We present a simple proof that the competitive ratio of any randomized online matching algorithm for the line exceeds ?{log?(n +1)}/15 for all n = 2?-1: i ? ?, settling a 25-year-old open question

Peserico, Enoch

Scquizzato, Michele

English

We present a simple proof that the competitive ratio of any randomized online matching algorithm for the line exceeds √log2(n+1)/15 for all n = 2i-1 : i ∈ N, settling a 25-year-old open question

Peserico E.

Scquizzato M.

Archivio istituzionale della ricerca - Università di Padova

Matching on the Line Admits Noo(√log n)-Competitive AlgorithmEnoch PesericoUniversità degli Studi di Padova, ItalyMichele Scquizzato #Università degli Studi di Padova, ItalyAbstractWe present a simple proof that the competitive ratio of any randomized online matching algorithmfor the line exceeds√log2(n+1)/15 for all n = 2i−1 : i ∈ N, settling a 25-year-old open question.2012 ACM Subject Classification Theory of computation → Online algorithmsKeywords and phrases Metric matching, online algorithms, competitive analysisDigital Object Identifier 10.4230/LIPIcs.ICALP.2021.103Category Track A: Algorithms, Complexity and GamesRelated Version Extended Version: https://arxiv.org/abs/2012.15593Funding Michele Scquizzato: supported, in part, by Univ. Padova grant BIRD197859/19.Acknowledgements We are indebted to Kirk Pruhs and the anonymous reviewers for their con-structive criticism and insightful observations.1 Online matching, on the lineIn online metric matching [7, 9] n points of a metric space are designated as servers. One byone n requests arrive at arbitrary points of the space; upon arrival each must be matchedto a yet unmatched server, at a cost equal to their distance. Matchings should minimizethe ratio between the total cost and the offline cost attainable if all requests were knownbeforehand. A matching algorithm is c(n)-competitive if it keeps this ratio no higher thanc(n) for all possible placements of servers and requests.It is widely acknowledged [1, 10, 14] that the line is the most interesting metric space forthe problem. Matching on the line models many scenarios, like a shop that must rent tocustomers skis of approximately their height, where a stream of requests must be servicedwith minimally mismatched items from a known store. Despite matching being specificallystudied on the line since at least 1996 [8], no tight competitiveness bounds are known.As for upper bounds, the line is a doubling space and thus admits an O(log n)-competitiverandomized algorithm [5]; a sequence of recent developments [1, 12, 13] yielded the sameratio without randomization. Better bounds have been obtained only by algorithms withadditional power, such as that to re-assign past requests [6, 11] or predict future ones [2].As for lower bounds, the competitive ratio is at least 4.591 for randomized algorithmsand 9 for deterministic ones since the cow-path problem is a special case of matching onthe line [8]. These bounds were conjectured tight [8] until a complex adversarial strategyyielded a lower bound of 9.001 for deterministic algorithms [4]. Beyond some Ω(log n) boundsfor restricted classes of algorithms [3, 10, 12], there has been no further progress on thelower-bound side before this work.EATCS© Enoch Peserico and Michele Scquizzato;licensed under Creative Commons License CC-BY 4.048th International Colloquium on Automata, Languages, and Programming (ICALP 2021).Editors: Nikhil Bansal, Emanuela Merelli, and James Worrell; Article No. 103; pp. 103:1–103:3Leibniz International Proceedings in InformaticsSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany103:2 Matching on the Line Admits No o(√log n)-Competitive Algorithm2 An Ω(√log n)-competitiveness boundWe prove a simple Ω(√log n) lower bound on the competitive ratio of randomized onlinematching algorithms for the line.For any n = 2i −1 with i ∈ N consider the [0, n+1] interval; for each positive integer j ≤ nplace a server at point j, and place n requests over log2(n+1) rounds as follows. On the rthround (for 1 ≤ r ≤ log2(n+1)) partition the interval into (n+1)/2r subintervals of length 2r,choose within each uniformly and independently at random an origin point, and place arequest on the closest integer multiple of 2−n breaking ties arbitrarily. “Discretizing” requestsinstead of directly using the corresponding origins prevents some technical difficulties – seeour remark at the end.We prove in Lemma 1 that the expected distance between the ℓth leftmost server and theℓth leftmost origin is O(√log n), so servers and requests can be matched with an expectedoffline cost O(n√log n). Conversely, we prove in Lemma 2 that any online matching algorithmALG incurs an expected Ω(n) cost in any given round, for a total cost Ω(n log n). The tworesults can be combined to prove that on some request sequence ALG incurs Ω(√log n) timesthe offline cost.▶ Lemma 1. The expected distance between the ℓth leftmost origin and the ℓth leftmost serveris at most√log2(n+1) + 3.Proof. Let Sℓ be ℓth leftmost server and gℓ be the number of origins to its left. Note thatif gℓ equals respectively ℓ or ℓ − 1, the ℓth origin is the first immediately to the left, or tothe right of Sℓ; and since the first round placed one origin in every subinterval of size 2,such an origin is within distance 3 of Sℓ. By the same token, denoting by δℓ the quantity|gℓ − (ℓ − ℓn+1 )|, the ℓth leftmost origin is within distance 2δℓ + 3 of Sℓ. Note that δℓ isthe absolute deviation from the mean of rℓ, since rℓ is the sum of n independent indicatorrandom variables each denoting whether a given origin was placed to the left of Sℓ, withtotal expectation nn+1 ℓ = ℓ −ℓn+1 (by construction, the expected density of origins is constantthroughout the main interval). At most one such variable in a given round has variancegreater than 0, albeit obviously at most 1/4: that corresponding to the origin placed in asubinterval holding Sℓ strictly in its interior. Adding the individual variances we obtain thatthe variance of rℓ, i.e. the expectation of δ2ℓ , is at most log2(n+1)/4; and since by Jensen’sinequality E[δℓ] ≤ E[δ2ℓ ]12 , the expected distance between Sℓ and the ℓth leftmost origin is atmost√log2(n+1) + 3. ◀▶ Lemma 2. Any randomized online matching algorithm incurs an expected cost greaterthan (n+1)/12 in each round.Proof. Consider an origin placed uniformly at random in a subinterval of size 2r during therth round. Assume m unmatched servers in the interior points of that subinterval divide itinto m + 1 segments of (integer) length d0, . . . , dm. Then the probability the correspondingrequest falls within a segment of length d is d/2r, in which case the expected distance of therequest from the segment’s closer endpoint is d/4. Adding over all the sr segments in allthe round’s subintervals, applying Jensen’s inequality, and noting that sr does not exceedthe number of subintervals (i.e. (n+1)/2r) plus the total number of unmatched servers (i.e.(n+1)/2r−1 − 1), the expected cost to service all requests in the round is at least:sr∑h=1dh4 ·dh2r ≥14 · 2r sr(n+1sr)2>(n+1)24 · 2r ·2r3(n+1) =n+112 . ◀E. Peserico and M. Scquizzato 103:3We can then easily prove the following:▶ Theorem. The competitive ratio of any randomized online matching algorithm for the lineexceeds√log2(n+1)/15 for all n = 2i − 1 : i ∈ N.Proof. Let CA(σ) be the expected cost incurred by a randomized online matching algorithmALG on a request sequence σ, and CO(σ) the offline cost; and let pσ be the probability ofgenerating σ through the origin-request process described earlier. Since ∀ai, bi > 0 we havethat (∑i ai)/(∑i bi) is a convex linear combination of the individual ratios ai/bi, focusingon the case√log2(n+1)/15 ≥ 1 for which√log2(n+1) + 3 + 2−n < (5/4)√log2(n+1):maxσ:pσ ̸=0CA(σ)CO(σ)≥∑σ:pσ ̸=0CA(σ)pσ∑σ:pσ ̸=0CO(σ)pσ>(n+1) log2(n+1)/12n(√log2(n+1) + 3 + 2−n)>√log2(n+1)15 . ◀▶ Remark. Without discretized requests the term∑σ:pσ ̸=0 CA(σ)pσ in the theorem’s proofwould have been an integral, potentially ill-defined (for example, if ALG serviced requestsfor rational points in an interval with one server and for irrational points with another).References1 Antonios Antoniadis, Neal Barcelo, Michael Nugent, Kirk Pruhs, and Michele Scquizzato.A o(n)-competitive deterministic algorithm for online matching on a line. Algorithmica,81(7):2917–2933, 2019.2 Antonios Antoniadis, Christian Coester, Marek Eliás, Adam Polak, and Bertrand Simon.Online metric algorithms with untrusted predictions. In Proceedings of the 37th ICML, pages345–355, 2020.3 Antonios Antoniadis, Carsten Fischer, and Andreas Tönnis. A collection of lower bounds foronline matching on the line. In Proceedings of the 13th LATIN, pages 52–65, 2018.4 Bernhard Fuchs, Winfried Hochstättler, and Walter Kern. Online matching on a line. Theor.Comput. Sci., 332(1-3):251–264, 2005.5 Anupam Gupta and Kevin Lewi. The online metric matching problem for doubling metrics.In Proceedings of the 39th ICALP, pages 424–435, 2012.6 Varun Gupta, Ravishankar Krishnaswamy, and Sai Sandeep. Permutation strikes back: Thepower of recourse in online metric matching. In Proceedings of the 23rd APPROX, pages40:1–40:20, 2020.7 Bala Kalyanasundaram and Kirk Pruhs. Online weighted matching. J. Algorithms, 14(3):478–488, 1993.8 Bala Kalyanasundaram and Kirk Pruhs. Online network optimization problems. In OnlineAlgorithms: The State of the Art, pages 268–280. Springer-Verlag, 1998. From the DagstuhlSeminar on Online Algorithms, 1996.9 Samir Khuller, Stephen G. Mitchell, and Vijay V. Vazirani. On-line algorithms for weightedbipartite matching and stable marriages. Theor. Comput. Sci., 127(2):255–267, 1994.10 Elias Koutsoupias and Akash Nanavati. The online matching problem on a line. In Proceedingsof the 1st WAOA, pages 179–191, 2003.11 Nicole Megow and Lukas Nölke. Online minimum cost matching with recourse on the line. InProceedings of the 23rd APPROX, pages 37:1–37:16, 2020.12 Krati Nayyar and Sharath Raghvendra. An input sensitive online algorithm for the metricbipartite matching problem. In Proceedings of the 58th IEEE FOCS, pages 505–515, 2017.13 Sharath Raghvendra. Optimal analysis of an online algorithm for the bipartite matchingproblem on a line. In Proceedings of the 34th SoCG, pages 67:1–67:14, 2018.14 Rob van Stee. SIGACT news online algorithms column 27: Online matching on the line, part1. SIGACT News, 47(1):99–110, 2016.ICALP 2021

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Matching on the Line Admits No o(?log n)-Competitive Algorithm

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