XOR-Sampling for Network Design with Correlated Stochastic Events

Xiaojian Wu∗ Yexiang Xue∗ Bart Selman Carla P. GomesDepartment of Computer Science, Cornell University, Ithaca, NY

xiaojian@cornell.edu {yexiang,selman,gomes}@cs.cornell.edu

Abstract

Many network optimization problems can be for-mulated as stochastic network design problems inwhich edges are present or absent stochastically.Furthermore, protective actions can guarantee thatedges will remain present. We consider the prob-lem of finding the optimal protection strategy un-der a budget limit in order to maximize some con-nectivity measurements of the network. Previousapproaches rely on the assumption that edges areindependent. In this paper, we consider a more re-alistic setting where multiple edges are not inde-pendent due to natural disasters or regional eventsthat make the states of multiple edges stochasti-cally correlated. We use Markov Random Fieldsto model the correlation and define a new stochas-tic network design framework. We provide a novelalgorithm based on Sample Average Approxima-tion (SAA) coupled with a Gibbs or XOR sampler.The experimental results on real road network datashow that the policies produced by SAA with theXOR sampler have higher quality and lower vari-ance compared to SAA with Gibbs sampler.

1 IntroductionMany problems such as influence maximization [Kempe etal., 2003], river network design [Neeson et al., 2015] and pre-disaster preparation [Peeta et al., 2010] can be formulated asstochastic network design problems, in which the edges ofthe network are present or absent stochastically. The goalis to take protection actions, which reinforce the connectiv-ity of a subset of edges with a limited budget, to maximizethe connectivity of the network under stochastic events. Forexample, in the road network design problem, as shown inFigure 1, edges can be present or absent from the networkstochastically when a natural disaster strikes. The problemthen is to invest smartly under a limited budget to strengthena few key road segments, so that ambulance centers can stillremain connected to population centers under the attacks ofmultiple natural disasters.

∗equal contribution.

Despite the worst-case PP-hard complexity to evaluatethe connectivity values under a fixed protection plan [Dyerand Stougie, 2006], approximate algorithms based on Sam-ple Average Approximation (SAA) have been developed tofind nearly optimal solutions [Sheldon et al., 2010; Wu etal., 2015], which convert the original stochastic optimizationproblem into a deterministic one, by optimizing the connec-tivity objectives over a fixed set of sample scenarios.

The success of previous approaches depends on the coreassumption that the state of each edge (e.g., present or absent)is independent of those of other edges. Under this assump-tion, high-quality samples can be drawn with simple schemessuch as flipping a biased coin for each edge. Also, it has beenshown that often a small number of samples (e.g., 10) is suf-ficient to find a good policy [Sheldon et al., 2010], despitethe fact that a much larger number of samples is needed fortheoretical guarantees [Kleywegt et al., 2002].

Nevertheless, the states of edges are often correlated inpractice. For example, in the road network design problemshown in Figure 1, edges tend to fail concurrently when a re-gional event (e.g., a natural disaster such as an earthquake ora snow storm) disables the connectivity of all edges in a givenarea. In this paper, we study this more realistic setting wherethe states of multiple edges can be correlated. We proposeto use a general probabilistic model – Markov Random Field(MRF) – to model the joint distribution of edge states, whichcan be built with domain knowledge or learned from data.

We make several contributions: (1) We propose a newstochastic network design problem framework, taking intoaccount correlated stochastic events based on MRF; (2) Weextend the SAA framework with a novel sampling schemeto convert the original stochastic network design probleminto a deterministic network design problem, which is for-mulated as a Mixed Integer Program (MIP) and solved byCPLEX 12.6. As we will show, also mentioned in pa-per [Kumar et al., 2012], the MIP can only scale to a smallnumber of samples (e.g., 30 samples). Consequently, asmall but good quality set of samples is critical to producehigh-quality solutions; (3) Another key contribution of thiswork is the coupling of XOR-sampling [Gomes et al., 2007;Ermon et al., 2013] with SAA; (4) We show that XOR-sampling coupled with SAA outperforms the standard Gibbssampler when coupled with SAA. As shown in paper [Ermonet al., 2013], the XOR sampler can produce MRF samples

Figure 1: In an exemplified road network design problem,the goal is to maintain connectivity between ambulance cen-ters (green dots) and population centers (blue dots) in case ofnatural disasters. The stochasticity of multiple edges is cor-related. For example, one natural disaster such as earthquakemay affect the connectivity of all edges in its range (shown inred circles).

with better quality but often has worse scalability than theGibbs sampler. However, in our setting, the bottleneck of theruntime is in solving the MIP, not the time required by theXOR sampler. This is because the MIP solver can only han-dle relatively small samples (therefore they have to be care-fully selected). Given the relatively small sample sizes, thetime required by the XOR sampler to produce such samplesis negligible. (5) We test our algorithms on the flood prepara-tion problem with real-world road networks [Wu et al., 2016].The goal is to maintain the connectivity of road networks foremergency medical services (EMS). The results show thatSAA with XOR sampling produces policies that are neverinferior in quality compared to SAA with Gibbs. In otherwords, the policies produced by the SAA with XOR samplingconsistently have similar or better quality than SAA withGibbs sampling. Furthermore, in some cases, a small numberof samples (e.g., 10) from the XOR sampler, is enough for theSAA to converge. Also, the solution quality with XOR sam-pling is consistently good, not very sensitive to the varianceof samples used in SAA. In comparison, the solution qual-ity produced by a Gibbs sampler has a higher variance whenthe number of samples used in SAA is small. These observa-tions suggest that, on large-scale networks, the SAA methodcan still produce a good policy if we use high-quality samplessuch as those from XOR sampling.

2 Problem StatementOne important application of the stochastic network designproblem is road network design [Peeta et al., 2010]. Naturaldisasters such as earthquakes and floods break road segmentsand destroy the connectivity of the whole road network. Thepaper [Wu et al., 2016] introduces the problem of flood prepa-ration for EMS as shown in Fig. 1. In this problem, duringfloods, roads may be washed out and paths connecting ambu-lance centers to patients are cut off. To maintain good connec-tivity of the road network in case of an emergency, spatiallycritical roads should be identified and protected beforehand,at certain costs. The question is how to find these spatiallycritical roads, given a budget limit, such that most of the pa-

tients can be reached when floods happen. This problem is anexample of the stochastic network design problem.

A stochastic network design problem is defined on a di-rected graph G = {V,E} with a source node s. Each nodeis associated with a weight w(v) (e.g., population density).Each edge e ∈ E is associated with a binary random vari-able θe that describes the state of the edge during disas-ters. θe = 1 means that the edge e is present or not de-stroyed. θe = 0 means that e is absent or destroyed. Letθ = {θe, e ∈ E} denote the vector of random variables ofall edges, each random variable taking value in {0, 1}|E| andPr(θ) denote their joint probability distribution. Pr(θ) maybe constructed from domain knowledge or learned from real-world data. Note that we use the bold symbol θ for randomvariables and the unbold symbol θ for their values. For agiven θ, the indicator function I(s v|θ) indicates whethernode v is reachable from the source s in the network by apath of all present edges. If node v is reachable, it equals to1. Otherwise, it equals to 0. The expected weighted numberof nodes that are reachable from the source s is written as

Eθ

[∑v∈V

I(s v|θ)w(v)

]=

∑θ∈{0,1}|E|

Pr(θ)∑v∈V

I(s v|θ)w(v)

(1)To be general, one protection action can protect one or

multiple edges. If an edge is protected, it is always present.Formally, we are given a collection of M edge sets E ={E1, E2, ..., EM} where each Es ⊆ E represents an actionto protect all edges in Es and is associated with a cost cs. Apolicy π is a subset of E that indicates all edges being pro-tected. Protecting an edge from u to v can be equivalentlyviewed as adding an edge (u, v) that is always present in thenetwork. That is, if (u, v) is protected by π, u is connected tov no matter θ(u,v) = 1 or 0, which makes the indicator func-tion depend on π as I(s v|θ, π). In other words, protectionactions only change the network structure but not Pr(θ), theprobability of the stochastic events such as natural disastersthat is independent of the network structure. The decisionmaking problem is to find the policy π, subject to a budgetlimit B, to maximize the expected weighted number of nodesthat are reachable from s, that is

maxπ⊆E

Eθ

[∑v∈V

I(s v|θ, π)w(v)

]s.t.

∑Es∈π

cs ≤ B (2)

This problem definition is general and subsumes many spe-cial cases. For example, in the real-world problem in ourexperiments, only a subset of edges are present or absentstochastically while the remaining edges are known to bepresent. For this special case, we can set the protection costsof those edges that are always present to zero so they are pro-tected automatically in the optimal solution.

Multiple Source Case In problem (2), it is assumed thatthere is only one source. It can be extended to multiplesources. For example, there are several ambulance centers,each of which is represented by a source. The house of apatient is encouraged to be reachable from multiple centersin case that ambulences in one center are busy and can’t re-spond. In this new problem formulation, instead of having

a single source s, we are given a set of sources S. Connect-ing to more sources increases the objective value accordingly.For each source in S, we count the weighted number of nodesthat are reachable from the source separately. The objectiveis to maximize the expected total weighted number of nodesthat are reachable from each source in S, that is,

maxπ⊆E

∑s∈S

Eθ

[∑v∈V

I(s v|θ, π)w(v)

]s.t.

∑Es∈π

cs ≤ B

(3)

In this case, each node in S issues a separate network flowto each other node. Later, we will introduce the algorithmto solve single source network design problems and it can beextended to solve multiple source problems.

Model Pr(θ) We propose to use the Markov random field(MRF) to model the joint probability distribution Pr(θ) of allvariables. MRF is a general model for the joint distribution ofmultiple correlated random variables. Using notions in MRF,we define the probability of edge states Pr(θ) as:

Pr(θ) =1

Z

∏α∈I

ψα({θ}α) (4)

where θα is a subset of variables in θ that the function ψαdepends on. ψα : {θ}α → R+ is a potential function,which maps every assignment of variables in {θ}α to a non-negative real value. I is an index set. Z is a normalizationconstant, which ensures that the probability adds up to one:Z =

∑θ∈{0,1}|E|

∏α∈I ψα({θ}α). In this paper, we will

also use Pr(θ) to represent the unnormalized probability den-sity:

Pr(θ) =∏α∈I

ψα({θ}α = {θ}α)

A potential function ψα({θ}α) defines the correlation be-tween all variables in the subset {θ}α. As a simple example,suppose {θ}α contains the subset of variables for all networkedges within a small spatial region. All roads in set {θ}αmay be either all present or all absent, depending on whethera regional event (e.g., an earthquake) happens or not. In thiscase, ψα({θ}α) will take high values when all variables are1 or 0 simultaneously. The structure of the MRF or the set Ican be built from domain knowledge and potential functionscan be learned from real-world data. The focus of this paperis not on how to construct the MRF but is how we solve prob-lem in Equation 3 in general when Pr(θ) is given in the formshown in Equation 4.

Theorem 1. For given policy π, evaluating the objective ofproblem (3) is #P complete.

Proof. This is a consequence of the #P -completeness of thegraph reliability problem [Dyer and Stougie, 2006], whichis to compute the probability that two given vertices is con-nected if the edges are present or absent independently. Ourproblem simplifies to the graph reliability problem when allpotential functions are singular.

maxy

1

N

N∑i=1

∑v∈V

w(v) ziv (1)

subject to∑(r,v)∈E

xirv −

∑(u,r)∈E

xiur =

{ ∑k 6=s z

ik if r = s

−zir if r 6= s∀r ∈ V, i =1 :N

(2)

0 ≤ xie ≤ (|V | − 1)

∑Es:e∈Es

yEs + θie

∀Es ∈ E, i =1 :N

(3)

0 ≤ xie ≤ |V | − 1 i = 1 : N, ∀e ∈ E (4)∑

Es∈EcsyEs ≤ B (5)

yEs ∈ {0, 1} ∀Es ∈ E ziv ∈ [0, 1] ∀i = 1 : N, ∀v ∈ V (6)

Figure 2: A compact MIP encoding of problem (5).

Other Applications The stochastic network design prob-lem (2) can be used to model other network optimizationproblems in a more realisitc way by allowing correlation be-tween random events. Here are two examples.

Barrier removal problem [Wu et al., 2014a; 2017] is animportant problem in computational sustainability where thegoal is to maximize the expected amount of habitat that onerandomly selected group of fish can reach. Previous problemformulation assumes that the probability that a group of fishis capable of passing a barrier is independent of the proba-bilities of passing other barriers. However, in practice, theprobability for a group of fish to pass a barrier mainly de-pends on the structure of the barrier and the capability of thefish. Therefore, it is more realistic that a fish is able to passmultiple structurally similar barriers or pass none of them.Our algorithm introduced later can be applied to solving thebarrier removal problem with correlated passage probabilitiesas well.

Influence maximization problem [Kempe et al., 2003] isa well-known problem in social computing, in which peoplewho are infected by an influence will continue to infect theirfriend stochastically. The independent cascade model as-sumes that each person will infect another person with someprobability independent from infections from other sources.Nevertheless, the infections coming from multiple friends canbe correlated, since friends from the same social group tendto post messages of similar content.

3 Our MethodIn this chapter, we introduce the algorithm to solve the sin-gle source network design problems (2) with Pr(θ) definedby (4). The algorithm can be extended to solve the multi-ple source network problem (3) with a little modification tothe MIP formulation. We use the Sample Average Approxi-mation [Kleywegt et al., 2002] (SAA) method to solve prob-lem (2). The basic idea is to draw N samples {θ1, ...., θN}of θ from the distribution Pr(θ) and convert the stochastic

optimization problem (2) into the following deterministic op-timization problem:

maxπ⊆E

1

N

N∑i=1

∑v∈V

I(s v|θi, π)w(v) s.t.∑Es∈π

cs ≤ B

(5)

Theorem 2. Let π∗ be the optimal policy of problem (5)with N samples and πOPT be the optimal policy of prob-lem (2). The number of samples for π∗ to converge to πOPTis Θ(|E|) [Kleywegt et al., 2002].

Problem (5) can be formulated as a MIP. Using the max-flow encoding from one source to multiple targets, we intro-duce a more compact MIP formulation than the one in pa-per [Wu et al., 2015] which usesO(|V |) times more variables(|V | is the number of nodes). This new compact formulationis shown in Fig. 2. The basic idea is as follows. For each sam-ple θi, and for each node v, we define a binary variable ziv ,which represents whether node v is reachable from s (= 1) inthe ith sample or not (= 0). Variables ziv can be relaxed to becontinous variable as shown in Fig. 2 because the value of zivis either 0 or 1 in the optimal solution. The objective functionin Equation 5 can be written as:

1

N

N∑i=1

∑v∈V

w(v) ziv

We then defines an indicator variable yEsfor the action to

protect all edges inEs. Protecting edges inEs costs cs, there-fore we have the budget constraint (5) in Fig. 2.

Next, we enforce flow constraints (2,3,4 in Figure 2) toguarantee that zik = 1 if and only if node k is reachable froms in the ith sample. The idea is to inject

∑k 6=s z

ik unit of flow

into the source s, and to pull out zik unit of flow (zik is either 1or 0, depending on whether node k is reachable from s in theith sample or not) at all nodes other than the source s. Herewe use xie as flow variables to represent the flow on edge ein the i-th sample. Constraint (2) is used to enforce a flowof this kind. If a node k is reachable from s (zik = 1), oneunit of flow will be consumed by node k. If k is not reachablefrom s (zik = 0), k will not consume any flow. Constraint(4) restricts the amount of flow on each edge between 0 and|V | − 1, which must be true since the total amount of flow is∑k 6=s z

ik and we have

∑k 6=s z

ik ≤ |V | − 1. Constraint (3)

ensures that no flow is allowed on edge e unless e is presentin the i-th sample (θe = 1) or e is protected by at least oneaction Es that covers e and the corresponding yEs

= 1.SAA is sensitive to the number of samples N used to ap-

proximate the objective function. Although in general a largeN is preferred in order to reduce the variance, the number ofvariables in the MIP encoding in Figure 2 scales linearly withrespect to N , which prevents us from using a large numberof samples (e.g., millions of samples) in practice. Under theindependent edge sampling case, despite theoretical results[Kleywegt et al., 2002] claim that O(|E|) number of sam-ples are required to obtain probabilistic guarantees, in prac-tice, people found that a small number of samples (e.g., 10)are sufficient to find a good policy [Sheldon et al., 2010].

We call this small set of samples as good samples or well-representative samples. That is, the number of samples isrelatively small but these samples can be used to find goodpolicies. To make SAA work, it is crucial to obtain a small-size (N small) but well-representative set of samples, so thatnot only the quality of the sample approximation is good, butalso the size of the sample set is small enough for MIP toscale. It is unclear and potentially more difficult to obtaingood samples when edges are correlated. In the followingsections, we introduce two candidate methods to attempt toobtain these so-called good samples {θ1, ..., θN}.

3.1 Gibbs SamplingGibbs sampling is a special case of the Metropolis-Hastingsalgorithm to obtain a sequence of samples which approxi-mates a given distribution Pr(θ). The high-level idea is tobuild a Markov chain whose transition probability is

Pr(θi|θ1, .., θi−1, θi+1, .., θ|E|) =Pr(θ1, . . . , θ|E|)

Pr(θ1, .., θi−1, θi+1, .., θ|E|).

It has been proved that the stationary distribution ofthe Markov chain converges to the target probability dis-tribution Pr(θ). Notice that we only need the un-ormalized density Pr(.), rather than Pr(.) to computePr(θi|θ1, .., θi−1, θi+1, .., θ|E|). This is the crucial fact thatmakes the transition of Gibbs sampling efficient in practice.

3.2 XOR SamplingSampling with XOR constraints [Gomes et al., 2007; 2006;Ermon et al., 2013; Chakraborty et al., 2013] was developed,thanks to the rapid progress in solving NP-complete problemsin the last decade. Despite the intractability in the worst case,modern MIP and Satisfiability Testing (SAT) solvers are ca-pable of exploiting the problem structure, and can scale upto millions of variables [Pipatsrisawat and Darwiche, 2010;Sontag et al., 2008; Riedel, 2008; Poon and Domingos, 2006;Gogate and Dechter, 2011]. XOR sampling transforms theproblem of sampling, which is an #P-complete problem,into solving satisfiability problems with additional XOR con-straints.

With XOR sampling, the resulting sampling distributioncan be proved to be arbitrarly close to uniform in the un-weighted case [Gomes et al., 2007; Chakraborty et al., 2013],and to be within a constant factor of the true probability dis-tribution in the weighted case [Ermon et al., 2013]. This is asignificant improvement compared with MCMC based tech-niques, which often require an exponential amount of samplesto reach its stationary distribution on many problems with in-tricate combinatorial structure.

XOR sampling is a special fit to our SAA framework. Therationale behind XOR sampling is to trade computational effi-ciency with provable near-optimal quality in the samples ob-tained. XOR sampling is more expensive than Gibbs sam-pling because it has to solve combinatorial problems at eachsampling step. The recent progress in constraint program-ming makes it possible to solve large-scale NP-completeproblems efficiently, therefore the benefit often outweightsthe cost. This is especially true in our SAA framework. First,

the MIP encoding for SAA prevents us from optimizing overmillions of examples simultaneously. Therefore, the sampleswe use in the optimization has to be small, but representativeenough for SAA to balance over all likely scenarios. Second,since the samples are obtained prior to the SAA optimization,we can afford to spend some time on obtaining good samples.We can even run several sampling programs in parallel. In ex-periments, we found that the time spent at the sampling stepwas negligible compared with solving the MIP.

General Idea To illustrate the high-level idea of XOR sam-pling, let us first consider the unweighted case where the un-normalized density Pr(θ) is either 1 or 0. Suppose thereare 2l different θ assignments which make Pr(θ) = 1. Wecall all these assignments “satisfying assignments”. An idealsampler draws samples from the set of satisfying assignments∆ = {θ : Pr(θ) = 1} uniformly at random; i.e., each mem-ber in ∆ has 2−l probability to be selected.

At a high level, XOR sampling obtains near-uniform sam-ples by finding one satisfying assignment subject to addi-tional randomly generated XOR constraints. To be specific,we keep adding XOR constraints to the original problem offinding satisfying assignments. We can prove that in expecta-tion, each newly added XOR constraint rules out about half ofthe satisfying assignments at random. Therefore, if we startwith 2l satisfying assignments in ∆, after adding l XOR con-straints, we will be left with only one satisfying assignment inexpectation, by which time we return this assignment as ourfirst sample. Because we can prove that the assignments areruled out at random in each step, we can guarantee that thelast assignment left must be a randomly chosen one from theoriginal set ∆. Please see [Gomes et al., 2007] for details.

For the weighted case, our goal is to guarantee that theprobability of sampling one θ is propotional to the unnormal-ized density Pr(θ). We implement a horizontal slice tech-nique to transform a weighted problem into an unweightedone, taking a 2-approximation. For the easiness of illus-tration, consider the simple case where P0 = minθ Pr(θ),Pk = maxθ Pr(θ) and Pk = 2kP0. Let δ = (δ0, . . . , δk−1) ∈{0, 1}k be a binary vector of length k. We sample (θ, δ) uni-formly at random from the following set ∆w using the afro-mentioned unweighted XOR sampling:

∆w = {(θ, δ) : Pr(θ) ≤ 2i+1P0 ⇒ δi = 0}.

If we sample (θ, δ) uniformly at random from ∆w and thenonly return θ, it can be proved that the probability of sam-pling θ is propotional to P02i−1 when Pr(θ) is sandwichedbetween P02i−1 and P02i. We refer the readers to [Ermon etal., 2013] for details.

4 Related WorkOur work is related to an active research field on reliabilityof physical communication networks under natural disastersand malicious coordinated attacks [Rahnamay-Naeini et al.,2011; Neumayer and Modiano, 2010]. In those papers, prob-abilistic models are built based on spatial point processes toevaluate the reliability of networks with geographically cor-

related failures. However, the problem of optimizing the re-liability of networks by taking protection actions is not dis-cussed and the proposed models are too complex to be usedto define solvable decision-making problems for network pro-tection.

Sheldon et al. (2010) formulate the problem of maximizingthe spread of cascades as stochastic network design and de-velop an algorithm combining SAA and MIP. In their model,all variables are assumed to be independently distributed sosamples of each variable can be drawn independently. TheirMIP works only when the network is a DAG and has only onesource while our MIP works for general directed graphs andan arbitrary number of sources. Wu et al. (2014b) solve theriver network design problem with a single source and mul-tiple sources by rounded dynamic programming algorithms[Wu et al., 2014a; 2014b]. However, their algorithms onlywork for tree-structured networks, and they also assume thatstochastic events are all independent. Recently, several fastapproximate algorithms based on the primal-dual techniquehave been developed to solve several deterministic networkdesign problems similar to our problem (5) [Wu et al., 2015;2016], but they do not guarantee to produce the optimal solu-tions. It is an interesting future work to see how we can com-bine their algorithms and XOR sampling techniques to trade-off the optimality with efficiency on large-scale networks.Xue et al. (2017) use an optimization algorithm by addingXOR constraints to solve a related landscape connectivityproblem. Their optimization problem takes into account ofspatial capture re-capture model, which is in a different set-ting from ours. More importantly, we use XOR samplingwithin the SAA framework, which is different from their ap-proach. 0

5 ExperimentsWe test our algorithm on a real-world problem, the so-calledFlood Preparation problem for the emergency medical ser-vices (EMS), on road networks introduced in paper [Wu et al.,2016]. Edges of the graph represent road segments. Nodesrepresent either the intersections of roads or locations whereEMS are requested through phone calls, which are sampledbased on the patient population density. Some edges or roadsegments are above roadway stream crossing structures, suchas culverts, which are vulnerable to floods and earthquakes.The failure of a crossing structure will disrupt the road seg-ment above it and cause the road to become impassable. Inour study area, only a subset of edges are associated withcrossings, each of which defines a binary variable θe. Thevalue of θe indicates whether the crossing under e fails (= 0)or not (= 1). A protection action can replace a crossing with asturdy type such that it will not fail during floods. The goal isto decide which crossings (or which edges out of the subset ofedges above crossings) to protect beforehand so patients re-main connected to ambulance centers even during floods, thatis, we will decide for each edge above a crossing whether toprotect it or not. To measure the connectivity during floods,we use the objective of the multiple source case introduced insection 2. Each source represents an ambulance center. Sinceeach center usually has a limited number of ambulances avail-

(a) Strong, 20 sources (b) Weak, 20 sources

(c) Weak, 2 sources (d) Weak, 10 sources

Figure 3: SAA with the XOR sampler produces better andmore stable policies (e.g., larger mean and smaller error bars)than SAA with the Gibbs sampler on small networks. Y axisis the average of 10 policy values (×103). Variances are il-lustrated by vertical bars that indicate the interval [mean - std,mean + std] (std: standard deviation). “Strong”: strong cor-relation. “Weak”: weak correlation. Legend: budget sizes asthe percentage of the total costs of protecting all edges.

able (sometimes only one ambulance available), a locationreachable from multiple centers has a lower risk of no am-bulance responding during floods than a location reachablefrom only one center. The problem can also be treated as asingle source problem in which it is sufficient to connect anode to only one center or being able to connect to more thanone center doesn’t further increase the objective. While ourmethodology applies to both cases, we only show the resultsfor the multiple source case.

We use a similar way as in paper [Neumayer and Modiano,2010; Rahnamay-Naeini et al., 2011] to synthetically gener-ate failure correlation between these crossing structures. Theidea is as follows. Each crossing may fail with a small chanceindependently and multiple spatially nearby crossings tend tofail concurrently if affected by a natural disaster. To generatedisaster locations, instead of using spatial point processes asin paper [Neumayer and Modiano, 2010] due to the lack oflandscape features, we uniformly generate a small number ofdisaster centers and define their effect regions with randomlygenerated radiuses. All variables within a region are corre-lated with different levels of strengths (strong or weak).

Small Networks We test our algorithm on a small networkof 502 edges and 20 crossings extracted from the road net-works, which consists of 20 binary random variables. 20 issmall enough to compute the exact objective value of Equa-tion (3) for any given policies so we can compare different

Figure 4: Runtime of MIP versus sample sizes. Lines inthe same color represent different budget sizes. The runtimevalue is the average of 30 experiments, each with indepen-dently drawn samples.

policies exactly. We test on a different number of sources.For each problem instance, we vary the sample size from 10to 180 and budget size from 10% to 40% of the total cost toprotect all crossings. For sample size N , we produce 10 poli-cies, each computed with N samples independently drawnfrom the Gibbs and XOR sampler. The objective value ofeach policy is computed with an exact solver. The mean andvariance over 10 policy values are reported. The results offour “representative” instances are shown in Fig. 3, and theresults of other experimental instances show similar trends.In most of the sample sizes and testing instances, the meanvalues produced by SAA with the XOR sampler is as goodor better than those obtained by SAA with the Gibbs sampler.Moreover, the policy produced by the XOR sampler is morestable in the sense that the variance of policy values is smallerthan the variance produced by the Gibbs sampler, as shownin Fig. 3(a-c). The results by the Gibbs sampler are compara-ble to the results by the XOR sampler in some instances. InFig. 3(d), both methods produced almost identical means andvariances. In Fig. 3(a), the XOR sampler gives smaller meanswith 10 samples but quickly outperforms the Gibbs samplerwith more samples. In Fig. 3(a)(c), for the 10% budget cases,the Gibbs sampler produces larger means but appears veryunstable. In most of these results, as the number of samplesincreases, the variance by the XOR sampler quickly reducesto 0 while the variance by the Gibbs sampler remains large(e.g., in Fig. 3(b) and (c)).

Large Networks We test on a large road network of 1, 562edges and 81 binary random variables where the exact pol-icy value is not available. Fig. 4 shows that the MIP quicklybecomes unscalable as the sample size and the number ofsources increase. CPLEX takes around 2.5 hours to solve theMIP of 30 samples and 20 sources and cannot finish within 4hours for 50 samples and 20 sources. Therefore, to work with20 or more sources or networks of larger sizes, it is importantthat good policies can be obtained by a small number of sam-ples (e.g., 30 samples for 20 sources). For 10 source case, it isinteresting that the runtime decreases a little bit after a certainnumber of samples for some budget sizes.

Also, for both samplers, we do experiments to find out thenumber of samples that are sufficient for the sample aver-

(a) The Gibbs sampler (b) The XOR sampler

Figure 5: The effect of sample sizes on estimation of policyvalues. Each line corresponds to a policy. For one policy, thevalue of Y axis is the difference of two subsequent estimatedpolicy values. For both the XOR and the Gibbs samplers, thedifference reduces to a small value with 5, 000 samples. TheXOR sampler converges better and faster comparing to theGibbs sampler.

age method to obtain a reasonably good estimation of pol-icy value. To test whether a sequence of values converges,we borrow the idea from Cauchy sequence test to see if thedifference of two subsequent values converges to zero. Theresults are shown in Fig. 5. We create a sequence of sam-ple sizes {N1, N2, ...} varying from 10 to 5000. In Fig. 5,the value reported on y-axis at x = Ni equals the differencebetween the average of Ni policy values and the average ofNi−1 policy values, each policy value estimated by an inde-pendent drawn sample of θ. If the difference goes to zero,it implies the sequence converges. As one can see from thefigure, the value difference for both the XOR and the Gibbssampling narrows down to a small interval while the valuedifference for the XOR sampler converges to a much smallerrange near 0 and much faster than for the Gibbs sampler. Theresults also imply that the empirical mean of 5000 samplescan produce good estimations of the policy value.

At last, we compare the solution qualities of our algorithmon large networks. Two representative results are shown inFig. 6. Again, the results of instances that are not shown inthe figure have similar trends. Since we can not calculatethe exact value of a policy (e.g., an expectation) on the largenetwork, we estimate the expectation using 5000 empiricalvalues of the policy, each calculated by an independent sam-ple. As demonstrated, 5000 samples are sufficient to obtaina good estimation. In the figure, the XOR sampler outper-forms the Gibbs sampler with better and stable policy values,in a similar way as for the small networks. In conclusion, theXOR sampler produces solutions with higher qualities and isless sensitive to sample variance.

6 ConclusionPrevious approaches for solving stochastic network designproblems rely on a restricted assumption that the states ofedges are independently distributed. An SAA based algo-rithm can produce high-quality solutions by taking advan-tage of this restricted assumption, although they only scal-ing up to a small number of samples. In this paper, we re-

(a) 1 source (b) 10 sources

Figure 6: SAA with the XOR sampler produces better andmore stable policies than SAA with the Gibbs sampler onlarge networks with 1 and 10 sources. Axes have the samemeanings as in Fig. 3.

lax the independence assumption and propose a more generalproblem framework by allowing correlated stochastic eventsto affect the states of edges in networks. Novel algorithmshave been developed based on the sample average methodand various samplers. We show that our algorithm can pro-duce high-quality solutions with a small number of samples ifthese samples are obtained from a XOR sampling technique.Testing on a real-world flood preparation problem for EMS,empirical results show that SAA with the XOR sampler pro-duces better and more stable solutions compared with SAAwith a Gibbs sampler.

AcknowledgementsThis research was supported by the National Science Foun-dation under Grant No. CNS-0832782, CNF-1522054, CNS-1059284, ARO grant W911-NF-14-1-0498 and the AtkinsonCenter for a Sustainable Future.

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