A Stronger Impossibility for Fully Online Matching
Alexander Eckl, Anja Kirschbaum, Marilena Leichter, Kevin Schewior
aa r X i v : . [ c s . D S ] F e b A Stronger Impossibility for Fully Online Matching ∗ Alexander Eckl , Anja Kirschbaum , Marilena Leichter , Kevin Schewior Advanced Optimization in a Networked Economy (AdONE), Technical University of Munich, Germany Department of Mathematics and Computer Science, University of Cologne, Germany
Abstract
We revisit the fully online matching model (Huang et al., J. ACM, 2020), an extension ofthe classic online matching model due to Karp, Vazirani, and Vazirani (STOC 1990), which hasrecently received a lot of attention (Huang et al., SODA 2019 and FOCS 2020), partly due toapplications in ride-sharing platforms. It has been shown that the fully online version is harderthan the classic version for which the achievable competitive ratio is at most 0 . − / e ≈ . . In the fully online matching model due to Huang et al. [1], the vertices of an unweighted graph G = ( V, E ) arrive online over time. Every vertex has a deadline at which it departs . At any time,only the subgraph of G induced by the vertices that have already arrived is known. An onlinealgorithm can (possibly randomly) match two adjacent vertices of the subgraph if both of themhave neither been matched nor departed.This model extends the classic model due to Karp, Vazirani, and Vazirani [3] in which G isbipartite, initially all vertices from one side arrive, and then the vertices from the other side arrive.The vertices from the former side only depart at the very end, and the vertices from the latterside depart even before the next vertex from that side arrives, i.e., they must be matched eitherupon arrival or never. The classic model and its variants have been extensively studied prior tothe work of Huang et al. [1] and have many applications, e.g., in online advertising [5]. However, ascenario not addressed by these models is, e.g., that of ride-sharing platforms in which customersand compatible drivers have to be matched, but both customers and drivers enter and leave thesystem at arbitrary times. This aspect is addressed by the fully online model.In both models, the performance of an online algorithm is measured through standard com-petitive analysis: A randomized online algorithm is called α -competitive if, for all instances, theexpected number of matches it performs is at least an α fraction of the number of matches anomniscient optimum could have performed. It is well known [3, 6] that for the classic model thecompetitive ratio achievable by randomized algorithms is 1 − / e ≈ . o (1) terms as | V | → ∞ , it can be obtained through the Ranking algorithm. The same holds true for the fractional relaxation of the model in which a (w.l.o.g. deterministic) algorithm may fractionally match vertices,thus obtaining a fractional matching [4]. In their seminal work on fully online matching, Huang et ∗ Funding: This work was supported in part by the Alexander von Humboldt Foundation with funds from theGerman Federal Ministry of Education and Research (BMBF) and by the Deutsche Forschungsgemeinschaft (DFG),GRK 2201.
Email addresses: [email protected] (Alexander Eckl), [email protected] (Anja Kirschbaum), [email protected] (Marilena Leichter), [email protected] (Kevin Schewior) . . . this work bip. frac. UB0 . . . Figure 1: Known bounds on the best-possible competitive ratio α for fully online matching. Wedistinguish between bipartite and general graphs as well as fractional and integral algorithms.al. [1] prove that a generalization of the Ranking algorithm to general graphs is 0 . .
5, and that no fractional algorithm can achieve a guarantee betterthan 0 . − / e .In follow-up work, Huang et al. [7, 2] revisit the fractional and bipartite cases. The state ofthe art is a 0 . . . In this work, we give an improvement of the impossibility from 0 . . λ + 1, and exactly one of its children is a leaf. Which of the childrenthe designated leaf is, however, remains unknown until the root has departed. While the optimumcan match the leaf vertex to the root, the online algorithm cannot do better than matching thefirst-level vertices to the root with identical fractional value, leaving some fraction of the leaf vertexpermanently unmatched. The construction is then repeated with the λ non-leaf children playing therole of the root, for a total of h levels. Finally, level h of the tree is augmented with the later-arrivingside of the “triangle” construction of Karp, Vazirani, and Vazirani [3]. It can be shown that theextra tree puts the algorithm in a worse position and that choosing λ = 7 and h → ∞ yields thebound of 0 . k vertices (corresponding to the root) on the first level and( λ + 1) k vertices on the second level, connected by a biclique. After the vertices on the previouslevel depart, it is revealed which of the λk vertices form a biclique with ( λ + 1) λk new vertices onthe next level. Again, this process is repeated for a total of h steps and the “triangle” construction2s added at the end. Note that now we are also allowed to choose any rational value for λ as longas we make k large enough. In fact, choosing λ ≈ . . · − .The second idea is replacing λ with different factors γ , . . . , γ ℓ for ℓ levels between the first h levels and the “triangle” construction. After taking the limit h → ∞ , we express the preciseresulting competitive ratio as a function of λ, γ , . . . , γ ℓ . Unfortunately, this function is alreadynon-convex for ℓ = 0, and the explicit formula gets quite complicated even for small values of ℓ .We therefore have to resort to numerical optimization. Specifically, we obtain the bound of 0 . . In Section 2, we formally describe the graph used in our construction with general parameters.In Section 3, we derive a closed formula for the impossibility for ℓ = 3 and optimize this boundnumerically. In this section, we formally describe the construction that is used to show the improved impossibility.The construction is parameterized by integers h, ℓ ≥ k ≥ λ > γ j > j ∈ [ ℓ ] := { , . . . , ℓ } . We first define the bipartite graph G underlying theconstruction and then specify the arrival and departure times of the vertices.The vertex set of G can be partitioned into 2 · ( h + ℓ +1) disjoint sets U i , V i for all i ∈ { , . . . , h + ℓ } .We have | U i | = | V i | for each such i . The set U consists of k vertices. With increasing index, thenumber of vertices increases for the first h steps by a factor of λ and in the j -th subsequent step bya factor of γ j . Formally, | U i | = λ i k ∀ i ∈ { , . . . , h } , | U h + j | = γ j | U h + j − | ∀ j ∈ { , . . . , ℓ } . We note that by the continuous nature of the variables γ j and λ , some of the sizes | U i | = | V i | might take non-integer values. We avoid this by choosing the parameter k in dependence of h largeenough such that all of these numbers are integer. We introduce edges such that ( U i , V i ∪ U i +1 ) isa complete bipartite graph for all i ∈ { , . . . , h + ℓ − } . The edges between U h + ℓ and V h + ℓ form anexception. Due to this exception, we define A := U h + ℓ and B := V h + ℓ . In Section 3, we describe howthe adversary chooses an ordering of A = { a , . . . , a | A | } and B = { b , . . . , b | B | } . Then a i and b j areconnected by an edge if and only if i ≥ j . The subgraph of G induced by A ∪ B has been previouslyused to show impossibilities for online matching problems [3, 4]. This completes the description of G . To complete the description of our online construction, we specify the arrival and departuretimes of the vertices in the graph G . At the beginning, all vertices in U arrive simultaneously.Let i ∈ { , . . . , h + ℓ − } and assume all vertices in U j for j ≤ i and in V j for j ≤ i − U i have departed again. Then, all vertices in U i +1 and V i (by definition, along with their edges to U i ) arrive simultaneously. Next, the vertices in U i depart simultaneously. At this point, it is impossible for an algorithm to differentiate betweenthe neighbors of U i , in particular identifying which of them belong to U i +1 and which to V i . InSection 3, we describe how the adversary can assign these vertices to U i +1 and V i based on the3 V U V U h + ℓ − V h + ℓ − U h + ℓ = A V h + ℓ = B Figure 2: Visualization of the bipartite graph G .matching decisions of the algorithm. If i ≤ h + ℓ −
2, the initial assumption is reestablished for thenext-larger index, and the arrivals and departures happen as described. Otherwise, A = U h + ℓ hasnow arrived, and all other previously arrived vertices have departed. Then the vertices in B arrivein order of b , . . . , b | B | , and every vertices departs again immediately after its arrival. Finally, thevertices A depart simultaneously. Note that all edges of the graph respect release and departuretimes in the sense that, for all of them, there is a time when both their endpoints have arrived butnot departed.The graph has a perfect matching which is the union of the following perfect matchings: Fromthe complete bipartite subgraphs induced by U i ∪ V i we choose an arbitrary perfect matching. For A and B we choose the perfect matching containing ( a i , b i ) for all i ∈ [ | A | ]. We also remark that U := S h + ℓi =0 U i and V := S h + ℓi =0 V i , however, do not correspond to the two sides of the bipartite graph. Using the construction from the previous section, we show our result in this section.
Theorem 3.1.
The competitive ratio of any fractional algorithm for fully online matching is atmost . , even on bipartite graphs. To prove this result, we have to bound both the value of the offline optimum and that of anyonline algorithm. Clearly, since G has a perfect matching and its vertex set is U ∪ V where | U | = | V | ,we have OPT = | U | = | V | . Now let ALG be any fractional online algorithm. For all vertices in our graph, we determine thefraction to which they are matched. For simplicity, we make two assumptions that are without lossof generality. First, we assume that the algorithm only performs matches along an edge wheneverone of its endpoints departs. Furthermore, when a vertex departs, we assume that the algorithmfully matches it unless all of its neighbors are fully matched. It can be seen by a simple exchangeargument that the latter assumption is indeed without loss of generality (see also [1]).Note that we only need to consider the matching value distributed at the departure times ofvertices in U ∪ · · · ∪ U h + ℓ − ∪ B . We do so in order of departure of the respective vertices, so we4tart with U i − for i ∈ { , . . . , h + ℓ } . Recall that, when these vertices reach their deadlines, their notdeparted neighbors N + ( U i − ) := U i ∪ V i − are indistinguishable for the algorithm. For simplicity, wewould like to analyze the water-filling algorithm [4] ALG wf which also treats these vertices equally,i.e., matches them all to the same fraction. It may, however, conceivably help the algorithm tomatch vertices to different fractions . In the following, we first show that this is (essentially) not thecase.We will define an adversary that assigns the vertices in N + ( U i − ) to U i and V i − (respecting | U i − | = | V i − | ) based on the matching decisions of the algorithm. A first approach may be to justassign those vertices to V i − that have been matched the least. We will, however, later need a lower and an upper bound on the value to which U i has been matched. The latter approach only gives usthe lower bound. We define p i − such that the total matching value of the vertices in U i − just afterthe departure of U i − is p i − · | U i − | . In other words, p i − is the average matching value across all(again, conceivably different) matching values of vertices in U i − at that point. We define p = 0.Note that, upon departure of U i − , ALG wf would assign a total fractional matching value of d := (1 − p i − ) · | U i − | | U i || N + ( U i − ) | (1)to any set of | U i | vertices in N + ( U i − ). Our adversary simply chooses an assignment of N + ( U i − )to U i and V i − such that p i · | U i | is as close as possible to d . The following lemma shows that theerror becomes vanishingly small. Lemma 3.2.
Let i ∈ [ h + ℓ ] and define n i := | U i | . For any given distribution of matching value to N + ( U i − ) by the algorithm, there is an adversary that chooses sets U i and V i − such that p i = 1 − p i − λ + 1 + ε ( i ) , for i ≤ h and p i = 1 − p i − γ j + 1 + ε ( h + j ) , for i = h + j, j ∈ [ ℓ ] , where the error term ε ( i ) fulfills | ε ( i ) | ≤ n i . Proof.
For now, let i ∈ [ h ] and let a distribution of the matching value from U i − to N + ( U i − ) begiven. Let m ( U i ) and m ( V i − ) be the total matching value received by U i and V i − , respectively.The adversary decides on a partition U i ˙ ∪ V i − of N + ( U i − ). We let the adversary partition thevertices such that m ( U i ) is as close as possible to d as in Equation (1). Note that here d = (1 − p i − ) | U i − | | U i || N ( U i − ) | = | U i | − p i − λ + 1 . Note that this value does not depend on the explicit partition since | U i | has a prescribed size whichis not chosen by the adversary. It is clear that d corresponds to the total matching value assignedto U i if all vertices in N + ( U i − ) have received the same amount of matching value. It turns outthat, independently of the assignment by the algorithm, it is always possible to assign sets U i and V i − with the correct cardinalities such that d − ≤ m ( U i ) ≤ d + 1 . (2)To verify this claim, assume for contradiction that ˆ U i is the set with minimal distance (cid:12)(cid:12)(cid:12) m ( ˆ U i ) − d (cid:12)(cid:12)(cid:12) > . Additionally, let ˆ V i − = N + ( U i − ) \ ˆ U i be the complementary set in the par-tition. Firstly, assume that m ( ˆ U i ) is larger than d . Let u ∈ ˆ U i be the element of ˆ U i with maximum5atching value and let v ∈ ˆ V i − be the element in ˆ V i − with minimum matching value. It musthold that 1 ≥ m ( u ) > m ( v ) ≥
0, otherwise the matching m ( ˆ U i ) cannot exceed the weighted average d . This implies 0 < m ( u ) − m ( v ) ≤
1. Hence when we exchange u and v in the sets ˆ U i and ˆ V i − , wehave m ( ˆ U i \{ u }∪{ v } ) − d = m ( ˆ U i ) − m ( u )+ m ( v ) − d and 0 ≤ m ( ˆ U i ) − d − ( m ( u ) − m ( v )) < m ( ˆ U i ) − d, so (cid:12)(cid:12)(cid:12) m ( ˆ U i \ { u } ∪ { v } ) − d (cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12) m ( ˆ U i ) − d (cid:12)(cid:12)(cid:12) , a contradiction to the minimality of the distance.Secondly, when m ( ˆ U i ) is smaller than d , let u ∈ ˆ U i be the element of ˆ U i with minimum matchingvalue and let v ∈ ˆ V i − be the element in ˆ V i − with maximum matching value. It holds 0 ≤ m ( u ) 1, and hence we can again switch u and v to receive (cid:12)(cid:12)(cid:12) m ( ˆ U i \ { u } ∪ { v } ) − d (cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12) m ( ˆ U i ) − d (cid:12)(cid:12)(cid:12) .Therefore, we have proven Equation (2) by contradiction. We divide it by | U i | to receive1 − p i − λ + 1 − | U i | ≤ m ( U i ) | U i | ≤ − p i − λ + 1 + 1 | U i | . By the definition of p i as m ( U i ) / | U i | , we finally have p i = 1 − p i − λ + 1 + ε ( i )for some error term with | ε ( i ) | ≤ n i .We repeat the above arguments for j ∈ [ ℓ ] with the corresponding growth parameters to obtain p h + j = 1 − p h + j − γ j + 1 + ε ( h + j )with | ε ( h + j ) | ≤ n h + j . Via induction, we obtain a closed-form description of p i (see appendix): p i = 1 λ + 2 − (cid:18) − λ + 1 (cid:19) i ! + ε ( i ) , ∀ i ∈ [ h ] , (3)with | ε ( i ) | ≤ in i . Computing explicit formulas for p j , j > h is quite cumbersome. We detail thealgebraic transformations for ℓ = 3 in the appendix.We are interested in the matching value the algorithm gives to vertices in V . Let us define q i as the total matched fraction of the vertices in V i for all i ∈ { , . . . , h + ℓ − } . Note that B = V h + ℓ is matched differently, something we will consider at a later point. Lemma 3.3. Let i ∈ { , . . . , h + ℓ − } . Then it holds that q i = p i +1 + ε ( i ) , where p i +1 := p i +1 − ε ( i +1) and | ε ( i ) | ≤ i +3 n i .Proof. Again, we only prove the cases i ∈ { , . . . , h − } ; the cases of larger i are analogous withadapted growth parameters. Let a distribution of the matching value from U i to their neighbors begiven. Since the total matching value in U i is (1 − p i ) n i , it holds that(1 − p i ) n i = p i +1 n i +1 + q i n i . q i n i = (1 − p i ) n i − p i +1 n i +1 = (1 − p i ) n i − (cid:18) − p i λ + 1 + ε ( i +1) (cid:19) λn i = n i (cid:18) − p i λ + 1 − λε ( i +1) (cid:19) . With n i = | U i | = | V i | we can simplify to q i = − p i λ +1 − λε ( i +1) . Inserting Lemma 3.2 gives q i = p i +1 − ( λ + 1) ε ( i +1)= ( p i +1 − ε ( i +1)) + ( ε ( i +1) − ( λ + 1) ε ( i +1))= p i +1 + ε ( i ) , where it is easy to see that | ε ( i ) | ≤ i +3 n i .In other words, we express the q i in terms of p i +1 without the error terms ε ( i +1) for all i ∈ { , . . . , h + ℓ − } . We can write the amount of matching value the vertex sets V i (except B )contribute to the algorithmic value as h + ℓ − X i =0 q i | V i | . Finally, we consider the vertices in A . Recall that their total matching value is p h + ℓ · | A | beforeany vertex in B has arrived. For simplicity, we define p A := p h + ℓ . We also define ρ such that theadditional matching value that A receives (that is, at the departure times of vertices in B ) is ρ · | A | .In the following, we will give an upper bound on ρ .Again, an analysis of ALG wf would be simpler but some carefulness is required because thematching value p A · | A | is not necessarily distributed uniformly across A . Fortunately, a simpleadversary for choosing the ordering of A and B suffices here. Lemma 3.4. The adversary can choose the ordering of A and B such that ρ ≤ − exp( − (1 − p A )) + 2 | A | . Proof. As described in Section 2, the vertices in B arrive, and immediately depart again, sequen-tially. The vertices are labeled b , . . . , b | B | in this order, and any vertex b i is adjacent to the vertices a i , . . . , a | A | . Note that the algorithm only learns about the identity of vertex a i after the departureof b i . Here, the adversary simply chooses a i in every round such that it has the minimum currentmatching value out of all remaining unlabeled vertices in A .We now bound ρ · | A | , the fractional matching value placed by the algorithm on edges between A and B , by the matching value ω · | A | placed by ALG wf on the same instance: Assuming thatthe matching value in A is equally distributed before B arrives, i.e., all vertices have exactly p A matching value, this algorithm simply matches b i equally among all vertices a i , . . . , a | A | . We claimthat ρ ≤ ω . In the following, we denote the matching value placed on a ∈ A by our algorithm andALG wf at the time of the departure of b i by m i ( a ) and ω i ( a ), respectively. We omit the index i ifwe refer to the final value.Let η ∈ [ | B | ] the final index for which ALG wf is able to assign matching value to A . Note thatevery time i appears before b η , ω i ( A ) ≥ m i ( A ), as ALG wf always matches the current vertex b i fully7able 1: Results of the numerical optimization. All numbers are rounded to the sixth decimal digit. ℓ λ · · · γ ℓ − γ ℓ − γ ℓ impossibility0 7.233629 · · · – – – 0.6317441 2.58117 · · · – – 8.0532 0.6297482 3.14832 · · · – 2.39011 7.8746 0.6296783 2.87586 · · · ... ... . . . ... ... ... ... 10 2.94419 · · · i ≥ η it holds that 1 = ω ( a i ) ≥ m ( a i ), so ω ( { a i : i ≥ η } ) ≥ m ( { a i : i ≥ η } ).Now let i ′ < η be the maximal index i so that ω ( a i ) < m ( a i ). By the adversary’s choice of a i ′ , itholds that m i ′ ( { a i : i > i ′ } ) ≥ m ( a i ′ ) |{ a i : i > i ′ }| > ω ( a i ′ ) |{ a i : i > i ′ }| = ω i ′ ( { a i : i > i ′ } ) . Using ω ( { a i : i ≤ i ′ } ) + ω i ′ ( { a i : i > i ′ } ) = ω i ′ ( A ) ≥ m i ′ ( A ) = m ( { a i : i ≤ i ′ } ) + m i ′ ( { a i : i > i ′ } ) , we have ω ( { a i : i ≤ i ′ } ) ≥ m ( { a i : i ≤ i ′ } ) and as i ′ < η is the last vertex with ω ( a i ) < m ( a i ), weknow that for all remaining vertices ALG wf assigns more matching value. Therefore, ω ( { a i : i ′ η no value is assigned. So the average fractional matching value ω fulfills ω | A | ≤ η , whichholds with equality exactly if the last active vertex b η is fully matched.Let us again refer to the matching value placed by ALG wf on a vertex a i by ω ( a i ). After b η hasdeparted, all remaining vertices a η , . . . , a | A | are already fully matched. Hence, for all i ≥ η , we have ω ( a i ) = 1. At the same time, ω ( a i ) , i ≥ η is composed of the initial value p A plus the entire valueassigned by the vertices b , . . . , b η . For all i ≥ η we can obtain:1 = ω ( a i ) ≥ p A + η − X j =1 | A | − j + 1) = p A + H | A | − H | A |− η +1 , where H n is the n -th harmonic number which we estimate by ln( n ) ≤ H n ≤ ln( n + 1). Hence, wehave 1 ≥ p A + ln( | A | ) − ln( | A | − η + 2) ≥ p A + ln (cid:18) | A || A | (1 − ω ) + 2 (cid:19) , where we used ω | A | ≤ η . We continue rearranging:1 − p A ≥ ln (cid:18) | A || A | (1 − ω ) + 2 (cid:19) = ⇒ exp( − (1 − p A )) ≤ − exp( − (1 − p A )) + 2 | A | . This completes the proof. 8e have now computed all necessary values for our formula. We double-count the matchingvalue by counting the fractional value to which each vertex is matched and establish for the entirevalue of ALG: 2 ALG = h +2 X i =0 q i | V i | + | U \ A | + p A | A | + 2 ρ | A | . For ℓ = 0, by inserting our formulas into ALG / OPT and taking the limit h → ∞ , we obtain incongruence with Huang et al. [1] λ − λ · (cid:18) − exp (cid:18) − λ + 1 λ + 2 (cid:19)(cid:19) + λ + 1 λ · ( λ + 2)as an upper bound on the competitive ratio. Interestingly, this function is non-convex as its deriva-tive has a local maximum at λ ≈ . q j when j > h is quite cumbersome, and the same holds forthe explicit formula of the resulting lower bound on the competitive ratio. We showcase the resultof this computation for ℓ = 3. We refer to the appendix for all details on the computation. Theerror terms in the formula vanish as we take the limit h → ∞ . The final formula for our upperbound on the competitive ratio depends only on γ , γ , γ and λ : λ + γ ( γ + 1)( λ − λ + ¯ γ ( λ − λ + γ λ + 2)( γ + 1)( λ + ¯ γ ( λ − γ ( λ − λ + ¯ γ ( λ − · γ ( λ + 2) + 1( γ + 1)( γ + 1)( λ + 2)+ γ γ ( λ − λ + ¯ γ ( λ − · γ ( γ + 1)( λ + 2) + ( λ + 1)( γ + 1)( γ + 1)( λ + 2)+ γ γ γ ( λ − λ + ¯ γ ( λ − · " − exp (cid:18) − γ + 1 (cid:18) γ + γ ( λ + 2) + 1( γ + 1)( γ + 1)( λ + 2) (cid:19)(cid:19) , using the abbreviation ¯ γ := P i =1 Q ij =1 γ j . We use the numerical computing software Matlabfor the numerical optimization. Using a trusted-region algorithm for unconstrained multivariateminimization we receive the values shown in Table 1. In particular, we obtain an upper boundsmaller than 0 . References [1] Zhiyi Huang, Ning Kang, Zhihao Gavin Tang, Xiaowei Wu, Yuhao Zhang, and Xue Zhu. Fullyonline matching. J. ACM , 67(3):17:1–17:25, 2020.[2] Zhiyi Huang, Zhihao Gavin Tang, Xiaowei Wu, and Yuhao Zhang. Fully online matching II:Beating ranking and water-filling. In IEEE Symposium on Foundations of Computer Science(FOCS) , pages 1380–1391, 2020.[3] R. M. Karp, U. V. Vazirani, and V. V. Vazirani. An optimal algorithm for on-line bipartitematching. In ACM Symposium on Theory of Computing (STOC) , pages 352–358, 1990.[4] Bala Kalyanasundaram and Kirk Pruhs. An optimal deterministic algorithm for online b-matching. Theor. Comput. Sci. , 233(1-2):319–325, 2000.95] Aranyak Mehta. Online matching and ad allocation. Foundations and Trends ® in TheoreticalComputer Science , 8(4):265–368, 2013.[6] Benjamin E. Birnbaum and Claire Mathieu. On-line bipartite matching made simple. SIGACTNews , 39(1):80–87, 2008.[7] Zhiyi Huang, Binghui Peng, Zhihao Gavin Tang, Runzhou Tao, Xiaowei Wu, and Yuhao Zhang.Tight competitive ratios of classic matching algorithms in the fully online model. In ACM-SIAMSymposium on Discrete Algorithms (SODA) , pages 2875–2886, 2019. A Computing closed forms for p i We use induction to obtain the closed form p i = 1 λ + 2 − (cid:18) − λ + 1 (cid:19) i ! + ε ( i ) , ∀ i ∈ [ h ] , with an error term fulfilling | ε ( i ) | ≤ in i . Since we set p = 0 = λ +2 (cid:0) − ( − / λ +1 ) (cid:1) , the inductionbase holds. For the induction hypothesis we use Lemma 3.2. p i = 1 − p i − λ + 1 + ε ( i )= 1 λ + 1 − λ + 1 λ + 2 − (cid:18) − λ + 1 (cid:19) i − ! + ε ( i − ! + ε ( i )= 1 λ + 1 − λ + 2 λ + 1 + (cid:18) − λ + 1 (cid:19) i ! + ε ( i − λ + 1 + ε ( i )= 1 λ + 1 − λ + 2)( λ + 1) − λ + 2 (cid:18) − λ + 1 (cid:19) i + ε ( i − λ + 1 + ε ( i ) | {z } =: ε ( i ) = 1 λ + 2 − (cid:18) − λ + 1 (cid:19) i ! + ε ( i ) , with | ε ( i ) | ≤ | ε ( i − | λ + 1 + | ε ( i ) | ≤ i − λn i − + 1 n i = in i , where we used n i = λn i − . 10nalogously, applying Lemma 3.2 to the closed-form of p h , we obtain p h +1 = 1 γ + 1 λ + 1 λ + 2 − (cid:18) − λ + 1 (cid:19) h +1 !! + ε ( h +1) , (4) p h +2 = 1( γ + 1)( γ + 1)( λ + 2) γ ( λ + 2) + 1 − (cid:18) − λ + 1 (cid:19) h ! + ε ( h +2) , (5) p h +3 = ( γ γ + γ + 1)( λ + 2) − (cid:16) − λ +1 (cid:17) h ( γ + 1)( γ + 1)( γ + 1)( λ + 2)+ ε ( h +3) . (6) B Detailed algebraic computations In this section, we present the detailed computations to receive our upper bound formula which isoptimized at the end of Section 3. We recall the value of the optimal solution of our instance:2 OPT = | U | + | V | = 2 | U | = 2 h +3 X i =0 | U i | = 2 k (cid:18) λ h +1 − λ − (cid:19) + kλ h ( γ + γ γ + γ γ γ ) | {z } =:¯ γ ! = 2 kλ h λ − λ − / λ h + ¯ γ ( λ − . (7)The algorithmic solution value is given by2 ALG = h +2 X i =0 q i | V i | + | U \ A | + p A | A | + 2 ρ | A | = 2 ρ | A | + h +2 X i =0 | U i | + h − X i =0 q i | V i | + q h | V h | + q h +1 | V h +1 | + q h +2 | V h +2 | + p A | A | . We insert the values of p i , q i and the cardinalities n i :2 ALG = 2 ρ | A | (8)+ kλ h λ − λ − / λ h + γ ( γ + 1)( λ − h − X i =0 n i ( p i +1 − ε ( i + 1)) + h − X i =0 n i ε ( i ) (10)+ n h ( p h +1 − ε ( h +1)) + n h ε ( h ) (11)+ n h +1 ( p h +2 − ε ( h +2)) + n h +1 ε ( h +1) (12)+ n h +2 ( p A − ε ( h +3)) + n h +2 ε ( h +2) (13)+ n A ( p A − ε ( h +3)) + n A ε ( h +3) . (14)11y equations (3), (4)-(6), the terms p i − ε ( i ) do not contain any error terms (they are directlysubtracted). Hence all remaining error terms in the above formula sum up to: ε := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h +2 X i =0 n i ε ( i ) + n A ε ( h +3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ h +2 X i =0 ( i +3) + ( h +3) ≤ ( h +5) + h +52 + h +3 ∈ Ω( h ) . As 2 OPT is dominated by λ h and λ > ε vanishes for h → ∞ . We insert (4)-(6) into lines(11)-(14). Then, we divide all lines (8)-(14) without error terms by 2 OPT and take limits h → ∞ :(9)2 OPT = kλ h λ − λ − kλ h ( λ − / λ h + γ ( γ + 1)( λ − λ − / λ h + ¯ γ ( λ − → ( λ + γ ( γ + 1)( λ − λ + ¯ γ ( λ − , (10)2 OPT = kλ h λ + 2 λ − kλ h (cid:16) − / λh λ − − ( − / λ +1 ) h − / λh λ +1 (cid:17) λ − / λ h + ¯ γ ( λ − → λ − λ + 2 (cid:16) λ − (cid:17) λ + ¯ γ ( λ − λ + 2) ( λ + ¯ γ ( λ − , (11)2 OPT = ( λ + 1) (cid:0) − ( − / λ +1 ) h +1 (cid:1) ( λ + 2)( γ + 1) · λ h k ( λ − λ h k ( λ − / λ h + ¯ γ ( λ − → ( λ + 1)( λ − λ + 2)( γ + 1) λ + ¯ γ ( λ − 1) = part of (16) , (12)2 OPT = γ ( λ + 2) + 1 − ( − / λ +1 ) h ( γ + 1)( γ + 1)( λ + 2) · γ λ h k ( λ − λ h k ( λ − / λ h + ¯ γ ( λ − → γ ( λ + 2) + 1( γ + 1)( γ + 1)( λ + 2) γ ( λ − λ + ¯ γ ( λ − , (13) + (14)2 OPT = ( λ − λ h kγ γ + λ h kγ γ γ )2 λ h k ( λ − / λ h + ¯ γ ( λ − · ( γ γ + γ + 1)( λ + 2) − − / λ +1 ) h ( γ + 1)( γ + 1)( γ + 1)( λ + 1)= ( λ − γ γ λ − / λ h + ¯ γ ( λ − · γ ( γ + 1)( λ + 2) + ( λ + 1) + ( − / λ +1 ) h ( γ + 1)( γ + 1)( λ + 1) → γ γ ( λ − λ + ¯ γ ( λ − · γ ( γ + 1)( λ + 2) + ( λ + 1)( γ + 1)( γ + 1)( λ + 2) = (18) , ≤ | A | (1 − exp( − (1 − p A )) + / | A | )OPT= 2OPT + kλ h γ γ γ ( λ − kλ h ( λ − / λ h + ¯ γ ( λ − · " − exp − γ ( γ + 1)( λ + 2) + ( λ + 1) + ( − / λ +1 ) h ( γ + 1)( γ + 1)( γ + 1)( λ + 2) + ε ( h + 3) ! → γ γ γ ( λ − λ + ¯ γ ( λ − · (cid:20) − exp (cid:18) − γ ( γ + 1)( λ + 2) + ( λ + 1)( γ + 1)( γ + 1)( γ + 1)( λ + 2) (cid:19)(cid:21) = (19) . As h → ∞ , our upper bound on the competitive ratio of ALG therefore approaches λ + γ ( γ + 1)( λ − λ + ¯ γ ( λ − λ + γ λ + 2)( γ + 1)( λ + ¯ γ ( λ − γ ( λ − λ + ¯ γ ( λ − · γ ( λ + 2) + 1( γ + 1)( γ + 1)( λ + 2) (17)+ γ γ ( λ − λ + ¯ γ ( λ − · γ ( γ + 1)( λ + 2) + ( λ + 1)( γ + 1)( γ + 1)( λ + 2) (18)+ γ γ γ ( λ − λ + ¯ γ ( λ − · " − exp − γ + 1 · (cid:18) γ + γ ( λ + 2) + 1( γ + 1)( γ + 1)( λ + 2) (cid:19) ! ..