest benefit-cost ratio sj = argmaxs

F (A∪s)−F (A)c(s) , with a par-

tial enumeration scheme achieves the same constant fac-tor (1 − 1/e) approximation guarantee (Sviridenko 2004;Krause & Guestrin 2005a).Path Constraints. In the robotic lake monitoring exampledescribed above, the observation selection problem is evenmore complex: Here, the locations s ∈ V are nodes in agraph G = (V, E , d), with edge weights d : E → R encod-ing distance. The goal is to find a collection of paths Pi inthis graph, one for each of the k robots, providing highestinformation F (P1 ∪ · · · ∪ Pk), subject to a constraint onthe path length. Fig. 2(a) visualizes this setting. In this set-ting, the simple greedy algorithm – selecting most informa-tive observations, such that the robot always keeps enoughfuel to return to the goal – performs arbitrarily badly.

In (Singh et al. 2007), we show that the multiple robotproblem can be reduced to optimizing paths for a singlerobot. We prove that if we have any single robot algorithmachieving an approximation guarantee of η (i.e., the solu-tions are at most a fraction (1/η) from optimal), a simplesequential allocation strategy – iteratively optimizing onepath at a time – achieves (nearly) the same approximationguarantee (η+1) for the case of multiple robots.

Chekuri & Pal (2005) proposed an algorithm for op-timizing a single path P subject to a path-length con-straint, achieving near-maximum submodular objectivevalue F (P). This algorithm achieves an approximationguarantee of η = log |P∗|, where P∗ is the the optimal path.Unfortunately, the algorithm is only pseudopolynomial; itsrunning time is O((|V|B)log |V|). In (Singh et al. 2007),in addition to generalizing this algorithm to multiple robotswith guarantee η+1, we present a spatial decomposition ap-proach and devise branch and bound schemes, which makethe algorithm of Chekuri & Pal practical. In our approach,the lake is partitioned into a grid of cells. These cells areconsidered nodes in a new graph with far fewer nodes, andthe algorithm of Chekuri et.al. is applied on this new graph.Using an adapted version of our approach, computationalcost can be traded off with the achieved objective value.

Theorem 2 Let P∗ be the optimal solution for the singlerobot problem with a budget of B and spatial decompositioninto M cells of size L. Then our algorithm finds a solution Pwith information value of at least F (P) ≥ 1−1/e

1+log2 N F (P∗),whose cost is no more thanO(LB), in timeO((MB)log M ).Communication Constraints. Often, when placing sen-sors, the sensors must be able to communicate with eachother. In the case of wireless sensor networks (WSN), forexample, the sensors need to communicate through lossylinks which deteriorate with distance, causing message lossand increased power consumption. We can also model thisproblem by considering the potential locations as nodes in a

graph G = (V, E , d). In (Krause et al. 2006), we assign theexpected number of retransmissions (i.e., the expected num-ber of times a message has to be resent in order to warrantsuccessful transmission) between locations u and v as edgecost d(u, v). The cost of a set of locations A ⊂ V is then theminimum cost of connecting the locations A in the graph G.More precisely, the cost c(A) of a placement is the sum ofthe edge costs of the cheapest (Steiner) tree containing thenodes in A. Under this new cost function, we consider themaximization problem (1). Here, as in the path constraintcase, the greedy algorithm performs arbitrarily poorly. In(Krause et al. 2006), we present a randomized approxima-tion algorithm for approximately solving this problem. Thisalgorithm exploit an additional property of many observa-tion selection objectives: Intuitively, sensors which are farapart make roughly independent observations. Formally, re-quire that there are constants r ≥ 0 and 0 < γ ≤ 1, suchthat, if two sets of locations A and B are at least distance rapart, it holds that F (A∪B) ≥ F (A)+γF (B). In this case,we say that F is (r, γ)-local. This locality property is em-pirically satisfied for mutual information FMI as we show in(Krause et al. 2006). For geometric objectives, FG is (r, γ)local for γ = 1 and r = 2maxs diam(Rs).Theorem 3 Given a graph G = (V, E, d), and an (r, γ)-local monotone submodular function F , we can find a treeT with cost O(r log |V|) × c(T ∗), spanning a set A withF (A) ≥ Ω(γ) × F (A∗). The algorithm is randomized andruns in polynomial-time.Theorem 3 shows that we can solve the problem (1) to pro-vide a sensor placement for which the communication costis at most a small factor (at worst logarithmically) larger,and for which the expected sensing quality is at most aconstant factor worse than the optimal solution. Fig. 2(b)compares the performance of our approximation algorithm –pSPIEL – with the (intractable) optimal algorithm (exhaus-tive search), as well as two reasonable heuristics, when opti-mizing a WSN to monitor temperature in a building. pSPIELperforms much closer to the optimal solution.

We used pSPIEL to design a WSN for monitoring lightin the Intelligent Workplace at CMU. Fig. 2(d) compares amanual deployment of 20 sensor motes with 19 motes de-ployed by pSPIEL. Surprisingly, pSPIEL does not extendthe placement in the Western area of the building. Duringdata collection, the Western part was unused at night; henceaccurate indoor light prediction requires many sensors in theEastern part. Daylight intensity can be well predicted evenwithout sensors in the Western part. pSPIEL lead to a 30%decrease in RMS error, and reduced the communication cost.

ExtensionsOnline guarantees for observation selection. The(1 − 1/e) bound for the greedy algorithm is offline, as wecan state it before running the algorithm. However, we canexploit submodularity even further to compute – often muchtighter – online bounds on the optimal solution. We look atthe current solution A, obtained, for example, by the greedyor any other algorithm. For each observation s which hasnot been considered yet, let δs = F (A∪s)−F (A) be theimprovement in score we would get by adding observation

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(c) Submodular returns (d) PlacementsFigure 2: (a) Informative paths planned for 3 boats on Lake Fulmor. (b) pSPIEL achieves near-optimal cost-benefit ratio. (c) Submodularitygives tight online bounds for any selection. (d) WSN deployments in CMU’s Intelligent Workplace (M20: manual, pS19: using pSPIEL).

s. Assume we want to select k observations. Let s1, . . . , sk

be the k observations for which δs is largest. Then F (A∗) ≤F (A) +

∑kj=1 δsj . This bound allows us to get guarantees

about arbitrary observation selections A. Fig. 2(c) showsoffline and online bounds achieved for placing an increasingnumber of sensors, when optimizing the time to detectionwhen securing water networks. The online bound is muchtighter, and closer to the score achieved by the greedy al-gorithm. These bounds also allows us to use mixed-integerprogramming (MIP) (Nemhauser & Wolsey 1981), whichcan be used to solve for the optimal observation selection,or to acquire even tighter bounds on the optimal solution bycomputing the linear programming relaxation.

Fast implementation. Submodularity can also be used tospeed up the greedy algorithm. The key observation is thatthe incremental benefits δs(A) = F (A ∪ s) − F (A) aremonotonically nonincreasing inA. By exploiting this obser-vation, we can “lazily” recompute only the δs(A) only forthe “most promising” locations s (Robertazzi & Schwartz1989). On the large BWSN network, this lazy implemen-tation decreased the running time from 30 hours to 1 hour.

Robust Sensor Placement. Sensor nodes are susceptibleto failures, e.g., through loss of power. In (Krause et al.2007), we describe an approach for optimizing placementswhich are robust against failures. The key observation is thatsubmodular functions are closed under nonnegative linearcombinations. Hence, the expected placement score overall possible failure scenarios is submodular. Similarly, thisobservation can also be used to handle uncertainty in themodel (network links, parameter uncertainty, etc.).

Multicriterion Optimization. Often, we have multipleobjectives F1, . . . , Fm which we want to simultaneously op-timize. In the water network example, we want to simulta-neously maximize the likelihood of detecting an intrusion,as well as minimize the expected population affected byan intrusion. In general, two placements A and B mightbe incomparable, i.e., neither A nor B is better in all mcriteria. Hence, the goal in multicriterion optimization isto find Pareto-optimal placements A, i.e., those such thatthere cannot exist a placement B for which Fi(B) ≥ Fi(A)for 1 ≤ i ≤ m, and Fj(B) > Fj(A) for some j. Acommon technique for finding such Pareto-optimal solu-tions is scalarization, i.e., choosing a set of positive weightsλ1, . . . , λm > 0 and maximizing F (A) =

∑i λiFi(A).

Since positive linear combinations of submodular functionsare still submodular, the resulting scalarized problem is a

submodular maximization problem, which can be addressedusing algorithms surveyed in this paper.

Implications for AIObservation selection problems are ubiquitous in AI. Wenow discuss several examples of fundamental AI problemswhich could potentially be addressed using our algorithms.

Fault Diagnosis. A classical AI problem is probabilis-tic fault diagnosis as considered, e.g., by Zheng, Rish, &Beygelzimer (2005). Here, one wants to select a set of teststo probe the system (e.g., a computer network), in orderto diagnose the state XU of unobservable system attributesU (e.g., presence of a fault). As criterion of informative-ness, Zheng, Rish, & Beygelzimer (2005) use the informa-tion gain FIG(A) = I(XA;XU ) of the selected tests A ⊆ Vabout the target attributes U . Krause & Guestrin (2005c)show that for a wide class of probabilistic graphical mod-els, this criterion FIG is submodular. Often, the cost of atest depends on the tests already chosen (i.e., test A mightbe cheaper if test B has been chosen). Algorithms as sur-veyed in this paper can potentially be used to approximatelysolve problems with such complex constraints. Related ap-plications are test selection for expert systems, e.g., in themedical domain, as well as optimizing test cases for soft-ware testing coverage.

Robotic Exploration. An important problem in roboticsis Simultaneous Localization and Mapping (SLAM) (c.f.,Sim & Roy (2005)). There, robots make observations in or-der to simultaneously localize themselves, and detect land-marks to create a map. Commonly, probabilistic modelssuch as Kalman Filters and extensions thereof are used totrack the uncertainty about the state variables XU (landmarkand robot locations). Using such a probabilistic model, ob-servations A can be actively collected by controlling therobot, with the goal of maximizing the information gainI(XA;XU ) about the unobserved locations, as consideredby Sim & Roy (2005). Since the robot’s movement is con-strained by obstacles, algorithms as those surveyed in thispaper can potentially be used to plan such informative paths.Minimizing Human Attention. In this paper, we consid-ered problems where each observation is expensive to ac-quire. In many applications however, we have an abundanceof information at our disposal, and the scarce resource isthe attention of the human, to which this information shouldbe presented. One example is active learning (c.f., Hoi etal. (2006)), where the goal is to learn a classifier, but thetraining data is initially unlabeled. A human expert can be

requested to label a set of examples, but each label is expen-sive. In some applications, complex constraints are present;e.g., when labeling a stream of video images, labeling a se-quence of consecutive images is cheaper than labeling in-dividual images. Hoi et al. (2006) show that certain activelearning objectives are (approximately) submodular, henceour algorithms could potentially be used in this context.

Another example is information presentation. Here, wewant to, e.g., choose a subset of email requests to display,or automatically summarize a large document, etc. In theseproblems, each selection of emails or sentences achieves acertain utility to the user, which is to be maximized. Wealso have complex constraints (e.g., reading long emails ismore “expensive”; when reading an email in a communica-tion thread, the initiating email has to be presented as well,etc.), which could potentially be addressed using our algo-rithms. Here, the key open research challenge is to under-stand, which classes of utility functions are submodular.

Influence Maximization. Kempe, Kleinberg, & Tar-dos (2003) show that the problem of selecting a set of peoplein a social network with maximum influence is a submodu-lar optimization problem. Here, an initial set of people are,e.g., targeted by a marketing campaign. The probability thatany person (initially untargeted) in the network becomes in-fluenced, depends on how many of their neighbors are al-ready influenced, as well as the strength of their interaction.Rather than simply asking for the best set of k people to tar-get in order to maximize the influence on the whole network,one could use our algorithms to consider more complex con-straints. For example, a marketing tour (e.g., an author sign-ing books) could proceed through the country, and the costto influence the next person depends on the tour’s current lo-cation, and the problem would be to find the optimum suchtour (or multiple tours at the same time).

Monotonic Constraint Satisfaction. A key problem inAI is finding satisfying assignments to logical formulas,or maximizing the number of satisfied clauses. Considerthe special case, where we are given a boolean formulaf(x1, . . . , xn) = C1 ∧ · · · ∧ Cm, where each Ci is aclause with weight wi ≥ 0, and containing the disjunc-tion of any set of positive literals (i.e., the formula doesnot contain negations). For a subset A ⊆ 1, . . . , n ofindices of variables xi set to true, the function F (A) =∑

i:Ci satisfied by some xi∈A wi is monotonic and submodular.Hence, the problem of selecting a set A of k variables toset to true maximizing the weight of satisfied clauses is asubmodular maximization problem. Our algorithms couldalso allow us to find such assignments subject to more com-plex constraints among the variables, e.g., where setting x1

to true reduces the cost of setting x4 to true.

ConclusionsWe have reviewed recent work on optimizing observation se-lections. Many natural observation selection objectives aresubmodular. We demonstrated how submodularity can beexploited to develop efficient approximation algorithms withprovable quality guarantees for observation selection. Thesealgorithms can handle a variety of complex combinatorial

constraints, e.g., requiring that the selected observations lieon a collection of paths. Submodularity also provides a pos-teriori bounds for any algorithm, and can be extended to,e.g., handle failing sensors, and multi-criterion optimization.

We are convinced that even beyond the problems con-sidered in this paper, submodular optimization can help totackle important AI search problems, and submodularity isa concept which can be more widely considered in AI.

ReferencesBai, X.; Kumar, S.; Yun, Z.; Xuan, D.; and Lai, T. H. 2006.Deploying wireless sensors to achieve both coverage and connec-tivity. In MobiHoc.Chekuri, C., and Pal, M. 2005. A recursive greedy algorithm forwalks in directed graphs. In FOCS, 245–253.Cressie, N. A. C. 1991. Statistics for Spatial Data. Wiley.Dhariwal, A.; Zhang, B.; Stauffer, B.; Oberg, C.; Sukhatme,G. S.; Caron, D. A.; and Requicha, A. A. 2006. Networkedaquatic microbial observing system. In ICRA.Guestrin, C.; Krause, A.; and Singh, A. 2005. Near-optimalsensor placements in Gaussian processes. In ICML.Hoi, S. C. H.; Jin, R.; Zhu, J.; and Lyu, M. R. 2006. Batch modeactive learning and its application to medical image classification.In ICML.Kempe, D.; Kleinberg, J.; and Tardos, E. 2003. Maximizing thespread of influence through a social network. In KDD.Krause, A., and Guestrin, C. 2005a. A note on the budgetedmaximization of submodular functions. Technical report, CMU-CALD-05-103.Krause, A., and Guestrin, C. 2005b. Optimal nonmyopic valueof information in graphical models - efficient algorithms and the-oretical limits. In Proc. of IJCAI.Krause, A., and Guestrin, C. 2005c. Near-optimal value of infor-mation in graphical models. In UAI.Krause, A.; Guestrin, C.; Gupta, A.; and Kleinberg, J. 2006.Near-optimal sensor placements: Maximizing information whileminimizing communication cost. In IPSN.Krause, A.; Leskovec, J.; Guestrin, C.; VanBriesen, J.; andFaloutsos, C. 2007. Efficient sensor placement optimization forsecuring large water distribution networks. Submitted to the Jour-nal of Water Resources Planning an Management.Nemhauser, G. L., and Wolsey, L. A. 1981. Studies on Graphsand Discrete Programming. North-Holland. chapter MaximizingSubmodular Set Functions: Formulations and Analysis of Algo-rithms, 279–301.Nemhauser, G.; Wolsey, L.; and Fisher, M. 1978. An analysisof the approximations for maximizing submodular set functions.Mathematical Programming 14:265–294.Robertazzi, T. G., and Schwartz, S. C. 1989. An accelerated se-quential algorithm for producing D-optimal designs. SIAM Jour-nal of Scientific and Statistical Computing 10(2):341–358.Rossman, L. A. 1999. The epanet programmer’s toolkit for anal-ysis of water distribution systems. In Annual Water ResourcesPlanning and Management Conference.Sim, R., and Roy, N. 2005. Global a-optimal robot exploration inslam. In ICRA.Singh, A.; Krause, A.; Guestrin, C.; Kaiser, W.; and Batalin, M.2007. Efficient planning of informative paths for multiple robots.In IJCAI.Sviridenko, M. 2004. A note on maximizing a submodular setfunction subject to knapsack constraint. Ops Res Lett 32:41–43.Zheng, A. X.; Rish, I.; and Beygelzimer, A. 2005. Efficient testselection in active diagnosis via entropy approximation. In UAI.