A Tight Bound for Stochastic Submodular Cover
Lisa Hellerstein, Devorah Kletenik, Srinivasan Parthasarathy
SStochastic Submodular Cover
A Tight Bound for Stochastic Submodular Cover
Lisa Hellerstein [email protected]
Dept. of Computer Science and EngineeringNYU Tandon School of EngineeringBrooklyn, NY 11201 USA
Devorah Kletenik [email protected]
Dept. of Computer and Information ScienceBrooklyn College, City University of New YorkBrooklyn, NY 11210 USA
Srinivasan Parthasarathy [email protected]
IBM T. J. Watson Research CenterYorktown Heights, NY 10598 USA
Abstract
We show that the Adaptive Greedy algorithm of Golovin and Krause (2011) achievesan approximation bound of (ln(
Q/η )+1) for Stochastic Submodular Cover: here Q is the“goal value” and η is the smallest non-zero marginal increase in utility deliverable by anitem. (For integer-valued utility functions, we show a bound of H ( Q ) , where H ( Q ) is the Q th Harmonic number.) Although this bound was claimed by Golovin and Krause in the originalversion of their paper, the proof was later shown to be incorrect (Nan & Saligrama, 2017).The subsequent corrected proof of Golovin and Krause (2017) gives a quadratic bound of (ln(
Q/η )+1) . Other previous bounds for the problem are Q/η )+1) , implied by workof Im et al. (2016) on a related problem, and k (ln( Q/η )+1) , due to Deshpande et al. (2016)and Hellerstein and Kletenik (2018), where k is the number of states. Our bound generalizesthe well-known (ln m +1) approximation bound on the greedy algorithm for the classical SetCover problem, where m is the size of the ground set.
1. Introduction
The
Stochastic Submodular Cover problem was introduced by Golovin and Krause (2011). Theirinterest in this problem was motivated by applications in a variety of areas, including sensorplacement, viral marketing, and active learning. The problem combines and generalizes twopreviously studied generalizations of the classical NP-complete Set Cover problem:
SubmodularCover and
Stochastic Set Cover .In the Submodular Cover problem, introduced by Wolsey (1982), the input consists of a finiteset of items E , a utility function f : 2 E → R (given by an oracle), and a cost function c : E → R + .The function f is a polymatroid function , that is, f ( ∅ ) = 0 and f is both monotone and submodular . Monotonicity means that ∀ A ⊆ B ⊆ E, f ( A ) ≤ f ( B ) . Submodularity means that ∀ A ⊆ B ⊆ E, ∀ e ∈ E \ Bf ( A ∪{ e } ) − f ( A ) ≥ f ( B ∪{ e } ) − f ( B ) The goal in the Submodular Cover problem is to find a subset A ⊆ E such that f ( A ) = f ( E ) and (cid:80) e ∈ A c ( e ) is minimized.The fact that the Submodular Cover problem is a generalization of the classical Set Coverproblem can be seen as follows. In the classical Set Cover problem, the input consists of a groundset of elements D = { d ,...,d m } , a family of subsets of the ground elements, S = { S ,...,S n } a r X i v : . [ c s . D S ] F e b ellerstein, Kletenik, & Parthasarathy Classical Set CoverSubmodular CoverStochastic Set CoverStochastic Submodular CoverFigure 1: Relationship between cover problemswhose union is equal to D , and a cost function c : S → R + . View each S i ∈ S as an item, anddefine the polymatroid utility function f : 2 S → R ≥ such that for A ⊆ S , f ( A ) = | (cid:83) S i ∈ A S i | .Finding a min-cost subset of items A such that f ( A ) = f ( S ) is equivalent to finding a min-costcover of the ground elements.Wolsey’s greedy algorithm for Submodular Set Cover repeatedly selects the item yieldingthe largest increase in utility, per unit cost. Wolsey’s analysis of this algorithm shows thatit achieves an approximation bound of (ln( Q/η )+1) , where Q = f ( E ) and η is the smallestnon-zero marginal increase in utility deliverable by an item.The Stochastic Set Cover problem is a different generalization of the classical Set Cover prob-lem, which captures situations in which there is uncertainty about which ground elements will becovered by each “subset”. A motivating example for this problem is a sensor placement problem,where prior to placing a sensor in a given location, it is unclear which subset of locations will bemonitored by the sensor (Golovin & Krause, 2011; Goemans & Vondrák, 2006). The input to theStochastic Set Cover problem consists of a ground set D = { d ,...,d m } , a family S = { S ,...,S n } of stochastic subsets of the ground set, and a cost function c : S → R + . There is an integer k > ,such that each S i is an independent random variable, whose value can be any one of k differentsubsets of ground set D (the k subsets can be different for each i ). For each of the k n possible com-binations of values of the n variables S i , it holds that (cid:83) ni =1 S i = D . Choosing S i incurs cost c ( S i ) ,the value of S i is not revealed until S i is chosen, and each S i can be chosen at most once. Theproblem is to determine how to sequentially and adaptively select stochastic subsets S i , until theunion of the (values of the) chosen S i is equal to D , so as to minimize the expected cost incurred.Selecting “adaptively” means that the choice of the next subset at each step can depend on therevealed values of the S i ’s that were already chosen (but not on the values of the other S i ’s).A natural greedy policy for this problem is to repeatedly choose the stochastic subset whichis expected to cover the largest number of uncovered items, per unit cost. Liu et al. (2008)presented and analyzed this policy for a special case of Stochastic Set Cover. They claimedthat the expected cost incurred by this policy is at most a factor of (ln m +1) greater thanoptimal expected cost of any other sequential, adaptive policy for the problem. However, aswe describe in Appendix B, the analysis of Liu et al. was not correct. tochastic Submodular Cover The problem addressed in this paper, the Stochastic Submodular Cover problem, combinesand generalizes the Stochastic Set Cover problem and the Submodular Cover problem. Wegive an informal definition of the problem here, and a more formal definition in Section 1.1. Inthis problem, there is a set of stochastic items E , and a cost function c : E → R + . Each item canbe in one of k possible “states”. The states of the different items are independent. There is autility function f that assigns a real value to each subset A of items from E . This value dependsnot only on which items are in A , but also on the states of those items. The value of f on theempty set is 0. For any fixed assignment of states to items, f is monotone and submodular.On the set of all items in E , the value of f is guaranteed to be equal to a fixed value Q , calledthe goal value , regardless of the states of the items. The state of an item e ∈ E is not revealeduntil item e is chosen and its cost c ( e ) is incurred. Each item can be chosen at most once. Theproblem is to find a policy for sequentially selecting items from E until the value of f on theset of chosen items (in their revealed states) is equal to Q , so as to minimize the expected costincurred. The policy can be adaptive, meaning that the choice of the next item to select candepend on the states of the items already chosen (but not on the states of the other items).We say that an algorithm achieves a γ -approximation bound for this problem, where γ ≥ ,if it produces a sequential, adaptive policy whose expected cost is at most a factor of γ largerthan the minimum expected cost of any sequential, adaptive policy.Golovin and Krause (2011) proposed and analyzed a greedy algorithm for Stochastic Sub-modular Cover, called Adaptive Greedy. It repeatedly selects the item which would yieldthe largest expected increase in utility, per unit cost. This generalizes the greedy algorithmsdescribed above. Golovin and Krause also considered a setting where the Adaptive Greedyalgorithm does not make an exactly optimal choice in each greedy step, but instead makes an α -approximate greedy choice. They claimed that Adaptive Greedy achieves an approximationbound of α (ln( Q/η )+1) , but subsequently Nan and Saligrama (2017) showed that their proofof the bound was incorrect. (The error in Golovin and Krause’s proof is not the same as theone in the paper of Liu et al., and the proofs use different approaches.) In a later version oftheir paper, Golovin and Krause (2017) presented a corrected version of their proof, but witha weaker quadratic bound of α (ln( Q/η )+1) .Im et al. (2016) studied a problem called Weighted Stochastic Submodular Ranking, whichgeneralizes Stochastic Submodular Cover. Their work implies that Adaptive Greedy, with α = 1 , achieves a bound of Q/η ) + 1) for Stochastic Submodular Cover. The prooftechnique used by Im et al. relies on a latency argument with geometrically increasing timeintervals. They did not attempt to minimize the constant of 56 in their bound. While it ispossible that their analysis could be tightened up to reduce the constant somewhat, a latencyargument based approach is unlikely to yield optimal constant factors. (See the encyclopediaarticle of Im (2016) on Min-Sum Set Cover for a related discussion.)In two connected papers, Deshpande et al. (2016) and Hellerstein and Kletenik (2018) showedthat Adaptive Greedy, with α = 1 , achieves a bound of k (ln( Q/η )+1) , where k is the numberof states. Their analysis used an LP formulation of the problem.The analysis for both the Q/η )+1) bound and the k (ln( Q/η )+1) bound can easilybe extended to α ≥ , leading to an extra multiplicative factor of α in the bounds.In this paper, we prove that the Adaptive Greedy algorithm yields an α (ln( Q/η )+1) ap-proximation bound for Stochastic Submodular Cover; the bound is αH ( Q ) , where H ( Q ) isthe Q th Harmonic number, when the utility function is integer valued. The tightness of the
1. When Wolsey introduced the Submodular Cover problem, he called it Submodular Set Cover. Similarly,Stochastic Submodular Cover is sometimes called Stochastic Submodular Set Cover. ellerstein, Kletenik, & Parthasarathy bound follows from the fact that it generalizes the (ln m +1) bound for classical Set Cover.Unless P = N P , the classical Set Cover problem cannot be approximated to within a factorof (1 − o (1))(ln m ) (Dinur & Steurer, 2014; Feige, 1998).Our bound restores the bound originally claimed by Golovin and Krause for StochasticSubmodular Cover. It also generalizes the bound claimed by Liu et al. (2008) for their restrictedversion of Stochastic Set Cover. In this paper, we use the definition of the Stochastic Submodular Cover problem previously usedby Deshpande et al. (2016) and by Hellerstein and Kletenik (2018). This definition is slightlymore general than the original definition of this problem given by Golovin and Krause (2011).In Appendix A, we describe the differences in the definitions, and its relevance to the definitionof the Weighted Stochastic Submodular Ranking problem studied by Im et al. (2016). Theanalyses of Golovin and Krause, and of Im et al., still hold for the definition of StochasticSubmodular Cover used in this paper.The Stochastic Submodular Cover problem is a special case of a more general coveringproblem addressed by Golovin and Krause (2011, 2017), where item states are not necessarilyindependent. In this problem, the function f , together with the joint distribution on item states,are assumed to satisfy the properties of “adaptive submodularity” and “adaptive monotonicity”.These properties are defined in terms of the expected value of f on random assignments of statesto items. In their original paper, the α (ln( Q/η )+1) bound was claimed for this more generalproblem, and the bound was then applied to Stochastic Submodular Cover. Their correctedbound, which is α (ln( Q/η )+1) , is currently the best bound for their more general coveringproblem. Our analysis of Stochastic Submodular Cover relies heavily on the assumption ofindependent item states, as do the analyses that led to the previous Q/η ) + 1) and k (ln( Q/η )+1) bounds. We leave open the question of whether an α (ln( Q/η )+1) bound canbe achieved for the more general problem of Golovin and Krause.In their work on Stochastic Submodular Cover, Deshpande et al. (2016) and Hellersteinand Kletenik (2018) actually showed that Adaptive Greedy with α = 1 achieves a bound of k (ln( R/η )+1) . Here R is the largest increase in utility attainable from a single item, and thusis at most Q . This is a generalization of the (ln( R/η )+1) bound of Wolsey for deterministicSubmodular Cover. Although the bound we prove in this paper is tight in terms of
Q/η ,we leave open the question of whether the Q in our bound can be replaced by R . For someproblems, R is much less than Q .Hellerstein and Kletenik also proved an alternative bound of (ln( R E /η E )+1) . Here R E is thelargest expected marginal increase in utility from a single item, and η E is the smallest expectednon-zero marginal increase in utility from that same item. This latter bound is incomparableto the (ln( Q/η )+1) bound shown in this paper, if items can have non-zero utility in one state,and zero utility in another.To achieve the bound in this paper, we use an amortized approach. Liu et al. (2008) also usedan amortized approach in their incorrect proof, but the approach here is different. A majordifference is that Liu et al. only considered the Stochastic Set Cover problem, and their approachrelied on charging costs to elements of the ground set as they were covered. In our problem,Stochastic Submodular Cover, costs cannot be charged in this way, since there is no ground set,there is only an abstract submodular utility function. This makes amortization more challenging.An analysis due to Wan et al. (2010) for a deterministic generalization of Submodular Coverused an amortized approach with an abstract submodular utility function, but it did not need tochastic Submodular Cover to address the challenges arising from the stochastic nature of our problem and the branchingin the adaptive ordering.We note that Deshpande et al. also introduced a dual greedy algorithm for StochasticSubmodular Cover, and proved that it achieves a bound that is qualitatively different andincomparable to the bounds given above. The bound generalizes the dual greedy approximationbound of Hochbaum (1982) for the classical Set Cover problem, and Fujito’s extension of itto Submodular Cover (cf. (Fujito, 2000)). These bounds all assume that the dual greedyalgorithm can find the exactly optimal dual greedy choice at every step. It is not clear howto extend these results to approximately optimal dual greedy choices.
2. Preliminaries
For the aid of the reader, a table of frequently used notation is presented in Appendix C.
We now present a formal definition of the Stochastic Submodular Cover problem andintroduce associated notation.Let E = { e ,...,e n } be a finite set of items . Let O = { o ,...,o k } be a finite set of states . A realization is a function ϕ : E → O , assigning state ϕ ( e ) to item e . A subrealization is a function ψ : E → O ∪{∗} which is a partial assignment of states to items (we assume ∗ (cid:54)∈ O ). If ψ ( e ) ∈ O then ψ ( e ) is the state assigned to e , while if ψ ( e ) = ∗ , then the state of e is unassigned. We define dom ( ψ ) (cid:44) { e ∈ E | ψ ( e ) ∈ O } . Realization ϕ has a corresponding relation rel ( ϕ ) (cid:44) { ( e,o ) ∈ E × O | ϕ ( e ) = o } . Subrealization ψ has a corresponding relation rel ( ψ ) (cid:44) { ( e,o ) ∈ E × O | e ∈ dom ( ψ ) ∧ ψ ( e ) = o } .Given subrealizations ψ and ψ (cid:48) , we write ψ ⊂ ψ (cid:48) to denote that rel ( ψ ) ⊂ rel ( ψ (cid:48) ) and ψ ⊆ ψ (cid:48) to denote that rel ( ψ ) ⊆ rel ( ψ (cid:48) ) . When ψ ⊆ ψ (cid:48) we call ψ (cid:48) an extension of ψ .Given a subrealization ψ and a pair ( e,o ) ∈ E × O where e (cid:54)∈ dom ( ψ ) , we use ψ ∪{ ( e,o ) } todenote the extension ψ (cid:48) of ψ such that rel ( ψ (cid:48) ) = rel ( ψ ) ∪{ ( e,o ) } .Let D be a distribution whose domain is the set of all realizations ϕ : E → O . Let Φ be arandom variable distributed according to distribution D . Thus Φ is a random assignment ofstates to items.In what follows, all probabilities and expectations are with respect to distribution D for Φ . Assumption 1 (Independence) . We assume that in distribution D , item states are independent,i.e., for all realizations ϕ : E → O , P [Φ = ϕ ] = (cid:89) e ∈ E P [Φ( e ) = ϕ ( e )] . Let F = { f ϕ | ϕ : E → O } be a family of utility functions f ϕ : 2 E → R ≥ , whose elements f ϕ have a one-to-one correspondence with the set of realizations ϕ . We assume that the functions f ϕ in F have the following two properties: Assumption 2 (Pointwise Polymatroid) . Every function in F is a polymatroid function. Assumption 3 (Sufficiency) . Given E (cid:48) ⊆ E and realizations ϕ , ϕ such that ϕ ( e ) = ϕ ( e ) for all e ∈ E (cid:48) , we have f ϕ ( E (cid:48) ) = f ϕ ( E (cid:48) ) . Assumption 3 (Sufficiency) says that the value of f ϕ on any subset of items E (cid:48) ⊂ E dependsonly on the states that ϕ assigns to items in E (cid:48) , and not to the states it assigns to items outsideof E (cid:48) .Define f to be the mapping from subrealizations ψ to R ≥ such that f ( ψ ) = f ϕ ( dom ( ψ )) ,for all realizations ϕ such that ψ ⊆ ϕ . By the Sufficiency Assumption, f is well-defined. ellerstein, Kletenik, & Parthasarathy Assumption 4 (Coverability) . There is a value Q ∈ R + such that for all realizations ϕ , f ( ϕ ) = Q . We call Q the goal value of f . We say that subrealization ψ is a cover of f if f ( ψ ) = Q .Relative to ψ , the marginal increase in utility due to observing e ∈ E \ dom ( ψ ) in state o , is f ψ ( e,o ) (cid:44) f ( ψ ∪{ ( e,o ) } ) − f ( ψ ) We use F ψ ( e ) to denote the random marginal increase in utility from e ∈ E \ dom ( ψ ) , relativeto ψ , that will result from observing the state Φ( e ) of e . That is, F ψ ( e ) (cid:44) f ψ ( e, Φ( e )) An adaptive covering policy π for f is a function which maps every subrealization ψ suchthat f ( ψ ) < Q to an item π ( ψ ) ∈ E \ dom ( ψ ) . Policy π specifies a sequential order in which to select the items of a cover for f . The orderis adaptive, meaning that the choice of the next item depends on the states of the alreadyselected items.The selection of items proceeds in iterations . In each iteration, an item e is selected andits state is observed. Each item can only be selected once. As the iterations proceed, the setof selected items, and their states, can be represented by a subrealization ψ : dom ( ψ ) is theset of selected items, and for each e in that set, ψ ( e ) is its observed state. Initially ψ is suchthat rel ( ψ ) = ∅ . Policy π specifies the item to select in each iteration, namely item e = π ( ψ ) .Selection of items, and observations of their states (with associated updating of ψ ) , continuesuntil ψ is a cover for f . Executing policy π on realization ϕ means following this procedurewhen the observed state of each selected item e is ϕ ( e ) .The adaptive sequential order specified by policy π can be represented by a decision tree T ( π ) , whose nodes are identified with subrealizations ψ . The root of the tree corresponds tothe initial iteration, identified with realization ψ where rel ( ψ ) = ∅ . The leaves are identifiedwith the possible covers of f constructed by the policy. An example tree is shown in Figure 2.Each non-leaf node ψ of the tree corresponds to an iteration where ψ represents the states ofthe items selected so far. The children of node ψ correspond to the possible updated values for ψ at the end of the iteration, after the state of selected item π ( ψ ) is observed. Letting e = π ( ψ ) ,for each o ∈ O there is a child node identified with ψ (cid:48) = ψ ∪ { ( e,o ) } , if there is a non-zeroprobability of observing state o for item e under distribution D (i.e., if P [Φ( e ) = o ] > ). InFigure 2, the edge from ψ to child ψ (cid:48) is labeled with the selected item e , and with the associatedincrease in utility f ( ψ (cid:48) ) − f ( ψ ) .By the Independence Assumption, for subrealization ψ we have ∀ e ∈ E \ dom ( ψ ) : E [ F ψ ( e ) | ψ ⊂ Φ] = E [ F ψ ( e )] Let L ( π ) be the set of leaf nodes of T ( π ) and let N ( π ) be the set of non-leaf nodes. Ifsubrealization ψ is a non-leaf node then f ( ψ ) < Q , and if it is a leaf node then f ( ψ ) = Q .The execution of π on random realization Φ corresponds to a random walk on T ( π ) whichstarts at the root and proceeds downwards towards a random leaf of the tree. We regard thewalk as visiting not just a sequence of nodes, but also the associated sequence of subrealizations.Formally, the associated sequence of subrealizations is ψ , ψ , ... , ψ k for some k > where f ( ψ ) = ∅ , f ( ψ k ) = Q , and for i = 1 ,...,k − , f ( ψ i ) < Q and ψ i +1 = ψ i ∪{ ( π ( ψ i ) , Φ( π ( ψ i )) } . The
2. We consider only deterministic policies, but our approximation bound for Adaptive Greedy also holdsrelative to a randomized policy r with minimum expected cost. Under the standard assumptions on thesources of randomness, executing r is equivalent to first choosing a policy π at random from a distribution D over deterministic policies, such that π is independent of Φ , and then executing π . Since r is optimal,every policy in the support of D has the same expected cost as r . Since our approximation bound holdswith respect to each π in the support of D , it also holds with respect to r . tochastic Submodular Cover final subrealization in this sequence, ψ k , is identified with the leaf at the end of the path in T ( π ) corresponding to the sequence. We refer to the final subrealization of the sequence asthe cover constructed by executing π on Φ and define the random variable Ψ( π ) (cid:44) the cover constructed by executing π on random realization Φ . 𝝍 𝝍 𝝍 𝝍 e , e , e , e , 0 e , e , N(σ) = { 𝝍 , 𝝍 𝝍 𝝍 } L(σ) = { 𝝍 , 𝝍 𝝍 } 𝝍 𝝍 𝝍 Figure 2: Tree representation of a policy π with Q = 10 . Annotation ( e,δ ) on edge ( ψ i ,ψ j ) indicates that e is selected at node ψ i and δ = f ( ψ j ) − f ( ψ i ) .For ψ ∈ N ( π ) and e = π ( ψ ) , E [ F ψ ( e )] is the expected increase in utility at node ψ of T ( π ) ,immediately prior to observing the state of e .Consider the execution of π on random realization Φ . If ψ ∈ N ( π ) is visited during thisexecution, then clearly ψ ⊂ Ψ( π ) and ψ ⊂ Φ . If ψ is not visited, then there must be an ancestor ψ (cid:48) of ψ such that the item π ( ψ ) chosen at node ψ is assigned to different states by ψ andby Ψ( π ) . Thus the following conditions are equivalent for ψ ∈ N ( π ) : (1) ψ is visited whenexecuting π on Φ , (2) ψ ⊂ Φ , and (3) ψ ⊂ Ψ( π ) .We define S ( π ) to be the random set of items selected by π when executed on randomrealization Φ . That is, S ( π ) (cid:44) dom (Ψ( π )) . We define C ( π ) (cid:44) (cid:88) e ∈ S ( π ) c ( e ) . Thus C ( π ) is a random variable whose value is the sum of the costs incurred by executing π on Φ .Formally, the inputs to the Stochastic Submodular Cover problem, consistent with thedefinitions and assumptions above, are as follows: (1) the set E = { e ,...,e n } of items, (2)the value of P [Φ( e ) = o ] under distribution D , for each e ∈ E and o ∈ O , (3) a cost function ellerstein, Kletenik, & Parthasarathy c : E → R + , and (4) an oracle that takes as input any subrealization ψ , and outputs the value f ( ψ ) . The problem is to find an adaptive covering policy π for f with minimum expected cost,where the expectation is with respect to the distribution D of Φ : min π E [ C ( π )] We will sometimes use the following intuitive perspective. Think of the execution of anadaptive covering policy π on Φ in terms of driving a car along a road of length Q , with stopsalong the way. The start of the execution corresponds to the start of the drive. During theexecution, when an item e is selected and its state is observed, there is a marginal increase δ inutility, and cost c ( e ) is incurred. This corresponds to driving a further distance of δ along theroad, at a cost of c ( e ) , before stopping again. At the end of execution, the sum of the increasesin utility is Q , which corresponds to having traveled the entire length Q of the road.
3. Adaptive Greedy Policy
The Adaptive Greedy algorithm of Golovin and Krause (2011, 2017), when used to solveStochstic Submodular Cover, executes the adaptive greedy policy defined as follows.Define the unit price of an item e ∈ E \ dom ( ψ ) with respect to subrealization ψ , denotedby υ ( e | ψ ) , as follows: υ ( e | ψ ) (cid:44) c ( e ) E [ F ψ ( e ) | ψ ⊂ Φ] = c ( e ) E [ F ψ ( e )] (1)The unit price is considered to be infinite if E [ f ψ ( e, Φ( e ))] = 0 .We denote the adaptive greedy policy by σ . The adaptive greedy policy with α = 1 is theadaptive covering policy that, for subrealization ψ such that f ( ψ ) < Q , selects the item e withsmallest unit price, so σ ( ψ ) = argmin e ∈ E \ dom ( ψ ) υ ( e | ψ ) with ties broken deterministically.In some applications, though, finding the item with smallest unit price can be a compu-tationally hard problem. We may instead solve the problem only approximately, with anapproximation ratio α .More generally, then, for fixed α ≥ , the adaptive greedy policy σ selects some element σ ( ψ ) such that υ ( σ ( ψ ) | ψ ) ≤ α (cid:18) min e ∈ E \ dom ( ψ ) υ ( e | ψ ) (cid:19) (2)We use e ψ to denote the item σ ( ψ ) . We use υ ψ to denote the quantity υ ( e ψ | ψ ) . Thus υ ψ (cid:44) υ ( e ψ | ψ ) ≤ α (cid:18) min e ∈ E \ dom ( ψ ) υ ( e | ψ ) (cid:19) (3) tochastic Submodular Cover
4. Comparison of π ∗ and σ Let π ∗ be an optimal adaptive covering policy for an instance of Stochastic Submodular Cover.We will prove an approximation bound that compares E [ C ( σ )] , the expected cost of theadaptive greedy policy, to E [ C ( π ∗ )] , the expected cost of the optimal policy. To accomplishthis, we introduce different ways of accounting for the costs incurred while executing policies π ∗ and σ on the same random realization Φ .Given item e and policy π , let π − [ e ] denote the inverse image of π under { e } : this is theset of all subrealizations corresponding to nodes in N ( π ) at which item e is selected.Consider the execution of π ∗ and σ on the same random realization Φ . The two executionsgive rise to values for random variables S ( π ∗ ) (the items selected by policy π ∗ ), S ( σ ) (the itemsselected by policy σ ), and Ψ( σ ) (the cover constructed by σ ). We define the revenue collectedby π ∗ and σ , in their executions on realization Φ , in terms of these random variables. Revenue collection by π ∗ Policy π ∗ collects its revenue from the items e ∈ E . For all e ∈ E , we define the revenue µ ( e ) that π ∗ collects from e as follows: µ ( e ) (cid:44) if e (cid:54)∈ S ( π ∗ ) c ( e ) if e ∈ S ( π ∗ ) \ S ( σ ) υ ψ F ψ ( e ) if e ∈ S ( π ∗ ) ∩ S ( σ ) , where ψ is thesubrealization in σ − [ e ] s.t. ψ ⊂ Ψ( σ ) (4)That is, if e was not selected by π ∗ , it does not contribute any revenue. If e was selected by π ∗ but not by σ , then its revenue equals c ( e ) . Otherwise, let ψ be the subrealization visited by σ where e was selected: the revenue in this case is defined to be the product of the unit price υ ψ and the marginal increase in utility obtained by σ when it observed e in state Φ( e ) .The total revenue collected by π ∗ is µ ∗ (cid:44) (cid:88) e ∈ E µ ( e ) (5) Revenue collection by σ Policy σ also collects revenue when executed on Φ . The revenue in this case is defined as follows.Let ρ ,ρ ,...,ρ | N ( σ ) | be all the subrealizations in N ( σ ) indexed in non-decreasing order of theirutilities so that f ( ρ ) ≤ f ( ρ ) ≤ ... ≤ f ( ρ | N ( σ ) | ) (ties are broken arbitrarily). We treat each ρ i as a marker located at point f ( ρ i ) on the interval [0 ,Q ] (corresponding to the road of length Q in our driving formulation). Let ρ | N ( σ ) | +1 be an extra marker placed at point Q . Define f ( ρ | N ( σ ) | +1 ) (cid:44) Q , and for i ∈ { ,..., | N ( σ ) |} define (cid:15) i (cid:44) f ( ρ i +1 ) − f ( ρ i ) . Let η denote the minimumnon-zero value of (cid:15) i , for i ∈ { ,..., | N ( σ ) |} , so η is the smallest non-zero increase in marginalutility deliverable by an item. When the utility function is integer valued, we have η ≥ .Observe that (cid:15) i is the length of the i th subinterval, which lies between ρ i and ρ i +1 . Also, if ψ is the subrealization corresponding to a node in T ( σ ) , then f ( ψ ) is on the boundary of one ellerstein, Kletenik, & Parthasarathy of these subintervals. We have | N ( σ ) | (cid:88) i =1 (cid:15) i = Q (6) (cid:15) | N ( σ ) | ≥ η (7)where (7) follows because f ( ρ | N ( σ ) | ) < Q and f ( ρ | N ( σ ) | +1 ) = Q .Policy σ collects its revenue from the markers ρ i . Viewing the execution of σ in terms ofthe drive along the road of length Q , revenue is collected from marker ρ i when the car reachesa distance of f ( ρ i ) from the start.We now describe the details of how σ collects revenue from ρ i . Let R i be the set of markersin the interval [0 ,f ( ρ i )] . Define: ψ (cid:32) i (cid:44) the event that, in the execution of σ on Φ , ψ was the last visited subrealization in R i That is, ψ (cid:32) i means that among all subrealizations that were visited when executing σ on Φ , ψ is the last one visited whose utility did not exceed ρ i .Note that if f ( ρ i ) = f ( ρ i +1 ) , then R i and R i +1 are equal and contain both ρ i and ρ i +1 . Foreach realization Φ and i ∈ { ,..., | N ( σ ) |} there is a unique ψ ∈ R i satisfying ψ (cid:32) i .Before completing the description of the revenue collection process for σ , we give an exampleto illustrate the above definitions. Example:
Suppose that σ is the policy π whose tree is shown in Figure 2. Here Q = 10 , f ( ψ ) = 0 , f ( ψ ) = 3 , and f ( ψ ) = f ( ψ ) = 1 . Breaking ties by increasing index, we have markers ρ = ψ , ρ = ψ , ρ = ψ , and ρ = ψ , and we place them at the points f ( ρ ) = 0 , f ( ρ ) = 1 , f ( ρ ) = 1 , and f ( ρ ) = 3 along the interval [0 , . This is shown graphically in Figure 3, alongwith the associated values of the (cid:15) i . The additional marker ρ , which is not shown in the figure,is located at the right end of the interval, at f ( ρ ) = 10 . Suppose that the execution of σ on Φ terminates with subrealization ψ , so Ψ( σ ) = ψ . In this case the subrealizations in N ( σ ) visitedduring the execution are ψ and ψ ; there is an increase of 3 units of utility after observing thestate of e , followed by an increase of 7 units after observing the state of e . Further, ψ (cid:32) i for i = 1 ,..., , while ψ (cid:32) i for i = 4 . The marginal increases in utility for this case are shown at thebottom of Figure 3.If, however, Ψ( σ ) = ψ , then after ψ , policy σ visits ψ and ψ before reaching a leaf. In thiscase, ψ (cid:32) i for i = 1 , and ψ (cid:32) i for i = 2 , , . Policy σ collects a random amount of revenue λ i from marker ρ i as follows: λ i (cid:44) υ ψ (cid:15) i , where ψ ∈ R i is such that ψ (cid:32) i (8)By the monotonicity property of the Pointwise Polymatroid Assumption, the event ψ (cid:32) i implies that there is no subrealization ψ (cid:48) ∈ N ( σ ) such that ψ (cid:48) ⊂ Ψ( σ ) and f ( ψ (cid:48) ) ∈ ( f ( ψ ) ,f ( ρ i )] .This implies the following fact. Fact 1. ∀ ψ ∈ N ( σ ) such that ψ ⊂ Ψ( σ ) : (cid:88) i s.t. ψ (cid:32) i λ i = υ ψ (cid:88) i s.t. ψ (cid:32) i (cid:15) i = υ ψ F ψ ( e ψ ) tochastic Submodular Cover 𝝆 = 𝝍 𝝆 = 𝝍 𝝆 = 𝝍 𝝆 = 𝝍 𝜺 = 1 𝜺 = 0 𝜺 = 2 𝜺 = 7 F 𝝍 (e ) = 3 F 𝝍 (e ) = 7 Figure 3: Revenue collection when σ is the policy π in Figure 2. Hybrid policy
We also define a hybrid policy that combines σ and π ∗ in a careful manner. In particular, let ψ be a subrealization such that ψ ∈ N ( σ ) . We define the hybrid policy π ψ whose execution on arandom realization Φ consists of two stages. In the first stage, the execution of π ψ on Φ followsthe same sequence as σ until it either visits ψ or terminates. As soon as it visits ψ , π ψ enters itssecond stage, during which it executes policy π ∗ , starting from the root of T ( π ∗ ) (i.e., as if it hasnot yet selected any items). However, during the execution of π ∗ , π ψ does not re-select an item e if it was already selected during the first stage. Instead, π ψ uses the previously observed stateof e (which is ψ ( e ) ) and proceeds with the execution of π ∗ as if π ∗ had observed that state for e . Note that when π ψ is executed on a random realization Φ such that ψ (cid:54)⊂ Ψ( σ ) (i.e., σ doesnot visit ψ when executed on Φ ), then π ψ selects the same items that would have been selectedby executing σ on Φ . When π ψ is executed on a realization Φ such that ψ ⊂ Ψ( σ ) , then theitems it selects include all items that would have been selected by executing π ∗ on Φ . It followsthat π ψ is an adaptive covering policy.Because σ selects item e ψ at node ψ of T ( σ ) , it follows that e ψ (cid:54)∈ dom ( ψ ) .
3. Technically, our description of π ψ should also specify the value of π ψ ( ψ (cid:48) ) for subrealizations ψ (cid:48) that do notappear as nodes in its decision tree. These values do not affect execution of π ψ and can be assigned arbitrarily. ellerstein, Kletenik, & Parthasarathy Further, by the definition of the hybrid policy, we have the following fact.
Fact 2. ∀ ψ ∈ N ( σ ) such that ψ ⊂ Ψ( σ ) , ∀ e ∈ E \ dom ( ψ ) : e ∈ S ( π ∗ ) iff e ∈ S ( π ψ ) (9)We define δ + ψ ( e ) to be the marginal utility obtained by π ψ when e is selected, if e is selectedin the second stage; if e is not selected by π ψ in the second stage, this value is .Formally: δ + ψ ( e ) (cid:44) (cid:40) F ψ (cid:48) ( e ) if ∃ ψ (cid:48) ∈ π ψ − [ e ] s.t. ψ ⊆ ψ (cid:48) , ψ (cid:48) ⊂ Ψ( π ψ )0 otherwise (10)
5. Approximation Analysis: Overview
The goal of our approximation analysis is to prove the following theorem:
Theorem 1.
The adaptive greedy policy σ achieves an approximation bound of ακ for Stochas-tic Submodular Cover, where κ = ln( Q/η ) + 1 when the utility function f is real valued and κ = H ( Q ) when the utility function is integer valued. That is, E [ C ( σ )] ≤ ακ E [ C ( π ∗ )] . We present the proof of the theorem in the following section. It is based on a sequence oflemmas. We first prove that the revenue collecting procedures defined above correctly accountfor the expected cost incurred by their associated algorithms. Specifically, in Lemma 1, we prove E [ C ( σ )] = (cid:80) | N ( σ ) | i =1 E [ λ i ] , meaning that the expected cost of the greedy policy σ is equal to the ex-pected revenue it collects from the markers λ i . In Lemma 2 we prove E [ C ( π ∗ )] = E [ µ ∗ ] , meaningthat the expected cost of the optimal policy π ∗ is equal to the revenue it collects from the items e .We then focus on comparing E [ µ ∗ ] to (cid:80) | N ( σ ) | i =1 E [ λ i ] . To make this comparison, consider λ i , therevenue collected by σ at marker ρ i . By the definition of λ i , its value depends on the sequence ofsubrealizations ψ visited by σ when executed on Φ . In particular, it depends on which ψ in thatsequence satisfies ψ (cid:32) i (i.e., which is the last subrealization in the sequence that satisfies f ( ψ ) ≤ f ( ρ i )) . For each i , we partition the sample space according to which ψ satisfies ψ (cid:32) i . In Lemmas3 and 4, we prove bounds relating the quantities E [ λ i | ψ (cid:32) i ] , E [ µ ( e ) | ψ (cid:32) i ] , and E [ δ + ψ ( e ) | ψ (cid:32) i ] .Lemma 3 shows that for all items e (cid:54)∈ dom ( ψ ) , α E [ µ ( e ) | ψ (cid:32) i ] ≥ E [ δ + ψ ( e ) | ψ (cid:32) i ] υ ψ . Recallthat δ + ψ ( e ) is the expected increase in utility from e in the second stage of π ψ (after the switchto π ∗ ), and υ ψ is the unit price paid by σ when it makes its greedy choice at node ψ of tree T ( σ ) .Lemma 3 is used to prove Lemma 4, which states that ∀ i ∈ { ,..., | N ( σ ) |} , E [ λ i | ψ (cid:32) i ] ≤ α E [ µ ∗ | ψ (cid:32) i ] Q − f ( ρ i ) (cid:15) i . For some intuition behind this statement, consider the execution of the greedypolicy, and the corresponding drive along the road of length Q , conditional on ψ (cid:32) i . After thecar reaches marker ρ i (has traveled a distance of f ( ρ i ) ), it travels an additional distance (cid:15) i untilit reaches the next marker ρ i +1 . If we view λ i as the revenue paid to finance this travel, then theexpected revenue collected by σ , per unit distance, in traveling from ρ i to ρ i +1 , is E [ λ i | ψ (cid:32) i ] /(cid:15) i .Consider this ratio to be the rate paid by σ in traveling from ρ i to ρ i +1 . Then consider whathappens in the drive corresponding to policy π ∗ when it reaches marker ρ i . At marker ρ i , theremaining distance to be traveled is Q − f ( ρ i ) . Since the total revenue collected by π ∗ is µ ∗ ,it will collect at most that much revenue in traveling the remaining distance Q − f ( ρ i ) . Thus E [ µ ∗ | ψ (cid:32) i ] is an upper bound on the expected revenue collected by π ∗ while traveling theremaining distance, and the ratio E [ µ ∗ | ψ (cid:32) i ] / ( Q − f ( ρ )) is an upper bound on the rate paid tochastic Submodular Cover by π ∗ during that travel. Lemma 4 says that this ratio (times α ) is an upper bound on therate paid by σ in traveling from marker ρ i to ρ i +1 .The next lemma, Lemma 5, states that ∀ i ∈ { ,..., | N ( σ ) |} , E [ λ i ] ≤ α E [ C ( π ∗ )] (cid:15) i Q − f ( ρ i ) . It isthe same as Lemma 4, except that the conditioning on ψ (cid:32) i is removed, and we use the factthat E [ µ ∗ ] = E [ C ( π ∗ )] . Having removed the conditioning, we no longer need to worry aboutthe branching in the decision tree for σ , and can simply think of σ as deterministically travelingstep by step from each marker ρ i to the next one, at a cost of E [ λ i ] for the step.The rest of the analysis is done in a standard way and leads to the desired approxi-mation bound for the adaptive greedy policy: E [ C ( σ )] ≤ α E [ C ( π ∗ )](ln( Q/η ) + 1) (and E [ C ( σ )] ≤ α E [ C ( π ∗ )] H ( Q ) when f is integer-valued).
6. Approximation Analysis
In this section, we will be considering the execution of the adaptive greedy policy σ , the optimalpolicy π ∗ , and the hybrid policy π ψ on the same random realization Φ . To make our expressionsmore compact, we will use the following notation to express, for each of these policies, theevent that a subrealization ψ (cid:48) is visited:• V σ ( ψ (cid:48) ) denotes the event ψ (cid:48) ⊂ Ψ( σ ) • V ∗ ( ψ (cid:48) ) denotes the event ψ (cid:48) ⊂ Ψ( π ∗ ) • V ψ ( ψ (cid:48) ) denotes the event ψ (cid:48) ⊂ Ψ( π ψ ) Consider running the greedy policy and the optimal policy on the same random realization Φ . We will use the following notation to express that item e is selected by the optimal policybut not the greedy policy, or that it is selected by both policies:• A ( e ) denotes the event e ∈ S ( π ∗ ) \ S ( σ ) • B ( e ) denotes the event e ∈ S ( σ ) ∩ S ( π ∗ ) The following lemma equates the expected cost of greedy policy σ to its expected revenue. Lemma 1. E [ C ( σ )] = | N ( σ ) | (cid:88) i =1 E [ λ i ] (11) Proof.
Recall that e ψ = σ ( ψ ) . We have ellerstein, Kletenik, & Parthasarathy | N ( σ ) | (cid:88) i =1 E [ λ i ]= E (cid:88) ψ ∈ N ( σ ) s.t V σ ( ψ ) υ ψ (cid:88) i s.t. ψ (cid:32) i (cid:15) i by the definition of λ i in (8) = E (cid:88) ψ ∈ N ( σ ) s.t V σ ( ψ ) υ ψ F ψ ( e ψ ) by Fact 1 = (cid:88) ψ ∈ N ( σ ) Pr[ V σ ( ψ )] υ ψ E [ F ψ ( e ψ )) |V σ ( ψ )]= (cid:88) ψ ∈ N ( σ ) Pr[ V σ ( ψ )] υ ψ E [ F ψ ( e ψ )] by the Independence Assumption = (cid:88) ψ ∈ N ( σ ) Pr[ V σ ( ψ )] c ( e ψ ) by the definition of υ ψ in (3) = (cid:88) e ∈ E (cid:88) ψ ∈ σ − [ e ] Pr[ V σ ( ψ )] c ( e )= (cid:88) e ∈ E Pr[ e ∈ S ( σ )] c ( e )= E [ C ( σ )] . The following lemma equates the expected cost of π ∗ to its expected revenue. Lemma 2. E [ C ( π ∗ )] = E [ µ ∗ ] Proof.
Recalling that A ( e ) denotes the event e ∈ S ( π ∗ ) \ S ( σ ) , define J (cid:44) (cid:32)(cid:88) e ∈ E Pr[ A ( e )] c ( e ) (cid:33) . Recalling that B ( e ) denotes the event e ∈ S ( π ∗ ) ∩ S ( σ ) , and using the definitions of µ and µ ∗ in (4) and (5), we have E [ µ ∗ ]= (cid:88) e ∈ E E [ µ ( e )]= J + (cid:32)(cid:88) e ∈ E Pr[ B ( e )] (cid:88) ψ ∈ σ − [ e ] Pr[ V σ ( ψ ) |B ( e )] υ ψ E [ F ψ ( e ) |B ( e ) ∧V σ ( ψ )] (cid:33) (12)To simplify the above expression, we first prove the following: ∀ e ∈ E, ψ ∈ σ − [ e ] : E [ F ψ ( e ) |B ( e ) ∧V σ ( ψ )] = E [ F ψ ( e ) |V σ ( ψ )] (13)To prove (13), assume e ∈ E and ψ ∈ σ − [ e ] . Since ψ ∈ σ − [ e ] , V σ ( ψ ) implies e ∈ S ( σ ) .Therefore, the condition B ( e ) ∧V σ ( ψ ) can be replaced by e ∈ S ( π ∗ ) ∧V σ ( ψ ) . We can refine this tochastic Submodular Cover conditioning further by taking into account not just the fact that e ∈ S ( π ∗ ) , but by consideringwhich subrealization ψ ∗ in π ∗ satisfies ψ ∗ x ∈ π ∗− [ e ] ∧V ∗ ( ψ ∗ ) . So: E [ F ψ ( e ) |B ( e ) ∧V σ ( ψ )]= E [ F ψ ( e ) | e ∈ S ( π ∗ ) ∧V σ ( ψ )]= (cid:88) ψ ∗ ∈ π ∗− [ e ] Pr[ V ∗ ( ψ ∗ ) | e ∈ S ( π ∗ ) ∧V σ ( ψ )] E [ F ψ ( e )) |V ∗ ( ψ ∗ ) ∧V σ ( ψ )]= (cid:88) ψ ∗ ∈ π ∗− [ e ] Pr[ V ∗ ( ψ ∗ ) | e ∈ S ( π ∗ ) ∧V σ ( ψ )] E [ F ψ ( e ) |V σ ( ψ )] (14) = E [ F ψ ( e ) |V σ ( ψ )] . (15)Line (14) holds due to the fact that ψ ∈ σ − [ e ] by assumption, and ψ ∗ ∈ π ∗− [ e ] , so e is not an ele-ment of either dom ( ψ ) or dom ( ψ ∗ ) . Whether σ visits node ψ of T ( σ ) depends only on the statesof the items in dom ( ψ ) , and whether π ∗ visits node ψ (cid:48) of T ( π ∗ ) depends only on the states ofthe items in dom ( ψ ∗ ) . Thus by the Independence Assumption, the value of Φ( e ) , and hence thevalue of E [ F ψ ( e )] , is independent of the conditions V σ ( ψ ) and V ∗ ( ψ (cid:48) ) , which justifies removalof the condition V ∗ ( ψ (cid:48) ) for (14). Line (15) holds because if π ∗ selects e when executed on Φ ,it visits exactly one realization ψ ∗ ∈ π ∗− [ e ] . Hence (cid:80) ψ ∗ ∈ π ∗− [ e ] Pr[ V ∗ ( ψ ∗ ) | e ∈ S ( π ∗ ) ∧V σ ( ψ )] = 1 .This completes the proof of (13).We can now use (13) to rewrite the expression for E [ µ ∗ ] in (12), yielding E [ µ ∗ ] = J + (cid:88) e ∈ E Pr[ B ( e )] (cid:88) ψ ∈ σ − [ e ] Pr[ V σ ( ψ ) |B ( e )] υ ψ E [ F ψ ( e ) |V σ ( ψ )]= J + (cid:88) e ∈ E Pr[ B ( e )] (cid:88) ψ ∈ σ − [ e ] Pr[ V σ ( ψ ) |B ( e )] υ ψ E [ F ψ ( e )] by the Independence Assumption = J + (cid:88) e ∈ E Pr[ B ( e )] (cid:88) ψ ∈ σ − [ e ] Pr[ V σ ( ψ ) |B ( e )] c ( e ) by the definition of υ ψ = J + (cid:32)(cid:88) e ∈ E Pr[ B ( e )] c ( e ) (cid:33) because (cid:88) ψ ∈ σ − [ e ] Pr[ V σ ( ψ ) |B ( e )] = 1= (cid:32)(cid:88) e ∈ E Pr[ e ∈ S ( π ∗ ) \ S ( σ )] c ( e ) (cid:33) + (cid:32)(cid:88) e ∈ E Pr[ e ∈ S ( π ∗ ) ∩ S ( σ )] c ( e ) (cid:33) = (cid:88) e ∈ E Pr[ e ∈ S ( π ∗ )] c ( e )= E [ C ( π ∗ )] . Recall the description of the hybrid policy π ψ , where ψ is a non-leaf node of T ( σ ) . At node ψ , σ selects item e ψ and e ψ (cid:54)∈ dom ( ψ ) . The tree T ( π ψ ) for the hybrid policy also contains node ψ , but the item selected at that node may be different from e ψ (since the hybrid policy switchesto π ∗ at this point).With that background, we are ready to prove the following technical lemma. ellerstein, Kletenik, & Parthasarathy Lemma 3.
Let ψ ∈ N ( σ ) and i ∈ { ,..., | N ( σ ) |} such that Pr[ ψ (cid:32) i ] > . Then ∀ e ∈ E \ dom ( ψ ) : α E [ µ ( e ) | ψ (cid:32) i ] ≥ E [ δ + ψ ( e ) | ψ (cid:32) i ] υ ψ (16) Proof.
Let e ∈ E \ dom ( ψ ) and let ψ and i be as given in the statement of the lemma.Define H (cid:44) { ψ h | ψ h ∈ π ψ − [ e ] ∧ ψ ⊆ ψ h } . Thus H is the set of nodes of T ( π ψ ) where e couldbe selected during the second stage of π ψ .We divide our analysis according to whether e is selected by both σ and π ∗ when they areexecuted on Φ , or just by π ∗ . We first prove the inequality in the statement of the lemma withrespect to the additional condition A ( e ) ( e is selected by π ∗ but not by σ ), and then with respectto the additional condition B ( e ) ( e is selected by both σ and π ∗ ). We do not need to considerconditions where e is not selected by π ∗ , because µ ( e ) and δ + ψ ( e ) are equal to 0 in these cases.• Additional condition A ( e ) We will show α E [ µ ( e ) |A ( e ) ∧ ψ (cid:32) i ] ≥ υ ψ E [ δ + ψ ( e ) |A ( e ) ∧ ψ (cid:32) i ] (17)Recall that A ( e ) denotes the event that e is in S ( π ∗ ) but not in S ( σ ) . The event ψ (cid:32) i implies that ψ ⊂ Ψ( σ ) . Thus by Fact 2, A ( e ) ∧ ψ (cid:32) i implies that e ∈ S ( π ψ ) . Further, since e (cid:54)∈ dom ( ψ ) , e must be selected in the second stage of π ψ .Let X ⊆ L ( σ ) denote the set of terminal subrealizations observable (covers constructed)by σ given that A ( e ) ∧ ψ (cid:32) i . Thus X is a subset of the possible values of Ψ( σ ) . We have α E [ µ ( e ) |A ( e ) ∧ ψ (cid:32) i ]= (cid:88) ψ h ∈ π ψ − [ e ] (cid:88) ψ x ∈ X Pr[ V ψ ( ψ h ) ∧ Ψ( σ ) = ψ x |A ( e ) ∧ ψ (cid:32) i ] αc ( e ) since µ ( e ) = c ( e ) when A ( e ) holds, and partitioning event ( A ( e ) ∧ ψ (cid:32) i ) w.r.t. which ψ h ∈ π ψ − [ e ] is visited by π ψ and which cover ψ x ∈ X is constructed by σ ≥ (cid:88) ψ h ∈ π ψ − [ e ] (cid:88) ψ x ∈ X Pr[ V ψ ( ψ h ) ∧ Ψ( σ ) = ψ x |A ( e ) ∧ ψ (cid:32) i ] υ ψ E [ F ψ ( e )] by the greedy choice (3) = (cid:88) ψ h ∈ π ψ − [ e ] (cid:88) ψ x ∈ X Pr[ V ψ ( ψ h ) ∧ Ψ( σ ) = ψ x |A ( e ) ∧ ψ (cid:32) i ] υ ψ E [ F ψ ( e ) |V ψ ( ψ h ) ∧ Ψ( σ ) = ψ x ] by the Independence Assumption since e (cid:54)∈ dom ( ψ h ) ∪ dom ( ψ x ) ≥ (cid:88) ψ h ∈ π ψ − [ e ] (cid:88) ψ x ∈ X Pr[ V ψ ( ψ h ) ∧ Ψ( σ ) = ψ x |A ( e ) ∧ ψ (cid:32) i ] υ ψ E [ F ψ h ( e ) |V ψ ( ψ h ) ∧ Ψ( σ ) = ψ x ] replacing F ψ by F ψ h , by submodularity since ψ ⊆ ψ h = υ ψ (cid:88) ψ h ∈ π ψ − [ e ] (cid:88) ψ x ∈ X Pr[ V ψ ( ψ h ) ∧ Ψ( σ ) = ψ x |A ( e ) ∧ ψ (cid:32) i ] E [ δ + ψ ( e ) |V ψ ( ψ h ) ∧ Ψ( σ ) = ψ x ] because V ψ ( ψ h ) implies e was selected by π ψ at node ψ h during its second stage = υ ψ E [ δ + ψ ( e ) |A ( e ) ∧ ψ (cid:32) i ] . tochastic Submodular Cover • Additional condition B ( e ) Because e (cid:54)∈ dom ( ψ ) , condition B ( e ) ∧ ψ (cid:32) e implies, by Fact 2, that π ψ visits some ψ h ∈ H .We will show α E [ µ ( e ) |B ( e ) ∧ ψ (cid:32) i ] ≥ υ ψ E [ δ + ψ ( e ) |B ( e ) ∧ ψ (cid:32) i ] (18)We first give the proof of (18) for the case where e = e ψ . If e = e ψ , we have E [ µ ( e ψ ) | ψ (cid:32) i ]= Pr[ e ψ ∈ S ( π ∗ ) | ψ (cid:32) i ] E [ µ ( e ψ ) | ψ (cid:32) i ∧ e ψ ∈ S ( π ∗ )] by the definition of µ =Pr[ e ψ ∈ S ( π ψ ) | ψ (cid:32) i ] E [ µ ( e ψ ) | ψ (cid:32) i ∧ e ψ ∈ S ( π ψ )] by Fact 2 = (cid:88) ψ h ∈ H Pr[ V ψ ( ψ h ) | ψ (cid:32) i ] E [ µ ( e ψ ) | ψ (cid:32) i ∧V ψ ( ψ h )] partitioning w.r.t. which ψ h ∈ H is visited = (cid:88) ψ h ∈ H Pr[ V ψ ( ψ h ) | ψ (cid:32) i ] υ ψ E [ F ψ ( e ψ ) | ψ (cid:32) i ∧V ψ ( ψ h )] by the definition of µ since ψ (cid:32) i ⇒ V σ ( ψ ) ≥ (cid:88) ψ h ∈ H Pr[ V ψ ( ψ h ) | ψ (cid:32) i ] υ ψ E [ F ψ h ( e ψ ) | ψ (cid:32) i ∧V ψ ( ψ h )] replacing f ψ by f ψ h , by submodularity = υ ψ E [ δ + ψ ( e ψ ) | ψ (cid:32) i ] by definition of δ + ψ . Thus (18) holds when e = e ψ .So suppose e (cid:54) = e ψ . In this case, define G to be the set of subrealizations ψ g satisfying thefollowing three properties: ψ g ∈ σ − [ e ] ψ g ⊃ ψf ( ψ ∪ ( e ψ ,ψ g ( e ψ ))) > f ( ρ i ) The second property says that in the greedy tree, node ψ is an ancestor of node ψ g . Thethird property concerns the child of node ψ that is on the path from ψ down to ψ g : this isthe node identified with subrealization ψ ∪ ( e ψ ,ψ g ( e ψ )) . It states that the value of f on thissubrealization is strictly greater than f ( ρ i ) . Therefore, ∀ ψ g ∈ G : V σ ( ψ g ) ⇒ ψ (cid:32) i (19)Conversely, if ψ (cid:32) i , then because e (cid:54)∈ dom ( ψ ) , condition B ( e ) implies that σ must visit some ψ g ∈ G . As before, condition B ( e ) ∧ ψ (cid:32) i implies that π ψ must visit some ψ h ∈ H . Therefore, B ( e ) ∧ ψ (cid:32) i ⇒ ∃ ! ψ g ∈ G, ∃ ! ψ h ∈ H such that V σ ( ψ g ) ∧V ψ ( ψ h ) where ∃ ! means the quantified element exists and is unique.Note that by the Independence Assumption, for ψ g ∈ G and ψ h ∈ H , since e (cid:54)∈ dom ( ψ g ) ∪ dom ( ψ h ) , E [ F ψ g ( e ) |V σ ( ψ g ) ∧V ψ ( ψ h )] = E [ F ψ g ( e )] (20)Before proceeding with the proof of (18), we show that the following equation holds for ψ g ∈ G : E [ µ ( e ) |B ( e ) ∧V σ ( ψ g )] = c ( e ) (21) ellerstein, Kletenik, & Parthasarathy The proof of (21) is as follows. Recalling that e (cid:54)∈ dom ( ψ ) , E [ µ ( e ) |B ( e ) ∧V σ ( ψ g )]= υ ψ g E [ F ψ g ( e ) |B ( e ) ∧V σ ( ψ g )] by the definition of µ ( e ) , since ψ g ∈ σ − [ e ]= (cid:88) ψ h ∈ H P [ V ψ ( ψ h ) |B ( e ) ∧V σ ( ψ g )] υ ψ g E [ F ψ g ( e ) |V σ ( ψ g ) ∧V ψ ( ψ h )] partitioning w.r.t. which ψ h ∈ H is visited by π ψ = (cid:88) ψ h ∈ H P [ V ψ ( ψ h ) |B ( e ) ∧V σ ( ψ g )] υ ψ g E [ F ψ g ( e ψ g )] by (20) and because ψ g ∈ σ − [ e ] , so e = e ψ g = c ( e ) because (cid:88) ψ h ∈ H P [ V ψ ( ψ h ) |B ( e ) ∧V σ ( ψ g )] = 1 and υ ψ g = c ( e ψ g ) / E [ F ψ g ( e ψ g )] . Having proved (21), we now prove (18), the desired inequality for condition B ( e ) . For ψ g ∈ G , define K g (cid:44) P [ V σ ( ψ g ) |B ( e ) ∧ ψ (cid:32) i ] We have α E [ µ ( e ) |B ( e ) ∧ ψ (cid:32) i ]= α (cid:88) ψ g ∈ G P [ V σ ( ψ g ) |B ( e ) ∧ ψ (cid:32) i ] E [ µ ( e ) |B ( e ) ∧V σ ( ψ g )] partitioning w.r.t. which ψ g ∈ G is visited, and because V σ ( ψ g ) ⇒ ψ (cid:32) i = α (cid:88) ψ g ∈ G K g c ( e ) by (21) and the definition of K g = α (cid:88) ψ g ∈ G K g (cid:88) ψ h ∈ H P [ V ψ ( ψ h ) |B ( e ) ∧V σ ( ψ g )] υ ( e | ψ h ) E [ F ψ h ( e )] partitioning w,r.t. which ψ h ∈ H is visited, and because υ ( e | ψ h ) = c ( e ) / E [ F ψ h ( e )]= α (cid:88) ψ g ∈ G K g (cid:88) ψ h ∈ H P [ V ψ ( ψ h ) |B ( e ) ∧V σ ( ψ g )] υ ( e | ψ h ) E [ F ψ h ( e ) |V σ ( ψ g ) ∧V ψ ( ψ h )] by the Independence Assumption = α (cid:88) ψ g ∈ G K g (cid:88) ψ h ∈ H P [ V ψ ( ψ h ) |B ( e ) ∧V σ ( ψ g )] υ ( e | ψ h ) E [ δ + ψ ( e ) |V σ ( ψ g ) ∧V ψ ( ψ h )] by the definition of δ + ψ ( e ) , since V ψ ( ψ h ) implies e was selected by π ψ at node ψ h during its second stage ≥ αυ ( e | ψ ) (cid:88) ψ g ∈ G K g (cid:88) ψ h ∈ H P [ V ψ ( ψ h ) |B ( e ) ∧V σ ( ψ g )] E [ δ + ψ ( e ) |V σ ( ψ g ) ∧V ψ ( ψ h )] since ψ ⊆ ψ h so by the submodularity property, F ψ ( e ) ≥ F ψ h ( e ) and hence υ ( e | ψ h ) ≥ υ ( e | ψ ) ≥ υ ψ (cid:88) ψ g ∈ G K g (cid:88) ψ h ∈ H P [ V ψ ( ψ h ) |B ( e ) ∧V σ ( ψ g )] E [ δ + ψ ( e ) |V σ ( ψ g ) ∧V ψ ( ψ h )] tochastic Submodular Cover since υ ψ = υ ( e ψ | ψ ) ≤ αυ ( e | ψ ) by the greedy choice (3) = υ ψ (cid:88) ψ g ∈ G K g E [ δ + ψ ( e ) |B ( e ) ∧V σ ( ψ g )]= υ ψ (cid:88) ψ g ∈ G P [ V σ ( ψ g ) |B ( e ) ∧ ψ (cid:32) i ] E [ δ + ψ ( e ) |B ( e ) ∧V σ ( ψ g )] by the definition of K g = υ ψ E [ δ + ψ ( e ) |B ( e ) ∧ ψ (cid:32) i ] . Finally, because µ ( e ) = 0 and δ + ψ ( e ) = 0 whenever e (cid:54)∈ S ( π ∗ ) , the lemma follows from theinequalities (18) and (17) proved for the two conditions, A ( e ) and B ( e ) . Fact 3.
For non-negative numbers α ,...,α n , and β ,...,β n such that (cid:80) ni =1 β i > : min i s.t. β i > α i β i ≤ (cid:80) ni =1 α i (cid:80) ni =1 β i (22)We combine the above fact with Lemma 3 to prove the following lemma. Lemma 4. ∀ i ∈ { ,..., | N ( σ ) |} : E [ λ i | ψ (cid:32) i ] ≤ α E [ µ ∗ | ψ (cid:32) i ] Q − f ( ρ i ) (cid:15) i (23) Proof.
Let E = { e | e ∈ E \ dom ( ψ ) ∧ E [ δ + ψ ( e ) | ψ (cid:32) i ] > } . E [ λ i | ψ (cid:32) i ] = υ ψ (cid:15) i ≤ min e ∈ E α E [ µ ( e ) | ψ (cid:32) i ] E [ δ + ψ ( e ) | ψ (cid:32) i ] (cid:15) i by Lemma ≤ (cid:80) e ∈ E \ dom ( ψ ) α E [ µ ( e ) | ψ (cid:32) i ] (cid:80) e ∈ E \ dom ( ψ ) E [ δ + ψ ( e ) | ψ (cid:32) i ] (cid:15) i by Fact ≤ α E [ µ ∗ | ψ (cid:32) i ] Q − f ( ψ ) (cid:15) i by definitions of µ ∗ and δ + ψ Lemma 5. ∀ i ∈ { ,..., | N ( σ ) |} : E [ λ i ] ≤ α E [ C ( π ∗ )] (cid:15) i Q − f ( ρ i ) (24) Proof.
Fix i ∈ { ,..., | N ( σ ) | . Let R = { ψ | ψ ∈ N ( σ ) ∧ Pr[ ψ (cid:32) i ] > } . The events { ψ (cid:32) i | ψ ∈ R } partition the sample space. Hence: E [ λ i ] = (cid:88) ψ ∈ R Pr[ ψ (cid:32) i ] E [ λ i | ψ (cid:32) i ] ≤ (cid:88) ψ ∈ R Pr[ ψ (cid:32) i ] α E [ µ ∗ | ψ (cid:32) i ] (cid:15) i Q − f ( ρ i ) by Lemma α E [ µ ∗ ] (cid:15) i Q − f ( ρ i )= α E [ C ( π ∗ )] (cid:15) i Q − f ( ρ i ) by Lemma ellerstein, Kletenik, & Parthasarathy Lemma 6. E [ C ( σ )] ≤ α E [ C ( π ∗ )] | N ( σ ) | (cid:88) i =1 (cid:15) i Q − f ( ρ i ) (25) Proof.
The lemma follows by combining Lemmas 1 and 5.With the above lemmas, we can now prove Theorem 1, which states that σ achieves anapproximation bound of ακ for Stochastic Submodular Cover, where κ = ln( Q/η )+1 whenthe utility function is real valued and κ = H ( Q ) when the utility function is integer valued. Proof of Theorem 1:
The theorem follows from Lemma 6 by bounding the summation | N ( σ ) | (cid:88) i =1 (cid:15) i Q − f ( ρ i ) using standard arguments, as follows.Let t denote | N ( σ ) | . Recall that (cid:15) i = f ( ρ i +1 ) − f ( ρ i ) . For i ∈ { ,...,t − } , (cid:15) i Q − f ( ρ i ) = (cid:90) f ( ρ i +1 ) f ( ρ i ) Q − f ( ρ i ) dx ≤ (cid:90) f ( ρ i +1 ) f ( ρ i ) Q − x dx. Therefore, t (cid:88) i =1 (cid:15) i Q − f ( ρ i ) ≤ (cid:15) t Q − f ( ρ t ) + (cid:90) f ( ρ t )0 Q − x dx = 1+ln Q/(cid:15) t because (cid:15) t = Q − f ( ρ t ) ≤ Q/η. (26)If f is integer-valued, then (cid:15) t ≥ and similarly, t (cid:88) i =1 (cid:15) i Q − f ( ρ i ) ≤ f ( ρ t ) − (cid:88) j =0 Q − j = 1+ H ( Q ) − H ( (cid:15) t ) ≤ H ( Q ) . (27) Acknowledgments
Partial support for this work came from NSF Award IIS-1909335 (for L. Hellerstein) and from aPSC-CUNY Award, jointly funded by The Professional Staff Congress and The City Universityof New York (for D. Kletenik). We thank the anonymous referees for their comprehensive anduseful comments. We thank Naifeng Liu for proofreading help.
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In the definition of Stochastic Submodular Cover given by Golovin and Krause (who called theproblem Stochastic Submodular
Coverage ), the input utility function is ˆ f : 2 E × O → R ≥ where E is the set of items and O is the set of states, such that ˆ f is a monotone and submodular setfunction. Since ˆ f is defined on all subsets of E × O , its domain includes “inconsistent” subsetscontaining pairs ( e,o ) and ( e,o (cid:48) ) , where o (cid:54) = o (cid:48) . Intuitively, this means that the input utilityfunction must assign values to collections of items in which a single item can simultaneouslybe in more than one state; further, these values must be compatible with the properties ofmonotonicity and submodularity. In contrast, our input utility function need only assign valuesto collections of items where each item has exactly one state. In fact, though, the value of ˆ f on ellerstein, Kletenik, & Parthasarathy the “inconsistent” subsets is irrelevant both to the remainder of the Golovin and Krause problemdefinition, and to the proof of their bound for Stochastic Submodular Cover. While in theirdefinition they require ˆ f to have values on these subsets, they effectively ignore these values.The definition of Stochastic Submodular Cover given by Golovin and Krause, with itemset E = { e ,...,e n } and state set O , is equivalent to a special case of the Weighted StochasticSubmodular Ranking (WSSR) problem, studied by Im et al. (2016). In particular, it is equiv-alent to the special case of the WSSR problem where there is a single function f : 2 D → [0 , with D = E × O , and such that for each X i , P [ X i = ( e,o )] = 0 if e (cid:54) = e i , otherwise P [ X i = ( e,o )] equals the probability P [Φ( e i ) = o ] in Stochastic Submodular Cover. Although the problemdefinition for the WSSR problem requires that f be defined on all subsets of D = E × O inthis case, including the “incompatible” subsets, the analysis of Im et al. applied to this casemakes no use of the value of f on such subsets. Appendix B. The Error in the Analysis of Liu et al., 2008
The paper of Liu et al. claims to show an (ln m +1) approximation bound for the adaptivegreedy policy, applied to Stochastic Set Cover. Consider the execution of this policy, whichalways selects the stochastic subset covering the largest expected number of items, per unitcost. Let S i ,...,S i k denote the sequence of chosen subsets. Each S i (cid:96) covers some number h of ground elements in S i (cid:96) not already covered by S i ,...,S i (cid:96) − . Consider the h elements to becovered one by one by S i (cid:96) . The price paid to cover each one is c i(cid:96) h .Consider a run of the greedy policy, and of an optimal policy OPT, on a random assignmentof values to the S i . Order the elements of the ground set in the order they are covered by thegreedy policy. Lemma 3 of Liu et al. gives an upper bound on price ( j ) , the price paid whenthe j th item in this order is covered. The bound is conditioned on ψ and Υ , where ψ is thesequence of S i chosen by the greedy policy (together with the observed values of those S i ), and Υ is the subset of S i chosen by both the greedy policy and OPT.For example, suppose the set of ground elements is { e ,...,e } and there are only three stochas-tic subsets, S ,S ,S . Consider the following realization of these subsets: S = { e ,e ,e ,e } , S = { e ,e ,e } , and S = { e ,e ,e } . Suppose that under this realization, the greedy policycovers the ground elements using all three subsets, in the following order: S ,S ,S . Supposethat the optimal policy, under the above realization, covers the elements using just S and S .Figure 4 shows the m = 6 elements of the ground set from left to right in the order they werecovered by the S i in the greedy policy (elements covered during the same step are listed in orderof their indices). Here ψ is equal to S ,S ,S with the realizations listed above, and Υ = { S ,S } .Figure 4: Greedy cover example tochastic Submodular Cover Right before the greedy policy covers its j th element, there remain m − j + 1 uncoveredelements. Fix j and let M last denote that set of elements. Let M com ⊆ M last denote the subsetof elements in M last that are first covered, during execution of the greedy policy, by subsetsin Υ . A crucial step of the proof of Lemma 3 is to show that m − j +1 = (cid:88) S i ∈ Υ cov ( S i )+ (cid:88) S i ∈S\ ψ optcov ( S i ) where cov ( S i ) is the number of elements in M last that are first covered by S i during execuionof the greedy policy, and optcov ( S i ) is the expected number of elements in M last \ M com firstcovered by S i during execution of the optimal algorithm. However, this equation does not hold.(It would be sufficient to show that the left hand side of the equation is bounded above by theright hand side, but that does not hold either.)Figure 4 with j = 1 is a counterexample. In this case, M last consists of all 6 ground elements.In this ψ , S covers 4 new elements, S covers 1 (namely e ), and S covers 1 (namely e ) .Therefore, cov ( S ) = 1 and cov ( S ) = 1 . However, S\ ψ is empty. So in this case, the left handside of the equation equals 6, while the right hand side equals 2. The problem is that in theoptimal policy, S and S are used to cover all 6 ground elements, but the greedy policy usesthem to cover only 2 elements, because the others were already covered by an S i not in Υ . Theconditioning on ψ and Υ does not imply the equality. While the condition can be strengthened,it is not clear how to do so in a way that doesn’t invalidate other parts of the proof. ellerstein, Kletenik, & Parthasarathy Appendix C. Table of Frequently Used Symbols
Symbol Description
E,e
Set of items, an individual item c ( e ) Cost of item eO Set of possible states for an item Φ Random realization of all item states ϕ A fixed realization; a possible outcome for Φ ψ A fixed subrealization dom ( ψ ) Items whose states are recorded in ψf Utility function mapping from set of allsubrealizations to R ≥ F ψ ( e ) Random marginal increase in f from observingstate of e , after observing sates in ψQ Value of f ( ϕ ) for all realizations ϕ (cover value) π,π ∗ ,σ A fixed policy; optimal policy; greedy policy π ψ Hybrid policy, runs σ and switches to π ∗ at ψπ ( ψ ) Item selected by π after observing states in ψT ( π ) Decision tree for policy πL ( π ) , N ( π ) Sets of subrealizations for leaf and non-leafnodes of T ( π ) , Ψ( π ) Random cover constructed by πC ( π ) Random cost of policy πµ ( e ) , µ ∗ Revenue from e in π ∗ ; total revenue in π ∗ λ i Revenue from marker ρ i in σe ψ Item chosen at node ψ of greedy tree T ( σ ) υ ( e | ψ ) , υ ψ unit price c ( e ) /F ψ ( e ) of e w.r.t. ψ ; unit pricepaid at node ψ of T ( σ ) ρ i , (cid:15) i Subrealization that is i th marker of σ ; differencein utility between ρ i and ρ i +1 δ + ψ ( e ) Equal to marginal utility from e in hybrid policy,if e selected during second stage, else equals 0 π − [ e ] Subrealizations for nodes of greedy tree where e is chosen ψ (cid:32) i ψ is last realization visited by σ such that f ( ψ ) ≤ f ( ρ i ) V σ ( e ) , V ψ ( e ) , V ∗ ( e ) e is chosen by: policy σ , policy π ψ , policy π ∗ A ( e ) , B ( e ) e is chosen by π ∗ but not by σ , e is chosen by bothTable 1: Frequently used symbolsis chosen by bothTable 1: Frequently used symbols