Fully-Dynamic Submodular Cover with Bounded Recourse
aa r X i v : . [ c s . D S ] S e p Fully-Dynamic Submodular Cover with Bounded Recourse
Anupam Gupta ∗ Roie Levin ∗ Abstract
In submodular covering problems, we are given a monotone, nonnegative submodular func-tion f : 2 N → R + and wish to find the min-cost set S ⊆ N such that f ( S ) = f ( N ) . When f is acoverage function, this captures SetCover as a special case. We introduce a general frameworkfor solving such problems in a fully-dynamic setting where the function f changes over time, andonly a bounded number of updates to the solution (a.k.a. recourse) is allowed. For concreteness,suppose a nonnegative monotone submodular integer-valued function g t is added or removedfrom an active set G ( t ) at each time t . If f ( t ) = P g ∈ G ( t ) g is the sum of all active functions,we wish to maintain a competitive solution to SubmodularCover for f ( t ) as this active setchanges, and with low recourse. For example, if each g t is the (weighted) rank function of amatroid, we would be dynamically maintaining a low-cost common spanning set for a changingcollection of matroids.We give an algorithm that maintains an O (log( f max /f min )) -competitive solution, where f max , f min are the largest/smallest marginals of f ( t ) . The algorithm guarantees a total recourseof O (log( c max /c min ) · P t ≤ T g t ( N )) , where c max , c min are the largest/smallest costs of elementsin N . This competitive ratio is best possible even in the offline setting, and the recourse boundis optimal up to the logarithmic factor. For monotone submodular functions that also havepositive mixed third derivatives, we show an optimal recourse bound of O ( P t ≤ T g t ( N )) . Thisstructured class includes set-coverage functions, so our algorithm matches the known O (log n ) -competitiveness and O (1) recourse guarantees for fully-dynamic SetCover . Our work simulta-neously simplifies and unifies previous results, as well as generalizes to a significantly larger classof covering problems. Our key technique is a new potential function inspired by Tsallis entropy.We also extensively use the idea of
Mutual Coverage , which generalizes the classic notion ofmutual information. ∗ Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213. This research was done underthe auspices of the Indo-US Virtual Networked Joint Center IUSSTF/JC-017/2017. Research supported in part byNSF awards CCF-1907820, CCF1955785, and CCF-2006953. Emails: {anupamg,roiel}@cs.cmu.edu . Introduction
In the
SubmodularCover problem, we are given a monotone, nonnegative submodular func-tion f : 2 N → Z + , as well as a linear cost function c , and we wish to find the min-cost set S ⊆ N such that f ( S ) = f ( N ) . This is a classical NP-hard problem: e.g., when f is a coverage functionwe capture the SetCover problem. Moreover, the greedy algorithm is known to be an (1+ ln f max ) -approximation, where f max is the maximum value of any single element [Wol82]. This bound is tightassuming P = NP , even for the special case of SetCover [Fei98, DS14].We consider this
SubmodularCover problem in a fully-dynamic setting, where the notion ofcoverage changes over time. At each time, the underlying submodular function changes from f ( t ) to f ( t +1) , and the algorithm may have to change its solution from S t to S t +1 to cover this newfunction. We do not want our solutions to change wildly if the function changes by small amounts.The goal of this work is to develop algorithms where this “churn” | S t △ S t +1 | is small (perhaps in anamortized sense), while maintaining the requirement that each solution S t is a good approximatesolution to the function f ( t ) . The change | S t △ S t +1 | is often called recourse in the literature.This problem has been posed and answered in the special case of SetCover —for clarity, from nowon this paper we consider the equivalent
HypergraphVertexCover (a.k.a. the
HittingSet )problem. In this problem, hyperedges arrive to and depart from an active set over time, and wemust maintain a small set of vertices that hit all active hyperedges. We know an algorithm thatmaintains an O (log n t ) -approximation which has constant amortized recourse [GKKP17]; here n t refers to the number of active hyperedges at time t . In other words, the total recourse—the totalnumber of changes over T edge arrivals and departures—is only O ( T ) . The algorithm and analysisare based on a delicate token-based argument, which gives each hyperedge a constant number oftokens upon its arrival, and moves these tokens in a careful way between edges to account for thechanges in the solution. What do we do in the more general SubmodularCover case, where thereis no notion of sets any more?In this work we study the model where a submodular function is added or removed from the active set G ( t ) at each timestep: this defines the current submodular function f ( t ) := P g ∈ G t g as the sum of functions in this active set. The algorithm must maintain a subset S t ⊆ N suchthat f ( t ) ( S t ) = f ( t ) ( N ) with cost c ( S t ) being within a small approximation factor of the optimal SubmodularCover for f ( t ) , such that the total recourse P t | S t △ S t +1 | remains small. To verifythat this problem models dynamic HypergraphVertexCover , each arriving/departing edge A t should correspond to a submodular function g t taking on value for any set S that hits A t , andvalue zero otherwise (i.e., g t ( S ) = [ S ∩ A t = ∅ ] ). Our main result is the following:
Theorem 1.1 (Informal) . There is a deterministic algorithm that maintains an e · (1 + log f max ) -competitive solution to Submodular Cover in the fully-dynamic setting where functions arrive/departover time. This algorithm has total recourse: O (cid:18) X t g t ( N ) ln (cid:16) c max c min (cid:17)(cid:19) , In the introduction we restrict to integer-valued functions for simplicity; all results extend to general submodularfunctions with suitable changes. See the technical sections for the full results and nuanced details. here g t ( N ) is the value of the function considered at time t , and c max , c min are the maximum andminimum element costs. Let us parse this result. Firstly, the approximation factor almost matches Wolsey’s result up to themultiplicative factor of e ; this is best possible in polynomial time unless P = NP even in the offlinesetting. Secondly, the amortized recourse bound should be thought of as being logarithmic—indeed,specializing it to HittingSet where each g t ( N ) = 1 , we get a total recourse of O ( T log( c max /c min )) over T timesteps, or an amortized recourse of O (log( c max /c min )) per timestep. Hence this recoursebound is weaker by a log-of-cost-spread factor, while generalizing to all monotone submodularfunctions. Finally, since we are allowed to give richer functions g t at each step, it is natural thatthe recourse scales with P t g t ( N ) , which is the total “volume” of these functions. In particular, thisproblem captures fully-dynamic HypergraphVertexCover where at each round a batch of k edges appears all at once (this is in contrast to the standard fully-dynamic model where hyperedgesappear one at a time). In this case the algorithm may have to buy up to k = g t ( N ) /f min newvertices in general to maintain coverage.We next show that for coverage functions (and hence for the HypergraphVertexCover problem),a variation on the algorithm from Theorem 1.1 can remove the log-of-cost-spread factor in terms ofrecourse, at the cost of a slightly coarser competitive ratio. E.g., for
HypergraphVertexCover the new competitive ratio corresponds to an O (log n ) guarantee versus an O (log D max ) guarantee,where D max being the largest degree of any vertex. Theorem 1.2 (Informal) . There is a deterministic algorithm that maintains an O (log f ( N )) - com-petitive solution to SubmodularCover in the fully-dynamic setting where functions arrive/departover time, and each function is -increasing in addition to monotone and submodular. Furthermore,this algorithm has total recourse: O (cid:18) X t g t ( N ) (cid:19) . Indeed, this result holds not just for coverage functions, but for the broader class of -increasing,monotone, submodular functions [FH05], which are the functions we have been considering, withthe additional property that have positive discrete mixed third-derivatives. At a high level, theseare functions where the mutual coverage does not increase upon conditioning. The most widely known algorithm for
SubmodularCover is the greedy algorithm; this repeatedlyadds to the solution an element maximizing the ratio of its marginal coverage to its cost. It is naturalto try to use the greedy algorithm in our dynamic setting; the main issue is that out-of-the-box,greedy may completely change its solution between time steps. In their result on recourse-bounded
HypergraphVertexCover , [GKKP17] showed how to effectively imitate the greedy algorithmwithout sacrificing more than a small amount of recourse. A barrier to making greedy algorithmsdynamic is their sequential nature, and hyperedge inserts/deletes can play havoc with this. So theygive a local-search algorithm that skirts this sequential nature, while simultaneously retaining thekey properties of the greedy algorithm. Unfortunately, their analysis hinges on delicately assigningedges to vertices. In the more general
SubmodularCover setting, there are no edges to track,and this approach breaks down.Our first insight is to return to the sequential nature of the greedy algorithm. Our algorithmmaintains an ordering π on the elements of N , which induces a solution to the SubmodularCover stack trace . We show that our local updates maintain the invariant that thisis the stack trace of an approximate greedy algorithm for the currently active set of functions. Wehope that this general framework of doing local search on the stack trace of an auxiliary algorithm finds uses in other online/dynamic algorithms.Our main technical contribution is to give a potential function to argue that our algorithm needsbounded recourse. The potential measures the (generalized) entropy of the coverage vector of thepermutation π . This coverage vector is indexed by elements of the universe N , and the valueof each coordinate is the marginal coverage (according to permutation π ) of the correspondingelement. Entropy is often used as a potential function (notably, in recent developments in onlinealgorithms) but in a qualitatively different way. In many if not all cases, these algorithms follow themaximum-entropy principle and seek high entropy solutions subject to feasibility constraints; thepotential function then tracks the convergence to this goal. On the other hand, in our setting thecost function is the support size of the coverage vector, and minimizing this roughly correspondsto low entropy . We show entropy decreases sufficiently quickly with every change performed duringthe local search, and increases by a controlled amount with insertion/deletion of each function g ,thus proving our recourse bounds.We find our choice of entropy to be interesting for several reasons. For one, we use a suitably chosen Tsallis entropy [Tsa88], which is a general parametrized family of entropies, instead of the classicalShannon entropy. The latter also yields recourse bounds for our problem, but they are substantiallyweaker (see Appendix D). Tsallis entropy has appeared in several recent algorithmic areas, forexample as a regularizer in bandit settings [SS14], and as an approximation to Shannon entropy instreaming contexts [HNO08]. Secondly, it is well known that for certain problems, minimizing an ℓ -objective is an effective proxy for minimizing sparsity [CRT06]. In our dynamics, the ℓ massstays constant since total coverage does not change when elements are reordered. However, entropyis a good proxy for sparsity, in the sense that it decreases monotonically (and quickly!) as ouralgorithm moves within the ℓ level set towards sparse vectors.In Section 2, we study the unit cost case to lay out our framework and highlight the main ideas. Weshow a log f max /f min competitive algorithm for fully-dynamic SubmodularCover with O ( P t g t ( N ) /f min ) total recourse. Then in Section 3, we turn to general cost functions. We again show a log( f max /f min ) competitive algorithm, this time with log( c max /c min ) · P t g t ( N ) /f min recourse. The algorithm tem-plate is the same as before, but with a suitably generalized potential function and analysis. Herewe also require a careful choice of the Tsallis entropy parameter, near (but not quite at) the limitpoint at which Tsallis entropy becomes Shannon entropy.In Section 4, we show how to remove the log( c max /c min ) dependence and achieve optimal recourse fora structured family of monotone, submodular functions: the class of -increasing (monotone, sub-modular) set functions [FH05]. These are set functions, all of whose discrete mixed third derivativesare nonnegative everywhere. Submodular functions in this class include measure coverage functions ,which generalize set coverage functions, as well as entropies for distributions with positive inter-action information (see, e.g., [Jak05] for a discussion). Since this class includes SetCover , thisrecovers the optimal O (1) recourse bound of [GKKP17]. For this result we use a more interestinggeneralization of the potential function from Section 2 that reweights the coverage of elements inthe permutation non-uniformly as a function of their mutual coverage with other elements of the3ermutation.In Appendix A, we show how to get improved randomized algorithms when the functions g t areassumed to be r -juntas. This is in analogy to approximation algorithms for SetCover underthe frequency parametrization. In Appendix B, we also show how to run an online “combiner”algorithm that gets the best of all worlds, with a competitive ratio of O (min( r, log f max /f min )) .Finally, in Appendix C, we demonstrate the generality of our framework by using it to recoverknown recourse bounded algorithms for fully-dynamic MinimumSpanningTree and
Minimum-SteinerTree . These achieve O (1) competitive ratios, and recourse bounds of O (log D ) where D is the aspect ratio of the metric. Our proofs here are particularly simple and concise. Submodular Cover.
While we introduced the problem for integer-valued functions, all resultscan be extended to real-valued settings by adding a dependence on f min , the smallest marginalvalue. Wolsey [Wol82] showed that the greedy algorithm, repeatedly selecting the element max-imizing marginal coverage to cost ratio, gives a f max /f min ) approximation for Submod-ularCover ; this guarantee is tight unless P = NP even for the special case of SetCover [Fei98, DS14]. Fujito [Fuj00] gave a dual greedy algorithm that generalizes the F -approximation for SetCover [Hoc82] where F is the maximum-frequency of an element. SubmodularCover has been used in many applications to resource allocation and placementproblems, by showing that the coverage function is monotone and submodular, and then applyingWolsey’s greedy algorithm as a black box. We automatically port these applications to the fully-dynamic setting where coverage requirements change with time. E.g., in selecting influential nodesto disseminate information in social networks [GBLV13, LG16, TWPD17, IIOW10], explorationfor robotics planning problems [KMGG08, JCMP17, BMKB13], placing sensors [WCZW15, RP15,ZPW +
17, MW16], and other physical resource allocation objectives [YCDW15, LCL +
16, TRPJ16].The networking community has been particularly interested in
SubmodularCover recently, since
SubmodularCover models network function placement tasks [AGG +
09, LPS13, KN15, LRS18,CWJ18]. E.g., [LRS18] want to place middleboxes in a network incrementally, and point out thatavoiding closing extant boxes is a huge boon in practice.The definition of m -increasing functions is due to Foldes and Hammer [FH05]. Bach [Bac13] gavea characterization of the class of measure coverage functions (which we define later) in terms of itsiterated discrete derivatives. This class generalizes coverage functions, and is contained in the class of -increasing functions. [IKBA20] give several additional examples of -increasing functions. Severalpapers [IKBA20, CM18, WMWP15, WWLP13] have given algorithms specifically for -increasingsubmodular function optimization. Online and Dynamic Algorithms.
There is a still budding series of work on recourse-bounded al-gorithms. Besides [GKKP17] which is most directly related to our work, researchers have studied theSteiner Tree problem [IW91, GK14, GGK16, ŁOP + + O (log n ) competitive and O ( F log n ) update time algorithm for fully-dynamic HypergraphVertexCover . An ongoing program ofresearch for the frequency parametrization of
HypergraphVertexCover [BHI18, BCH17, BK19,AAG +
19, BHN19, BHNW20] has so far culminated in an F (1 + ǫ ) competitive algorithm with poly( F, log c max /c min , /ǫ ) update time (where F is the frequency).4n recent work, [GL20] studied the problem of maintaining a feasible solution to Submodular-Cover in a related online model. That setting is an insertion-only irrevocable analog of thiswork, where functions g t may never leave the active set. Our results can be seen as an exten-sion/improvement when recourse is allowed: not only can our algorithm handle the fully-dynamiccase with insertions and deletions, but we improve the competitive ratio from O (log n · log f max /f min ) to O (log f max /f min ) , which is best possible even in the offline case.Our work is related to work on convex body chasing (e.g., [AGGT20, Sel20]) in spirit but not intechniques. For an online/dynamic covering problems, the set of feasible fractional solutions withindistance α of the optimal solution at a given time step form a convex set: our goal is similarlyto “chase” these convex bodies, while limiting the total movement traversed. The main differenceis that we seek absolute bounds on the recourse, instead of recourse that is competitive with theoptimal chasing solution. (We can give such bounds because our feasible regions are structured andnot arbitrary convex bodies). A set function f : 2 N → R + is submodular if f ( A ∩ B ) + f ( A ∪ B ) ≤ f ( A ) + f ( B ) for any A, B ⊆ N .It is monotone if f ( A ) ≤ f ( B ) for all A ⊆ B ⊆ N . We assume access to a value oracle for f that computes f ( T ) given T ⊆ N . The contraction of f : 2 N → R + onto N \ T is defined as f T ( S ) = f ( S | T ) := f ( S ∪ T ) − f ( T ) . If f is submodular then f T is also submodular for any T ⊆ N . We use the following notation. f ( t )max := max { f ( t ) ( j ) | j ∈ N } , (1.1) f ( t )min := min { f ( t ) ( j | S ) | j ∈ N, S ⊆ N , f ( t ) ( j | S ) = 0 } . (1.2) f max := max t f ( t )max , (1.3) f min := min t f ( t )min . (1.4)Also we let c max and c min denote the largest and smallest costs of elements respectively.We will sometime use the simple and well known inequalities: Fact 1.3.
Given positive numbers a , . . . , a k and b , . . . , b k : min i a i b i ≤ P i a i P i b i ≤ max i a i b i (1.5)Throughout this paper, we will use the convention that k denotes that range of indices from to k . Mutual Coverage.
We will use the notion of mutual coverage defined in [GL20]. Independently,[IKBA20] defined and studied the same quantity under the slightly different name submodularmutual information . Definition 1.4 (Mutual Coverage) . The mutual coverage and conditional mutual coverage withrespect to a set function f : 2 N → R + are defined as: I f ( A ; B ) := f ( A ) + f ( B ) − f ( A ∪ B ) , (1.6) I f ( A ; B | C ) := f C ( A ) + f C ( B ) − f C ( A ∪ B ) . (1.7)5e may think of I f ( A ; B | C ) intuitively as being the amount of coverage B “takes away” from thecoverage of A (or vice-versa, since the definition is symmetric in A and B ), given that C was alreadychosen. This generalizes the notion of mutual information from information theory: if N is a set ofrandom variables, and S ⊆ N , and if f ( S ) denotes the joint entropy of the random variables in theset S , then I is the mutual information. Fact 1.5 (Chain Rule) . Mutual coverage respects the identity: I f ( A ; B ∪ B | C ) = I f ( A ; B | C ) + I f ( A ; B | C ∪ B ) . This neatly generalizes the chain rule for mutual information.
We now present our first algorithm for unit-cost fully-dynamic
SubmodularCover . We will showthe following:
Theorem 2.1.
For any γ > e , there is a deterministic algorithm that maintains a γ (log f ( t )max /f ( t )min +1) -competitive solution to unweighted SubmodularCover in the setting where functions arrive/departover time. Furthermore, this algorithm has total recourse: · e ln γγ − e ln γ · P t g t ( N ) f min = O (cid:18) P t g t ( N ) f min (cid:19) . The algorithm and its analysis are particularly clean; we will build on these in the following sections.We begin by describing the algorithm. We maintain a permutation π of the elements in N , andassign to each element its marginal coverage given what precedes it in the permutation. We writethis marginal value assigned to element π i as F π ( π i ) := f ( π i | π i − ) . (2.1)We consider two kinds of local search moves:1. Swaps: transform π to π ′ by moving an element at position i to position i − on the conditionthat F ( π i ) ≥ F ( π i − ) . In words, this is a bubble operation (as in bubble-sort).2. γ -moves : transform the permutation π to π ′ by moving an element u from a position q tosome other position p < q on the condition that for all i ∈ { p, . . . , q − } , F π ′ ( π ′ p ) ≥ γ · F π ( π i ) . In words, when u moves ahead in line, it “steals” coverage from other elements along the way;we require that the amount covered by u after the move (which is given by F π ′ ( π ′ p ) since u now sits at position p ) to be at least a γ factor larger than the coverage before the move ofany element that u jumps over. (See Figure 1.)The dynamic algorithm is the following. When a new function g ( t ) arrives or departs, update thecoverages F π of all the elements in the permutation. Subsequently, while there is a local searchmove available, perform the move. Output the prefix of π of all elements with non-zero coverage.This is summarized in Algorithm 1, with a setting of γ > e .6 b c d e (a) Before Move e a b c d (b)
After Move
Figure 1:
Illustration of a legal γ -move. Each rectangle represents the marginal coverage of an element of thepermutation. The height of the item that moves must be at least γ times the height of anyone it cuts in line. Algorithm 1
FullyDynamicSubmodularCover π ← arbitrary initial permutation of elements N . for t = 1 , , . . . , T do t th function g t arrives/departs. while there exists a legal γ -move or a swap for π do Perform the move, and update π . Output the collection of π i such that F π ( π i ) > . Let us start by arguing that if the algorithm terminates, it must produce a competitive solution.
Lemma 2.2.
Suppose no γ -moves are possible, then for every index i such that F π ( π i ) > , andfor every index i ′ > i , the following holds. Let π ′ be the permutation where π i ′ is moved to position i . Then F π ( π i ) > F π ′ ( π i ′ ) γ (2.2) Proof.
Suppose there are elements π i and π i ′ such that condition (2.2) does not hold, i.e. F π ′ ( π j ) ≥ γ · F π ( π i ) . Since by assumption there are no swaps available, the permutation π is in non-increasingorder of F π ( π i ) values, so for all indices j > i it also holds that F π ( π i ′ ) ≥ γ · F π ( π j ) . Hence moving i ′ from its current position to position i is a legal γ -move, which is a contradiction. Corollary 2.3.
The output at every time step is γ · (log f ( t )max /f ( t )min + 1) -competitive.Proof. Lemma 2.2 implies that the solution is equivalent to greedily selecting an element whosemarginal coverage is within a factor of /γ of the largest marginal coverage currently available.Given this, the standard analyses of the greedy algorithm for SubmodularCover [Wol82] implythat the solution is γ · (log f ( t )max /f ( t )min + 1) competitive. We move on to showing that the algorithm must terminate with O ( g ( N ) /f min ) amortized recourse.For this analysis, we define a potential function parametrized by a number α ∈ (0 , to be fixed7ater: Φ α ( f, π ) := X i ∈ N ( F π ( π i )) α . As noted in the introduction, up to scaling and shifting, this is the Tsallis entropy with parameter α . We show several properties of this potential: Properties of Φ α :(I) Φ α increases by at most g t ( N ) · ( f min ) α − with the addition of function g t to the activeset. (II) Φ α does not increase with deletion of functions from the system. (III) Φ α does not increase during swaps. (IV) For an appropriate range of γ , the potential Φ α decreases by at least ( γ/ ( e ln γ ) − · ( f min ) α = Ω(( f min ) α ) with every γ -move. Lemma 2.4. If α = (ln γ ) − , then Φ α respects properties (I) – (IV) .Proof. For brevity, define h ( z ) := z α . Since α ∈ (0 , , this function is concave and non-decreasing.We start with property (I) . When a function g is added to the system, for some set of i ∈ [ n ] , itincreases k i := F π ( π ( i )) by some amount ∆ i . Observe that P i ∆ i = g ( N ) . By the concavity of h : X i h ( k i + ∆ i ) − X i h ( k i ) ≤ X i h (∆ i ) ≤ P i ∆ i f min · h ( f min ) = g ( N ) · ( f min ) α − . Property (II) follows since h is non-decreasing.For property (III) , we wish to show that if u immediately precedes v in π and F π ( u ) ≤ F π ( v ) , thenswapping u and v does not increase the potential. Let b π denote the permutation after the swap.Note that F b π ( u ) ≤ F π ( v ) and F b π ( v ) ≥ F π ( v ) , since u may only have lost some amount of coverageto v . Suppose this amount is k , i.e. k = F π ( u ) − F b π ( u ) = F b π ( v ) − F π ( v ) . Then: Φ α ( f, b π ) − Φ α ( f, π ) = h ( F b π ( u )) + h ( F b π ( v )) − h ( F π ( u )) − h ( F π ( v ))= h ( F π ( u ) − k ) + h ( F π ( v ) + k ) − h ( F π ( u )) − h ( F π ( v )) which is non-positive due to the concavity of h .It remains to prove property (IV) . Suppose we perform a γ -move on a permutation π . Let u be theelement moving to some position p from some position q > p , and let π ′ denote the permutationafter the move. For convenience, also define: v i := F π ( π i ) , (the original coverage of the i th set) a i := I f ( π i ; u | π i − ) = F π ( π i ) − F π ′ ( π i ) . (the loss in coverage of the i th set)Note that for all i
6∈ { p, . . . , q } , we necessarily have a i = 0 . Also note that F π ′ ( S ) = P i a i , and bythe definition of a γ -move, for any j we have P i a i ≥ γ · v j . Then the change in potential is: Φ α ( f, π ′ ) − Φ α ( f, π ) = (cid:16) X i a i (cid:17) α + X i ( v i − a i ) α − X i v αi ≤ (cid:16) X i a i (cid:17) α − X i a i · α · v α − i (2.3)8 (cid:16) X i a i (cid:17) α − (cid:16) X i a i (cid:17) α · α · γ − α (2.4) ≤ − (cid:16) γe ln γ − (cid:17) ( f min ) α . (2.5)Above, (2.3) holds since h is concave and thus h ( v i − a i ) − h ( v i ) ≤ ∇ h ( v i ) · ( − a i ) . Line (2.4) holdsby the definition of a γ -move, since P i a i ≥ γv j for every j ∈ { p, . . . , q } . Finally, (2.5) comes fromour choice of α = (ln γ ) − and the fact that P i a i ≥ f min .Finally, we return to proving the main theorem. Proof of Theorem 2.1.
By Lemma 2.2, if Algorithm 1 (using Definition 2.1 for F π ) terminates thenit is γ · (log f ( t )max /f ( t )min + 1) -competitive.By (I) – (IV) , the potential Φ α increases by at most g t ( N )( f min ) α − for every function g t insertedto the active set, decreases by ( f min ) α · ( γ/ ( e ln γ ) − per γ -move, and otherwise does not increase.By inspection, Φ α ≥ . The number of elements e with F π ( e ) > grows by only during γ -moves inwhich F π ( e ) was initially . Otherwise, this number never grows. We account for elements leavingthe solution by paying recourse upfront when they join the solution.Hence, the number of changes to the solution is at most: · P t g t ( N )( f min ) − α · e ln γ ( f min ) α ( γ − e ln γ ) = O (cid:18) P t g t ( N ) f min (cid:19) . Our algorithm gives a tunable tradeoff between approximation ratio and recourse depending on thechoice of γ . Note that if we wish to optimize the competitive ratio, setting γ = e (1 + δ ) gives arecourse bound of (cid:20)(cid:18) δ ) δ − ln(1 + δ ) (cid:19)(cid:21) P t g t ( N ) f min = O (cid:18) δ (cid:19) P t g t ( N ) f min as δ approaches . For simplicity one can set γ = e to get the bound in Theorem 2.1. We now turn to the general costs case and show the main result of our paper:
Theorem 3.1.
There is a deterministic algorithm that maintains an e · (log f ( t )max /f ( t )min + 1) - com-petitive solution to SubmodularCover in the setting where functions arrive/depart over time.Furthermore, this algorithm has amortized recourse: O (cid:18) g ( N ) f min ln (cid:18) c max c min (cid:19)(cid:19) per function arrival/departure. Given the last section, our description of the new algorithm is very simple. We reuse Algorithm 1,except we redefine: F π ( π i ) := f ( π i | π i − ) c ( π i ) . We will specify the last free parameter γ shortly.9 .2 Bounding the Cost To bound the competitive ratio, note that Eq. (2.2) did not use any particular properties of F π ,except that in the solution output by the algorithm, there are no γ -moves or swaps with respect to F π in permutation π . The analog of Corollary 2.3 is nearly identical: Corollary 3.2.
The output at every time step is γ · (log f ( t )max /f ( t )min + 1) -competitive.Proof. Lemma 2.2 implies that the solution is equivalent to greedily selecting an element whosemarginal coverage/cost ratio is within a factor of /γ of the largest marginal coverage/cost ratiocurrently available. The standard analyses of the greedy algorithm for SubmodularCover [Wol82]imply that the solution is γ · (log f ( t )max /f ( t )min + 1) competitive. To make our analysis as modular as possible, we will write a general potential function, parametrizedby a function h : R → R : Φ h ( f, π ) := X i ∈ N c ( π i ) · h ( F π ( π i )) . With foresight, we require several properties of h : Properties of h :(i) h is monotone and concave. (ii) h (0) = 0 . (iii) h satisfies x · h ′ ( x/γ ) ≥ (1 + ǫ γ ) h ( x ) . (iv) h satisfies y · h ( x/y ) is monotone in y .To bound the recourse, our goal will again be to show the following properties of our potentialfunction Φ h : Properties of Φ h :(I) Φ h increases by at most g t ( N ) f min · c max · h (cid:18) f min c max (cid:19) with the addition of function g t to the active set. (II) Φ h does not increase with deletion of functions from the system. (III) Φ h does not increase during sorting. (IV) For an appropriate range of γ , the potential Φ h decreases by at least ǫ γ · c min · h (cid:18) f min c min (cid:19) with every γ -move. 10ogether, the statements imply a total recourse bound of: P t g t ( N ) ǫ γ · f min · c max c min · h ( f min /c max ) h ( f min /c min ) Lemma 3.3. If h respects properties (i) – (iv) then Φ h respects properties (I) – (IV) .Proof. We start with property (I) . When a function g t is added to the system, for some set of i ∈ [ n ] ,it increases k i := F π ( π ( i )) by some amount ∆ i . Then the potential increase is: Φ h ( f ( t ) , π ) − Φ h ( f ( t − , π ) = X i ∈ [ n ] c ( π i ) · h (cid:18) k i + ∆ i c ( π i ) (cid:19) − X i ∈ [ n ] c ( π i ) · h (cid:18) k i c ( π i ) (cid:19) ≤ X i ∈ [ n ] c ( π i ) · h (cid:18) ∆ i c ( π i ) (cid:19) (3.1) ≤ X i ∈ [ n ] c max · h (cid:18) ∆ i c max (cid:19) (3.2) ≤ P i ∆ i f min · c max · h (cid:18) f min c max (cid:19) (3.3) = g t ( N ) f min · c max · h (cid:18) f min c max (cid:19) . Above step (3.1) is by properties (i) and (ii) , step (3.2) is by property (iv) and step (3.3) is byproperty (i) .Property (II) follows since h is non-decreasing.The proof of property (III) is similar to the one in Lemma 2.4. Suppose u immediately precedes v in π but F π ( u ) ≤ F π ( v ) , and let b π denote the permutation after the swap. We have that F b π ( u ) ≤ F π ( v ) and F b π ( v ) ≥ F π ( v ) , since u may only have lost some amount of coverage to v . Suppose this amountis k , i.e. k = F π ( u ) − F b π ( u ) = F b π ( v ) − F π ( v ) . Then: Φ h ( f, b π ) − Φ h ( f, π ) = c ( u ) · h ( F b π ( u )) + c ( v ) · h ( F b π ( v )) − c ( u ) · h ( F π ( u )) − c ( v ) · h ( F π ( v ))= c ( u ) (cid:18) h (cid:18) F π ( u ) − kc ( u ) (cid:19) − h ( F π ( u )) (cid:19) + c ( v ) (cid:18) h (cid:18) F π ( v ) + kc ( v ) (cid:19) − h ( F π ( v )) (cid:19) ≤ k · ( h ′ ( F π ( v )) − h ′ ( F π ( u ))) which is non-positive due to the concavity of h and the fact that F π ( v ) ≥ F π ( u ) .Finally the proof of property (IV) is also similar to the version in the last section. Suppose weperform a γ -move on a permutation π . Let u be the element moving to some position p from someposition q > p , and let π ′ denote the permutation after the move. Then: Φ h ( f, π ′ ) − Φ h ( f, π ) = c ( u ) · h P i ∈ [ n ] a i c ( u ) ! + X i ∈ [ n ] c ( π i ) · h (cid:18) v i − a i c ( π i ) (cid:19) − X i ∈ [ n ] c ( π i ) · h (cid:18) v i c ( π i ) (cid:19) ≤ c ( u ) · h P i ∈ [ n ] a i c ( u ) ! − X i ∈ [ n ] a i · h ′ (cid:18) v i c ( π i ) (cid:19) (3.4)11 c ( u ) · h P i ∈ [ n ] a i c ( u ) ! − c ( u ) X i ∈ [ n ] a i c ( u ) · h ′ P i ∈ [ n ] a i γ · c ( u ) ! (3.5) ≤ c ( u ) · h P i ∈ [ n ] a i c ( u ) ! − (1 + ǫ γ ) · c ( u ) · h P i ∈ [ n ] a i c ( u ) ! (3.6) ≤ − ǫ γ · c ( u ) · h P i ∈ [ n ] a i c ( u ) ! ≤ − ǫ γ · c min · h (cid:18) f min c min (cid:19) . (3.7)Above step (3.4) uses the concavity of h . Step (3.5) uses the fact that moving u was a legal γ -moveand thus P i ∈ [ n ] a i /c ( u ) ≥ γv i /c ( π i ) . Step (3.6) follows from property (iii) . Finally, (3.7) usesproperty (iv) again.With this setup, the proof of the main theorem boils down to identifying an appropriate function h , and a suitable choice of γ . Proof of Theorem 3.1.
Recall the general recourse bound is P t g t ( N ) ǫ γ · f min · c max c min · h ( f min /c max ) h ( f min /c min ) . A good choice for h is h ( x ) = x − δ / (1 − δ ) for δ = (ln( c max /c min ) + 1) − , and with γ = e . Properties (i) – (iv) are easy to verify. To see that this implies the bound O (cid:18) P t g t ( N ) f min ln (cid:18) c max c min (cid:19)(cid:19) , note that γ = e ≥ ((1 + δ ) / (1 − δ )) /δ , in which case ǫ γ = γ δ (1 − δ ) − ≥ δ . Furthermore, ( c max /c min ) δ = O (1) . -increasing functions In this section we show to remove the log( c max /c min ) dependence from the recourse bound when f is assumed to be from a structured class of set functions: the class of -increasing functions. Westart with some definitions. Following the notation of [FH05], we define the derivative of a set function f as: dfdx ( S ) = f ( S ∪ { x } ) − f ( S \{ x } ) . We notate the m -th order derivative of f with respect to the subset A = { i , . . . i m } as df /dA , andthis quantity has the following concise expression: dfdA ( S ) = X B ⊆ A ( − | B | f (( S ∪ A ) \ B ) . efinition 4.1 ( m -increasing [FH05]) . We say that a set function is m -increasing if all its m -thorder derivatives are nonnegative, and m -decreasing if they are nonpositive. We denote by D + m and D − m the classes of m -increasing and m -decreasing functions respectively. Note that D +1 is the class of monotone set functions, and D − is the class of submodular set functions.When f is the joint entropy set function, the m -th derivative above is also known as the interactioninformation , which generalizes the usual mutual information for two sets of variables, to m sets ofvariables.We will soon show improved algorithms for the class D +3 , that is the class of -increasing set functions.The following derivation gives some intuition for these functions: dfd { x, y, z } ( S ) = X B ⊆{ x,y,z } ( − | B | f (( S ∪ { x, y, z } ) \ B )= f ( x | S ) − f ( x | S ∪ { y } ) − ( f ( x | S ∪ { z } ) − f ( x | S ∪ { y, z } ))= I f ( x, y | S ) − I f ( x, y | S ∪ { z } ) . (4.1)Thus a function f in contained in D +3 if and only if mutual coverage decreases after conditioning.Bach [Bac13, Section 6.3] shows that a nonnegative function is in D +2 m − ∩ D − m simultaneously forall m ≥ if and only if it is a measure coverage function : each element i ∈ N is associated withsome measurable set S i under a measure µ , and f ( S , . . . , S t ) = µ (cid:0)S ti =1 S i (cid:1) . We now show our second main result:
Theorem 4.2.
There is a deterministic algorithm that maintains an O (log f ( N ) /f min ) -competitivesolution to SubmodularCover in the fully-dynamic setting where functions arrive/depart overtime, and each function is -increasing in addition to monotone and submodular. Furthermore, thisalgorithm has total recourse: O (cid:18) P t g t ( N ) f min (cid:19) . We reuse Algorithm 1 from previous sections, only this time we redefine F π more substantially. Giventwo permutations α and β on N , we define the ( i, j ) mutual affinity of ( α, β ) (and its conditionalvariant) as I α,β ( α i , β j ) := I f ( α i , β j | α i − ∪ β j − ) , I α,β ( α i , β j | S ) := I f ( α i , β j | α i − ∪ β j − ∪ S ) . Recall that I f denotes the Mutual Coverage. To give some insight into these definitions, observethat mutual affinity telescopes cleanly: Observation 4.3.
The chain rule implies that X j ∈ [ n ] I α,β ( α i , β j ) = f ( α i | α i − ) , We avoid the term set coverage function used by [Bac13], since we already use this terminology to mean thespecial case of a counting measure defined by a finite set system, as in
SetCover . i ∈ [ n ] I α,β ( α i , β j ) = f ( β j | β j − ) , X i,j ∈ [ n ] I α,β ( α i , β j ) = f ( N ) . Let ψ denote the ordering of N in increasing order of cost . Then: F π ( π i ) := X j ∈ [ n ] I π,ψ ( π i , ψ j ) c ( π i ) · c ( ψ j ) . (4.2)The following observation gives some intuition for this definition. Observation 4.4.
The expression I α,β ( α i , β j ) is nonzero only if (a) element α i precedes element β j in permutation α , and also (b) element β j precedes element α i in permutation β . In light of these remarks, F π ( π i ) decomposes the marginal coverage of π i into its mutual affinitywith all the elements ψ j ∈ N that are simultaneously cheaper than π i and that follow π i in thepermutation π , and weights each of these affinities by ( c ( π i ) c ( ψ j )) − .With this re-definition of F π , Algorithm 1 is fully specified (though it is still parametrized by γ ).We move to proving formal guarantees. Since our definition of F π (and thus the behavior of the algorithm) has significantly changed, weneed to reprove the competitive ratio guarantee. Our goal will be the following Lemma: Lemma 4.5.
If no swaps or γ -moves are possible, the solution is γ · log( f ( t ) ( N ) /f ( t )min ) -competitive. For the remainder of Section 4.3, we drop the superscript and refer to f ( t ) as simply f .Define a level L ℓ be the collection of elements u such that F π ( u ) ∈ [ γ ℓ , γ ℓ +1 ) . Note that sinceno swaps were possible, permutation π is sorted in decreasing order of F π , and thus L ℓ forms acontiguous interval of indices in π . Our proof strategy will be to show that for any level ℓ , the totalcost of all elements in L ℓ is at most γ · c ( Opt ) . Subsequently, we will argue that there are at most O (log f ( N ) /f min ) non-trivial levels. Lemma 4.6 (Each Level is Inexpensive) . If there are no swaps or γ -moves for π , then for any ℓ > , the total cost of all elements in L ℓ is at most γ · c ( Opt ) .Proof. Suppose some level L ℓ has cost c ( L ℓ ) > γ · c ( Opt ) . We will argue that in this case theremust be a legal γ -move available. To start, using Fact 1.3 we observe that: γ ℓ ≤ min u ∈L ℓ F π ( u ) (def) = min u ∈L ℓ P j ∈ [ n ] I π,ψ ( u, ψ j ) /c ( ψ j ) c ( u ) ≤ P u ∈L ℓ P j ∈ [ n ] I π,ψ ( u, ψ j ) /c ( ψ j ) P u ∈L ℓ c ( u ) , which by rearranging means: X u ∈L ℓ X j ∈ [ n ] I π,ψ ( u, ψ j ) c ( ψ j ) ≥ γ ℓ · c ( L ℓ ) > γ ℓ +2 · c ( Opt ) . (4.3)Let π < be the set of items preceding L ℓ in π , and π ≥ be the set of items in or succeeding L ℓ . Wealso define Opt ≥ := Opt ∩ π ≥ . 14 ℓ o o o o o o (a) Permutation π . L ℓ o o o o o o (b) Permutation π ′ , with Opt ≥ moved ahead of L ℓ . Figure 2:
Illustration of the proof of Lemma 4.6.
First, we imagine moving all the elements of
Opt ≥ , simultaneously and in order according to π , topositions just before L ℓ . Let π ′ be this new permutation. We first show that some o ∈ Opt hasa high value F π ′ ( o ) after this move. Then we show that this element o also constitutes a potential γ -move in π , a contradiction.For the first claim, we start by using Fact 1.3 again: max o ∈ Opt ≥ F π ′ ( o ) ≥ P o ∈ Opt ≥ P j ∈ [ n ] I π ′ ,ψ ( o, ψ j | π < ) /c ( ψ j ) P o ∈ Opt ≥ c ( o )= 1 c ( Opt ≥ ) · X o ∈ Opt ≥ X j ∈ [ n ] I π ′ ,ψ ( o, ψ j | π < ) c ( ψ j )= 1 c ( Opt ≥ ) · X j ∈ [ n ] f ψ ( ψ j | π < ) c ( ψ j ) (4.4) ≥ c ( Opt ≥ ) · X u ∈L ℓ X j ∈ [ n ] I π,ψ ( u, ψ j | π < ) c ( ψ j ) (4.5)The lines (4.4) and (4.5) above follow by Observation 4.3 (note that (4.5)) is an inequality because L ℓ may be a strict subset of π ≥ ). Since u ∈ L ℓ ⊆ π ≥ , this is: = 1 c ( Opt ≥ ) · X u ∈L ℓ X j ∈ [ n ] I π,ψ ( u, ψ j ) c ( ψ j ) (4.6) > γ ℓ +2 , (4.7)where line (4.7) used the inequality (4.3). In summary, F π ′ ( o ) is a factor γ bigger than any of the F π ( u ) values for u ∈ L ℓ .However, since permutation π ′ is obtained by moving many elements of π and not a single γ -move,we need to show that moving this element o alone gives a legal γ -move in π . In fact, observe thatwe have not yet used that f ∈ D +3 : we will use it now. Claim 4.7.
Let π o be the permutation derived from π where a single element o ∈ Opt ≥ is movedbefore L ℓ . Then F π o ( o ) ≥ F π ′ ( o ) .Proof. Start from π ′ and move an element o ′ ∈ Opt ≥ with o ′ = o back to its original position in π .Call this permutation π ′′ . Then: F π ′′ ( o ) = X j ∈ [ n ] I π ′′ ,ψ ( o, ψ j ) c ( o ) c ( ψ j ) X j ∈ [ n ] I π ′ ,ψ ( o, ψ j ) + ( I π ′′ ,ψ ( o, ψ j ) − I π ′′ ,ψ ( o, ψ j | { o ′ } )) c ( o ) c ( ψ j ) ≥ X j ∈ [ n ] I π ′ ,ψ ( o, ψ j ) c ( o ) c ( ψ j ) , where the second equality used I π ′ ,ψ ( o, ψ j ) = I π ′′ ,ψ ( o, ψ j | { o ′ } ) , and where the inequality usedequation (4.1), that I f ( a, b ) − I f ( a, b | c ) ≥ for -increasing functions. Repeating this processinductively until all elements but o have been returned to their original positions in π yields thepermutation π o , which proves the claim.Now we can complete the proof of Lemma 4.6. This means that if F π ′ ( o ) ≥ γ ℓ +2 , then there is alegal γ -move (namely the one which moves o ahead of L ℓ ), because by assumption every u ∈ π ≥ has F π ( u ) ≤ γ ℓ +1 . This contradicts the assumption that none existed.Next, we argue that the cost of all elements with very high or very low values of F π is small. Lemma 4.8 (Extreme Values Lemma) . If there are no swaps or γ -moves for π , the following hold: (i) There are no elements u such that < F π ( u ) ≤ f min / ( γ · ( c ( Opt )) ) . (ii) The total cost of all elements u such that F π ( u ) ≥ ( f ( N )) / ( f min · ( c ( Opt )) ) is at most √ γ · Opt .Proof of Lemma 4.8.
To prove item (i) , we observe that if permutation π has no local moves, thenevery element u must have high enough F π ( u ) to prevent any elements that follow it from cuttingit in line. Claim 4.9.
If there are no swaps or γ -moves for π , then for every π i ∈ N , and every j ∈ [ n ] suchthat I π,ψ ( π i , ψ j ) > we have: F π ( π i ) > f ( ψ j | π i − ) γ · ( c ( ψ j )) . Proof.
If there are no swaps or γ moves, then F π ′ ( ψ j ) < γ · F π i (where π ′ is the permutation obtainedfrom π by moving the element ψ j to the position ahead of u ). Expanding definitions: F π ′ ( ψ j ) (def) = X j ′ ∈ [ n ] I π ′ ,ψ ( ψ j , ψ j ′ ) c ( ψ j ) c ( ψ j ′ ) ≥ X j ′ ∈ [ n ] I π ′ ,ψ ( ψ j , ψ j ′ )( c ( ψ j )) = f ( ψ j | π i − )( c ( ψ j )) . In the first inequality we used that c ( ψ j ′ ) ≤ c ( ψ j ) by Observation 4.4, in the second equality weused Observation 4.3. Rearranging terms yields the claim.Now item (i) follows by setting ψ j to be the cheapest element that succeeds u in the permutation.Note that c ( ψ j ) ≤ c ( Opt ) .For item (ii) , let S be the set of indices i with F π ( π i ) ≥ ( f ( N )) / ( f min · ( c ( Opt )) ) . Then byFact 1.3: f ( N ) c ( Opt ) √ f min ≤ min i ∈ S p F π ( π i ) = min i ∈ S F π ( π i ) p F π ( π i ) (4.8) (def) = 1 c ( π i ) X j ∈ [ n ] I π,ψ ( π i , ψ j ) c ( ψ j ) p F π ( π i ) (4.9) ≤ c ( S ) · X i ∈ S X j ∈ [ n ] I π,ψ ( π i , ψ j ) c ( ψ j ) p F π ( π i ) c ( S ) ≤ c ( Opt ) √ f min f ( N ) · X i ∈ S X j ∈ [ n ] I π,ψ ( π i , ψ j ) c ( ψ j ) p F π ( π i ) ≤ √ γ · c ( Opt ) √ f min f ( N ) · X i ∈ S X j ∈ [ n ] I π,ψ ( π i , ψ j ) p f ( ψ j | π i − ) (4.10) ≤ √ γ · c ( Opt ) √ f min f ( N ) · X i ∈ S X j ∈ [ n ] I π,ψ ( π i , ψ j ) √ f min ≤ √ γ · c ( Opt ) f ( N ) · X i ∈ S X j ∈ [ n ] I π,ψ ( π i , ψ j )= √ γ · c ( Opt ) . (4.11)Above, step (4.10) is due to Claim 4.9 again and noting that if the denominator is 0 then thenumerator is also 0, and step (4.11) is due to Observation 4.3.Using these ingredients, we can now wrap up the proof of the competitive ratio. Proof of Lemma 4.5.
The number of levels ℓ such that γ ℓ lies between f min / ( γ ( c ( Opt )) ) and ( f ( N )) / ( f min · ( c ( Opt )) ) is O (log γ ( f ( N ) /f min ) ) = O (log( f ( N ) /f min )) . By Lemma 4.6, eachlevel in this range has cost at most O ( c ( Opt )) . By Lemma 4.8, there are no elements with non-zerocoverage in levels below this range, and the total cost of all elements above this range is O ( c ( Opt )) .Thus the total cost of the solution is O (log( f ( N ) /f min )) · c ( Opt ) . We follow our recipe of using the modified Tsallis entropy as a potential (with a reminder that theunderlying definitions of F π have changed) with α fixed to / : Φ / ( π ) := n X i =1 c ( π i ) p F π ( π i ) . It is worthwhile to interpret these quantities in the context of
HypergraphVertexCover . Inthis case, ψ i and ψ j correspond to vertices. Let Γ( v ) (and Γ( V ) ) denote the edge-neighborhood of v (and the union of the edge neighborhoods of vertices in the set V ), then I π,ψ ( π i , ψ j ) = | (Γ( π i ) ∩ Γ( ψ j )) \ (Γ( π i − ∪ ψ j − )) | , and if we use c ( e ) to denote the cost of the cheapest vertex hitting edge e , we can simplify c ( π i ) · p F π ( π i ) := vuut X e ∈ Γ( π i ) \ Γ( π i − ) c ( π i ) c ( e ) . In words, we are reweighting each hyperedge by the ratio of costs between the current vertex coveringit, and its cheapest possible vertex that could cover it. Intuitively, this means the potential will behigh when many elements are in sets that are significantly more expensive than the cheapest setsthey could lie in.This time the properties of Φ / we show are: 17 / = X i s c ( π i ) c ( ψ ) · I π ( π i ; ψ ) + c ( π i ) c ( ψ ) · I π ( π i ; ψ ) + c ( π i ) c ( ψ ) · I π ( π i ; ψ ) + c ( π i ) c ( ψ ) · I π ( π i ; ψ ) + . . . Figure 3:
Illustration of Φ / . Elements are arranged in order of π . Properties of Φ / :(I) Φ / increases by at most g ( N ) / √ f min with every addition of a function to the system. (II) Φ / does not increase with deletion of functions from the system. (III) Φ / does not increase during swaps. (IV) If γ > , then Φ / decreases by at least Ω( √ f min ) with every γ -move.Together, these will yield a recourse bound of P t g t ( N ) /f min . Lemma 4.10. Φ / satisfies property (I) .Proof. Consider a step in which f changes to f ′ because the function g was added to the system.For convenience, let b F π and b I π,ψ denote the quantities F π and I π,ψ after g has been added. Thenthe potential increase is: Φ / ( f ( t ) , π ) − Φ / ( f ( t − , π ) = X i ∈ [ n ] c ( π i ) b F π ( π i ) qb F π ( π i ) − X i ∈ [ n ] c ( π i ) F π ( π i ) p F π ( π i )= X i ∈ [ n ] X j ∈ [ n ] b I π,ψ ( π i , ψ j ) c ( ψ j ) qb F π ( π i ) − X i ∈ [ n ] X j ∈ [ n ] I π,ψ ( π i , ψ j ) c ( ψ j ) p F π ( π i ) Since F π ( π i ) can only increase when a function g is added to the active set, this is ≤ X i ∈ [ n ] X j ∈ [ n ] b I π,ψ ( π i , ψ j ) − I π,ψ ( π i , ψ j ) c ( ψ j ) p F π ( π i ) (def) = X i ∈ [ n ] X j ∈ [ n ] I g ( π i , ψ j | π i − ∪ ψ j − ) c ( ψ j ) p F π ( π i ) ≤ X j ∈ [ n ] P i ∈ [ n ] I g ( π i , ψ j | π i − ∪ ψ j − ) c ( ψ j ) p F π ( ψ j ) (4.12) = X j ∈ [ n ] g ( ψ j | ψ j − ) X j ′ ∈ [ n ] c ( ψ j ) I π,ψ ( ψ j , ψ j ′ ) c ( ψ j ′ ) − X j ∈ [ n ] g ( ψ j | ψ j − ) X j ′ ∈ [ n ] I π,ψ ( ψ j , ψ j ′ ) − (4.13) ≤ P j ∈ [ n ] g ( ψ j | ψ j − ) √ f min (4.14) = g ( N ) √ f min . Step (4.12) uses the relationship F π ( π i ) ≥ F π ( ψ j ) , which holds because summands are nonzeroonly if ψ j succeeds π i in permutation π (by Observation 4.4), or the numerator is 0, and thefact that elements are sorted in decreasing order of F π . Step (4.13) uses c ( ψ j ′ ) ≤ c ( ψ j ) , also byObservation 4.4. Finally step (4.14) uses Observation 4.3. Lemma 4.11. Φ / satisfies property (III) .Proof. Consider an index i such that swapping π i and π i +1 increases the potential. Then for somequantity δ := P j ∈ [ n ] ( I π,ψ ( π i , ψ j ) − I π ′ ,ψ ( π i , ψ j )) /c ( j ) > (since f ∈ D +3 ) we have: < ∆Φ / < c ( π i +1 ) s F π ( π i +1 ) + δc ( π i +1 ) − p F π ( π i +1 ) ! + c ( π i ) s F π ( π i ) − δc ( π i ) − p F π ( π i ) ! < δ p F π ( π i +1 ) − δ p F π ( π i ) (4.15) ⇒ F π ( π i +1 ) ≤ F π ( π i ) . Above, (4.15) holds since square root is a concave function and thus √ a + b − √ a ≤ b/ (2 √ a ) . Thisimplies that the local move was not a legal swap. Lemma 4.12. If γ > , then Φ / satisfies property (IV) .Proof. Suppose the local move changes the permutation π to π ′ by moving element u from position q to p . For notational convenience, define the following quantities: v i := F π ( π i ) ,a ij := I π ( π i , ψ j ) − I π ′ ( π i , ψ j ) (def) = [ p ≤ i ≤ q ] · ( I π ( π i , ψ j ) − I π ( π i , ψ j | { u } )) . Recall that by (4.1), the quantity a ij is (crucially) nonnegative when f is -increasing. Also, byexpanding the definition of Mutual Coverage, for all indices i ∈ { p, . . . , q } we can rewrite a ij as: a ij = f ( ψ j | π i − ∪ ψ j − ) − f ( ψ j | π i ∪ ψ j − ) − [ f ( ψ j | π i − ∪ ψ j − ∪ { u } ) − f ( ψ j | π i ∪ ψ j − ∪ { u } )]= I f ( u, ψ j | π i − ∪ ψ j − ) − I f ( u, ψ j | π i ∪ ψ j − ) . Thus, by the Chain Rule these terms telescopes such that: X i ∈{ p,...,q } a ij = I f ( u, ψ j | π p − , ψ j − ) (def) = I π ′ ,ψ ( u, ψ j ) . With this, we can bound the potential change after a local move as Φ / ( f, π ′ ) − Φ / ( f, π ) c ( u ) p F π ′ ( u ) + X i ∈ [ n ] c ( π i ) s v i − X j ∈ [ n ] a ij c ( π i ) c ( ψ j ) − X i ∈ [ n ] c ( π i ) √ v i ≤ c ( u ) p F π ′ ( u ) − c ( π i ) X i ∈ [ n ] √ v i · X j ∈ [ n ] a ij c ( π i ) c ( ψ j ) (4.16) ≤ c ( u ) p F π ′ ( u ) − √ γ · c ( π i ) p F π ′ ( u ) · X i ∈ [ n ] X j ∈ [ n ] a ij c ( π i ) c ( ψ j ) (4.17) = c ( u ) p F π ′ ( u ) − √ γ · c ( u ) p F π ′ ( u ) · X j ∈ [ n ] P i ∈ [ n ] a ij c ( u ) c ( ψ j ) Above, step (4.16) holds since square root is a concave function and thus √ a + b − √ a ≤ b/ (2 √ a ) .The next step (4.17) is due to the definition of γ -moves which ensure that v i ≤ F π ′ ( u ) /γ (note thatthis step also makes use of the nonnegativity of a ij ). Using the telescoping property of the a ij , wecan continue: = c ( u ) p F π ′ ( u ) − √ γ · c ( u ) p F π ′ ( u ) · X j ∈ [ n ] I π ′ ,ψ ( u, ψ j ) c ( u ) c ( ψ j ) (def) = − (cid:18) √ γ − (cid:19) vuut X j ∈ [ n ] c ( u ) · I π ′ ,ψ ( u, ψ j ) c ( ψ j ) ≤ − (cid:18) √ γ − (cid:19) s X j ∈ [ n ] I π ′ ,ψ ( u, ψ j ) (4.18) ≤ − (cid:18) √ γ − (cid:19) p f min . (4.19)Step (4.18) comes from Observation 4.4, and finally step (4.19) follows by Observation 4.3 andthe fact that f min is a lower bound on marginal coverage for any element with nonzero marginalcoverage.We wrap up with the proof of the main theorem. Proof of Theorem 4.2.
Set γ = 5 > . By Lemma 4.5, if Algorithm 1 (using Definition 4.2 for F π )terminates then it is O (log f ( N ) /f min ) -competitive.By (I) – (IV) , the potential Φ / increases by at most g t ( N / √ f min for every function g t inserted tothe active set, decreases by √ f min · (cid:0) √ γ/ − (cid:1) per γ -move, and otherwise does not increase. Byinspection, Φ α ≥ . The number of elements e with F π ( e ) > grows by only during γ -moves inwhich F π ( e ) was initially . Otherwise, this number never grows. We account for elements leavingthe solution by paying recourse upfront when they join the solution.Hence, the number of changes to the solution is at most: · P t g t ( N ) √ f min · √ f min ( √ γ −
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A Algorithm for r -bounded instances We can achieve a better approximation ratio if each function g is an r -junta for small r . Recall thatan r -junta is a function that depends on at most r variables. In this section we prove the theorem: Theorem A.1.
There is a randomized algorithm that maintains an r -competitive solution in ex-pectation to Fully Dynamic SubmodularCover in the setting where functions arrive/depart overtime, and these functions are each r -juntas. Furthermore this algorithm has total recourse: P t g t ( N ) f min . V g = { u ∈ N | g ( u ) = 0 } be the elements influencing g . Ourassumption says that | V g | ≤ r . Let S be the solution maintained by the algorithm. Each function g ∈ G ( t ) maintains a set U g ⊆ V g of elements assigned to it. We say that g is responsible for U g . Wealso define the following operation: Probing a function . Sample one element u ∈ V g \ S with probability: c ( u ) X v ∈ V g \ S c ( v ) . Add the sampled element u to the current solution S , and to U g .Given these definitions, we are ready to explain the dynamic algorithm. Function arrival . When a function g arrives, initialize its element set U g to ∅ . Then, while g ( S ) = g ( N ) , probe g t . Function departure . When a function g departs, remove all its assigned elements U g from S .This may leave some set of functions g , . . . , g s uncovered. For each of these functions g t in orderof arrival, while g t ( S ) = g ( N ) , probe g t .The P t g t ( N ) /f min recourse bound is immediate, since the total number of probes can be at most P t g t ( N ) /f min in total. It remains to bound the competitive ratio.We prove the following: Lemma A.2.
For any element u ∈ N : E X g ∈ G ( t ) : u ∈ V g X v ∈ U g c ( v ) ≤ r · c ( u ) . This will imply as a consequence: E [ c ( S )] ≤ X o ∈ Opt t E X g ∈ G ( t ) : o ∈ V g X v ∈ U g c ( v ) ≤ r · X o ∈ Opt c ( o ) = r · c ( Opt t ) . Proof of Lemma A.2.
The proof is by induction. Fix u ∈ N , and consider the functions g ∈ G ( t ) forwhich u ∈ V g . Let X i be the random variable that is the i th function probed. Let Y i be the randomvariable that is the element of N sampled during the i th probe. With this notation, the inductivehypothesis is: E X i ≥ j c ( Y i ) | X , Y , . . . X j − , Y j − ≤ r · c ( u ) . i = m , note that given X , Y , . . . , X m − , Y m − , the variable X m is determined.Suppose X m = g . Then: E [ c ( Y m ) | X , Y , . . . X m − , Y m − ] = E [ c ( Y m ) | X , Y , . . . X m − , Y m − , X m = g ]= X v ∈ V g c ( v ) (cid:16)P v ′ ∈ V g \ S t c ( v ′ ) (cid:17) · c ( v ) ≤ | V g | P v ′ ∈ V g \ S t c ( v ′ ) ≤ r · c ( u ) . For the inductive step, suppose the claim holds for j + 1 , and consider the case for j : E X i ≥ j c ( Y i ) | X , Y , . . . X j − , Y j − = E [ c ( Y j ) | X , Y , . . . X j − , Y j − ] + E X i ≥ j +1 c ( Y i ) | X , Y , . . . X j − , Y j − ≤ r P v ′ ∈ V g \ S t c ( v ′ ) + X u ′ = u E X i ≥ j +1 c ( Y i ) | X , . . . Y j − , Y j = u ′ · P (cid:0) Y j = u ′ | X , . . . , Y j − (cid:1) ≤ r P v ′ ∈ V g \ S t c ( v ′ ) + r · c ( u ) · − c ( u ) P v ′ ∈ V g \ S t c ( v ′ ) ! = r · c ( u ) . B Combiner Algorithm
We show that we can adapt the combiner algorithm of [GKKP17] to our general problem.
Theorem B.1.
Let A G be an O (log f ( N ) /f min ) -competitive algorithm for fully-dynamic Submod-ularCover with amortized recourse R G . Let A P D be an O ( r ) -competitive algorithm for fully-dynamic SubmodularCover when all functions are r -juntas with amortized recourse R P D . Thenthere is an algorithm achieving an approximation ratio of O (min(log( f ( N ) /f min ) , r ) for fully-dynamic SubmodularCover when all functions are r -juntas, and it has total recourse O ( R G + R P D ) .Proof. The idea is to partition the functions into different buckets based on their junta-arity, inpowers of up to log f ( N ) /f min . We run a copy of A P D which we call A ( ℓ ) P D on each bucket B ℓ separately, and run A G one single time on the set of remaining functions.Formally, for every index < ℓ < ⌈ log log( f ( N ) /f min ) ⌉ , maintain a bucket B ℓ representing the setof functions g such that g is a k -junta, for k ∈ [2 ℓ , ℓ +1 ) . Also maintain the bucket B G for anyremaining functions. When a functions arrives, we insert it into exactly one appropriate bucket andupdate the appropriate algorithm. Lemma B.2.
The total cost of the solution maintained by the algorithm is O (min(log f ( N ) , r ) .Proof. If r ≤ log( f ( N ) /f min ) , the algorithm never runs A G . Each algorithm A ( ℓ ) P D is O (2 ℓ +1 ) -competitive, and thus maintains a solution of cost no more than O (2 ℓ +1 ) c ( Opt ) . The largest25ucket index is ℓ max = ⌈ log r ⌉ . Hence the total cost of the solution is: ℓ max X ℓ =1 O (2 ℓ +1 ) · c ( Opt ) = O ( r ) · c ( Opt ) . If on the other hand, r > log( f ( N ) /f min ) , then the largest bucket index is ℓ max = ⌈ log log( f ( N ) /f min ) ⌉ .The total cost of the A P D algorithms is then ℓ max X ℓ =1 O (2 ℓ +1 ) · c ( Opt ) = O (log f ( N ) /f min ) · c ( Opt ) . Meanwhile, the total cost of the solution maintained by A G on the remaining functions has cost O (log f ( N ) /f min ) · c ( Opt ) . Thus the global solution maintained by the combiner algorithm is also O (log f ( N ) /f min ) -competitive.The recourse bound is immediate since each function g arrives to/departs from exactly one bucket,so at most one algorithm among { A ( ℓ ) P D } ℓ ∪ { A G } has to update its solution at every time step. C Further Applications
In this section, we show how to recover several known results on recourse bounded algorithmsusing our framework. We hope this is a step towards unifying the theory of low recourse dynamicalgorithms.
C.1 Online Metric Minimum Spanning Tree
In this problem, vertices in a metric space are added online to an active set. Let A t denote the activeset at time t . After every arrival, the algorithm must add/remove edges to maintain a spanningtree S t for A t that is competitive with the MinimumSpanningTree . We show:
Theorem C.1.
There is a deterministic algorithm for Online Metric
MinimumSpanningTree that achieves a competitive ratio of O (1) and an amortized recourse bound of O (log D ) , where D isthe ratio of the maximum to minimum distance in the metric. [GGK16] show how to get an O (1) worst case recourse bound. MinimumSpanningTree is a special case of
SubmodularCover in which N is the set of edgesof the graph, and f is the rank function of a graphic matroid. The main difference between thedynamic version of this problem and our setting is that here vertices arrive online along with alltheir incident edges . Hence not only is the submodular function changing, but N is also growing.We show that this detail can be handled easily.We define the submodular function f ( t ) to be the rank function for the current graphic matroid,i.e. f ( S ) = | A t | − c t , where c t is the number of connected components induced by S on the set ofvertices seen thus far. Note that f max = f min = 1 , so an edge having nonzero coverage is equivalentto the edge being in our current solution, S t .Now the algorithm is: 26 lgorithm 2 FullyDynamicMST π ← arbitrary initial permutation of edges. for t = 1 , , . . . , T do When vertex v t arrives, add edges incident to v t to tail of permutation in arbitrary order,and update f ( t ) . while there exists a legal γ -move or a swap for π do Perform the move, and update π . Output the collection of π i such that F π ( π i ) > .As in Corollary 2.3, if the algorithm terminates then it represents the stack trace of an approximategreedy algorithm for MinimumSpanningTree . Hence the solution is O (1) competitive. To boundthe recourse, we use the general potential Φ h from Section 3. As before, local moves decrease thepotential Φ h by ǫ γ · c min · h (cid:16) c min (cid:17) , so it suffices to show that the potential does not increase bytoo much when f ( t ) is updated. Exactly one new edge will have increased marginal coverage, andits coverage will increase from to . Thus the increase in potential is at most c max · h (1 /c max ) .Together, these imply an amortized recourse bound of: ǫ γ · c max c min · h (1 /c max ) h (1 /c min ) . Setting h ( x ) = x − δ / (1 − δ ) along with δ = (ln( c max /c min + 1)) − and γ = ǫ as in Theorem 3.1, wehave ǫ γ ≥ δ , and hence we get a recourse bound of O (ln( c max /c min )) = O (ln D ) . C.2 Fully-Dynamic Metric Minimum Steiner Tree
We show that we can also fit into our framework the harder problem of maintaining a tree thatspans a set of vertices in the fully-dynamic setting where vertices can both arrive and depart. Wemust produce a tree S t that spans the current set of active vertices A t , but we allow ourselves touse Steiner vertices that are not in the active set. We show: Theorem C.2.
There is a deterministic algorithm for Fully-Dynamic Metric
MinimumStein-erTree that achieves a competitive ratio of O (1) and an amortized recourse bound of O (log D ) ,where D is the ratio of the maximum to minimum distance in the metric. This guarantee matches that of [ŁOP + O (1) amortized recourse.Our algorithm is the same local search procedure as before, with one twist. We maintain a set ofvertices L we call the live set. This set is the union of the active terminals we need to span, andany Steiner vertices currently being used. We define f ( t ) similarly to before as f ( t ) ( S ) = | L | − c t ,where c t is the number of connected components induced by the edge set S on the set of verticesin L . Note that this function is submodular, because it is the rank function of the graphic matroidon the live vertex set L .Now when a vertex v departs, we mark it as a Steiner vertex but leave it in the live set. If at anypoint during the local search deg( v ) = 2 , we replace v with the edge that shortcuts between v ’s twoneighbors. If at any point point deg( v ) = 1 , we delete v and its neighboring edge.To show the competitive ratio we can rely on known results [IW91, GK14]. If Algorithm 3 terminates,the output tree is known as a γ -stable extension tree for the terminal set S .27 lgorithm 3 FullyDynamicSteinerTree π ← arbitrary initial permutation of edges. for t = 1 , , . . . , T do if vertex v t arrives then Add edges incident to v t to tail of permutation in arbitrary order, and update f ( t ) . else if vertex v t departs then Mark v t as a Steiner vertex. Run CleanSteinerVertices . while there exists a legal γ -move or a swap for π do Perform the move, and update π . Run
CleanSteinerVertices . Output the collection of π i such that F π ( π i ) > . procedure CleanSteinerVertices while there is a Steiner vertex v with deg( v ) = 2 do Let u and u be the neighbors of v . Add the edge ( u , u ) to the position of ( v, u ) in π . // this shortcuts v Delete all edges incident to v from N , remove v from the live set, and update f ( t ) . while there is a Steiner vertex v with deg( v ) = 1 do Delete all edges incident to v from N , remove v from the live set, and update f ( t ) . Lemma C.3 (Lemma 5 of [IW91]) . If T is a γ -stable extension tree for A t , then: c ( T ) ≤ γ · c ( Opt ( A t )) where Opt ( A t ) is the optimal Steiner tree for terminal set A t . Since we set γ = e , this gives us a competitive ratio of e = O (1) .It remains to show the recourse bound. Deleting degree and vertices requires a constant numberof edge changes, so this can be charged to each vertex’s departure. We show that the potentialargument from before is not hampered by the changes to the algorithm. Claim C.4.
The procedure
CleanSteinerVertices does not increase the potential.Proof.
When a degree Steiner vertex is deleted, the incident edge is removed from the permutationand no other edge’s marginal coverage changes.When a degree Steiner vertex is deleted, the edges ( v, u ) and ( v, u ) are replaced by the edge ( u , u ) . Recall that our choice of potential is: Φ h ( π ) = X e ∈ S t c ( e ) δ for < δ < . By triangle inequality, and concavity of h : d ( u , u ) δ ≤ ( d ( v, u ) + d ( v, u )) δ ≤ d ( v, u ) δ + d ( v, u ) δ . Thus this replacement only decreases the potential.Otherwise, the potential increases during vertex arrivals and decreases during γ -moves exactly asin Appendix C.1. We are left with the same recourse bound of O (ln D ) .28 Bounds using the Shannon entropy potential
We show that Shannon entropy also works as a potential, albeit with the weaker recourse bound of: O (cid:18) P t g t ( N ) f min ln (cid:18) c max c min · f max f min (cid:19)(cid:19) . Define the Shannon Entropy potential to be the expression: Φ ( f, π ) := X i ∈ N F π ( π i ) log c ( π i ) F π ( π i ) . In order to ensure that Φ remains nonnegative and monotone in each F π ( π i ) , scale c by /c min and f by / ( e · f max ) such that all costs are greater than and all coverages are less than /e . We willaccount for this scaling at the end.Note that Φ is Φ h from Section 3 with h ( x ) = x log(1 /x ) . This h satisfies properties (i) , (ii) and (iv) but not (iii) . Properties of Φ :(I) Φ increases by at most g t ( N ) · ln ( c max /f min ) with the addition of function g t to theactive set. (II) Φ does not increase with deletion of functions from the system. (III) Φ does not increase during swaps. (IV) If γ > e , then Φ decreases by at least f min ln( γ/e ) with every γ -move.The proofs that Φ satisfies properties (I) – (III) follows directly from Lemma 3.3, since these donot use property (iii) . It remains to show the last property. Lemma D.1. If γ > e , every γ -move decreases Φ by at least f min · ln( γ/ǫ ) .Proof. Suppose we perform a γ -move on a permutation π . Let u be the element moving to someposition p from some position q > p , and let π ′ denote the permutation after the move. Forconvenience, also define: v i := F π ( π i ) , (the original coverage of the i th set) a i := I f ( π i ; u | π i − ) = F π ( π i ) − F π ′ ( π i ) . (the loss in coverage of the i th set)Then: Φ ( f, π ′ ) − Φ ( f, π )= n X i =1 ( v i − a i ) ln c ( π i ) v i − a i + n X i =1 a i ln c ( u ) P ni =1 a i − n X i =1 v i ln c ( π i ) v i ≤ − n X i =1 a i ln (cid:18) c ( π i ) e · v i (cid:19) + n X i =1 a i ln c ( u ) P ni =1 a i (D.1)29 − n X i =1 a i ln (cid:18) γe · c ( u ) P i a i (cid:19) + n X i =1 a i ln c ( u ) P ni =1 a i (D.2) = − n X i =1 a i ln (cid:16) γe (cid:17) = − f min · ln (cid:16) γe (cid:17) . Step (D.1) follows because, by concavity of the function h ( x ) = x log x , we have h ( a + b ) − h ( a ) ≤ b · h ′ ( a ) . Step (D.2) follows because u moving to position p is a γ -move, hence P j a j /c ( u ) ≥ γ · v i /c ( π i ) .We now show the weaker recourse bound. By (I) , the potential Φ h increases by at most g t ( N ) · ln (cid:18) c max f min (cid:19) with the addition of function g t to the active set. By (IV) , it decreases by Ω( f min ) with everymove that costs recourse , and otherwise does not increase. Since we scaled costs by /c min andcoverages by / ( e · f max ) , this implies a recourse bound of: O (cid:18) P t g t ( N ) f min ln (cid:18) c max c min · f max f min (cid:19)(cid:19) ..