A Universal Error Measure for Input Predictions Applied to Online Graph Problems
Authors: Giulia Bernardini
Alexander Lindermayr
Alberto Marchetti-Spaccamela
Nicole Megow
Leen Stougie
Michelle Sweering
Venue: NeurIPS 2022 Accept (2022)
Keywords: learning-augmented algorithms
untrusted predictions
prediction error
online algorithms
network design
routing
TSP

Abstract: We introduce a novel measure for quantifying the error in input predictions. The error is based on a minimum-cost hyperedge cover in a suitably defined hypergraph and provides a general template which we apply to online graph problems. The measure captures errors due to absent predicted requests as well as unpredicted actual requests; hence, predicted and actual inputs can be of arbitrary size. We achieve refined performance guarantees for previously studied network design problems in the online-list model, such as Steiner tree and facility location. Further, we initiate the study of learning-augmented algorithms for online routing problems, such as the online traveling salesperson problem and the online dial-a-ride problem, where (transportation) requests arrive over time (online-time model). We provide a general algorithmic framework and we give error-dependent performance bounds that improve upon known worst-case barriers, when given accurate predictions, at the cost of slightly increased worst-case bounds when given predictions of arbitrary quality. 

A Universal Error Measure for Input Predictions Applied to Online Graph Problems
Giulia Bernardini
Alexander Lindermayr University of Bremen, Germany 1inderal@uni -uni--remen-nenneee.eeeeeeee de
University of Trieste, Italy giulia. bernardini@unite
Alberto Marchetti-Spaccamela
Nicole Megow
La Sapienza University of Rome, Italy alberto@diag, uniromal...
University of Bremen, Germany nicole. megoo@uni- bbremen de
Leen Stougie
Michelle Sweering CWI, Amsterdam, The Netherlands Michelle. Sww.rligewwe nl
CWI and Vrije Universiteit Amsterdam, The Netherlands Leen Leen. www.stttuuiiii......... .n
Abstract
We introduce a novel measure for quantifying the in input predictions. The error is based on minimum-cost hyperedge cover in suitably defined hypergraph and provides general template which we apply to online graph problems. The measure captures eπrоrs due to absent predicted requests as well as unpredicted actual requests; hence, predicted and actual inputs can be of arbitrary size. We achieve refined performance guarantees for previously studied network design problems the online-list model, such as Steiner tree and facility location. Further, we initiate the study of auggeettd algorithms for online routing problems, such as the online traveling salesperson problem and the online dial-a-ride problem, where (transportation) requests arrive over time (online-time model), We provide a general algorithmic framework and we give error-dependent performance bounds that improve upon known worst-case barriers, when given accurate predictions, the cost of slightly increased worst-case bounds when given predictions of arbitrary quality.
Introduction
We develop a novel measure for quantifying the error in input predictions and apply it to derive error-dependent performance guarantees of algorithms for online metric graph problems. Online graph problems are among the most fundamental online optimization problems, where an initially unknown input is revealed incrementally. The two main paradigms for the incremental information release are the online-time model and the online-list model. In the online-time model, initially unknown requests are revealed time and can be served any time, whereas in the online-list model, requests are revealed one-by-one and must be served immediately before the next request appears. In this work, we address specifically the following online routing and network design problems.
Online-time routing problems the classical Online Traveling Salesperson Problem (OLTSP) and Online Dial-a-Ride Problem (OLDARRP,, server move unit speed in given metric space. Transportation requests appear online over time, each defining start and end point in the metric space (in the TSP both points equal). The task is determine tour serve all requests (in
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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more involved any order) by moving to the corresponding start and end points, The objective is to minimize the makespan, i.e., the time point when all requests have been served and the server is back in the origin.
These problems are well-studied [7] 8 17 18 24). as well as other related variants 31.21 36).
Online-list network design problems In the Online Steiner Tree Problem, requests are terminal nodes that are revealed one-by-one in given metric space (lypically represented as a complete edge- weighted graph) and must be connected to a fixed root by selecting edges via other (Steiner) nodes. In the closely related Online Steiner Forest Problem, a request is composed of two nodes which have to be connected by the selected set of edges. In both problems, the objective is to minimize the total cost of selected edges. In the more general Online Facility Location Problem, a facility can be opened at every vertex at a certain one-time cost at any time, and arriving client vertices are connected upon arrival to the closest open facility at the cost of the shortest path to it. The goal is to minimize the opening and
connection cost. These problems are very well-studied 2 13
15
20
21 23
25
26
30
39
45 46].
The performance of online algorithms is typically assessed by worst-case analysis. An algorithm is called p-competitive if it computes, for any input instance, a solution with objective value within a multiplicative factor p of the optimal value that can be computed when knowing the full instance upfront. The competitive ratio of an algorithm is the smallest factor p for which is p-competitive. For the above problems (nearly) tight bounds on the competitive ratio are known. For OLTSP and OLDARP, there have been shown best possible 2-competitive algorithms 77 For the online
network design problems, the existence of ) ccompeiitive algorithms has been ruled out [25 30 39
and algorithms with (tight) logaritogai upper bounds have been shown [15 25 30
39:45 45
The assumption in online optimization of not having any prior knowledge about future requests seems overly pessimistic. In particular, given the success of machine-learning methods and data- driven applications, one may expect to have access to predictions about future requests. However, simply trusting such predictions might lead to very poor solutions, as these predictions come with no quality guarantee. The recent vibrant line of research initiated in (37 38) aims at incorporating such error-prone predictions into online algorithms, learning-and auggented algorithms with a performance that is close to that of an optimal offline algorithm when given accurate predictions (called consistency) and, at the same time. never being (much) worse than that of a best known algorithm without access to predictions (called robustness). Further, the performance of an algorithm shall degrade in a controlled way with increasing prediction error.
go beyond worst-case barriers. The goal are
In this paper, consider an input predictions model, i.e., there is given a set R of predictions for the actual online input R of a problem. We do not make any assumption on the quality of the prediction or on its size. In particular, ft might be substantially larger or smaller than R.
Defining an appropriate error measure is crucial task in this line of research. There is no common agreement (yet) in the literature on what constitutes a good error measure, The philosophy behind our error measure is the following. Any -aggmnnte algorithm needs to trust the predictions to some extent, as otherwise no improvement upon an online algorithm is possible. The error should then be able to sensitively bound how much any (reasonable) algorithm pays for erroneous predictions. Roughly, our error measure achieves this by approximating the extra cost that an algorithm trusting the predictions has when it serves the true instance; very informally, this is OPT T Several natural measures (for graph problems) have been proposed, such as the number of erroneous predictions [47). the fi-nrrm (e.g., distances between predicted and real points), perfect matching-bas based errors [12). Although we cannot expect that single error measure is appropriate for all problems, we propose universal template based on the cost of a hyperedge cover in bipartite hypergraph that is constructed in a problem-specific way.
R)) (R)
1.1 Our contributions
Cover error for input predictions Here we sketch the main idea of our error measure, which will be made precise in Section 2 We separately cover the errors incurred by unexpected actual requests, R. Rt and absent predicted requests, Rt R, as these pose a potential threat to an algorithm which trusts R. For each of the two eтror types we consider a suitable weighted bipartite hypergraph with erroneous requests on the left side and define an error measure combining the costs of minimum hyperedge covers of the left side of each hypergraph. Let concentrate on errors due to unexpected actual requests, being the nodes on the left side, with the predicted requests as the nodes on the right side. Each hyperedge links single node on the right side with subset of the nodes on the left side
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which it covers. Its contribution to the overall error, i.e., its cost in the hyperedge cover problem to cover all left side nodes, is related to the optimal cost for the subinstance induced by its nodes. E.g. in OLTSP this cost is the value of an optimal tour for some unexpected requests (left) when starting from some predicted request (right), which can he seen as a minimum detour that needs to be made from the predicted request to serve the unexpected requests. Bounding the number of left side requests in the hyperedges by & yields a hierarchical family {A of errors, with higher values of k giving errors that reflect more precisely the cost due to trusting wrong predictions.
The cover error fulfills several useful properties. First, it provides framework that may apply to various problems by assigning appropriate costs to the hyperedges. E.g. for the online-time model it allows integrate in a very precise way actual and predicted release dates, as we demonstrate in
Section B This is a feature which previous metric graph eπrors seem to miss [12 47). because they rely on counting incorrect predictions or disallow asymmetric cost functions. Our error also naturally supports different sizes of R and R, reflecting the input sequence length being unknown in almost all online optimization problems in the literature. Although previously studied error measures [12 do support this in theory, we will show in Section 2 that they fail to detect good predictions in certain scenarios, which results in imprecise weak performance bounds. In contrast, the cover error guarantees an almost optimal performance of the same algorithms in these cases. We therefore hope that the cover error will be useful for better analyzing existing learning-augnan augmented algorithms and other problems in the future.
Algorithms with error-sensitive performance bounds Our algorithmic results are twofold: we provide the first saaming-augmentedd algorithms for online-time routing problems, and we give new droorddpendencces for existing algorithms for online- list network design problems. The unifying element that we achieve these by problem-dependent implementations of our new cover error. We first introduce general framework for OLTSP and OLDARP, in which we delay the moment in which we start following the optimal predicted tour by multiplicative trust factor a E (0, 1). For robustness, before starting to follow the predictions any p-competitive online algorithm is executed in a black-box fashion. We prove an crro-ddepnnenccy w.n.. the first cover error in the hierarchy A (as properly defined in Section 2}. We denote by C* the cost of an optimal tour on the actual requests R. Theorem 1. OLTSP and OLDARP admit learning- augmeeted algorithms that use a p-competitive algorithu as subroutine and achieve competitive ratio that is, for any >> 0, bounded by min {(1 + ax) (1 + 3-A1) 1 2)
Hence, sufficiently good predictions help to beat the classic lower bound of 2 (8] for OLTSP.
When using the algorithm of (7) as a subroutine, we can further refine our algorithm by carefully aligning the used waiting strategies and prove an improved robustness guarantee.
Theorem 2. OLTSP and OLDARP admit a competitive algorithm of [7] as subroutine and achieves competitive ratio that is, for any >> O. bounded by min {(1 + a:) (1 + 3-11) C- ,2+ ).
algorithum that uses the 2-
In general, online algorithms for OLTSP and OLDARP aim for tackling the uncertainty of the input rather than efficient running times, which is also the case for the above discussed results. Yet, we show in Section 3.2
that we can trade efficiency with slightly larger constant factors in the guarantees.
Further, remark that simpler and tightened results possible for restricted metric spaces: in Section 3.2 we provide improved bounds for OLTSP on the positive half of the real line. Here, a minimalistic prediction, single value predicting the optimal makespan C",, suffices to obtain an almost tight consisteney-robustness tradeoff
We complement these theoretical bounds with empirical results (see Section 5) on simulated real- world taxi instances in the city road network of Manhattan, which indicate the superior performance of our new algorithms compared to classic methods in both general and relevant restricted scenarios.
Further, we consider online-list graph problems and analyze the algorithmic framework provided by Azar, Panigrahi and Touitou [2] w.r.t. our new cover error. For each problem, we specify hyperedge cost functions which follow the same paradigm and prove new error-dependent bounds.
Theorem 3.
The algorithms in (12) J for the online Steiner tree or online (capacitated) facility location
problem incur, for any parameter k 2 1, a cost of at most (1)) OPT O (log k) Ak--
Theorem 4. The algorithm in [2g for the online Steiner forest problem incurs, for any parameter k 2 1, cost of at most O(1) OPT + Okk)
Ag.
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These bounds hold sinultaneously for any k. The algorithm is still robust due to the robustness bound of ( provided by Azar et al. in [12]. On the technical side, we can exploit a technical lemma by [12) that allows us to split the analysis in two parts. The actual proofs for bounding the algorithm's s cost by the cost of an optimal solution and the cover error are completely different.
For certain input scenarios we substantially strengthen the bounds on the competitive ratio provided by Azar et al. [12]. Indeed, their bound never improves over 2[log( m [R) [ft] mmin [1 ()) which is not better than the best possible competitive ratio O (log) Ri) for classic online algorithms, if A] and R differ significantly. We show that there are input scenarios for which their error measure overestimates the actual error substantially and, thus, gives poor performance bounds while the algorithm performs actually well; more precisely, we prove a constant competitive ratio for the algorithm in [2 w.r.t. our error measure whereas the bound w.r.t. the previous measure is O (t)).
1.2 Further related work
While untrusted predictions have been successfully integrated into online models for many different problems, none of the previous approaches and models seem to capture the complexity of combined routing and scheduling decisions. Related research includes work scheduling (10 14 29 33 34]
40. 43). routing in metric spaces such as the k-server problem and more generally metrical task
systems 35), graph exploration 22] and online network design
47
Further, there is hardly any work on integrating untrusted predictions into online problems in the online-time model. The only exception seems to be the work by Antoniadis et al. [6] on online speed scaling with a prediction model that includes release dates and deadlines. The nature of the speed-scaling problem, however, is very different from the routing problems we consider. Other works on non-clairvoyant scheduling with jobs arriving over time (minimizing flow time [10 total completion time 34 and energy (14) assume predictions on the job sizes or priorities; release dates are known in advance in [4]. while 11 34) consider purely online problems w.r.t. job arrivals.
Very recently and independently of our work, two papers 27 28 were announced that also study OLTSP in the models in general metric spaces. For arbitrary input predictions, their result has no error-dependency and a weaker consistency-robustees sively study OLTSP on the real line. Assuming that the correct number of requests is known in advance, they study the power of predictions on the locations; their results are incomparable to ours.
setting. Hu et al. [28) consider OLTSP with different prediction
tradeoff compared to Theorem Gouleakis et al. (27) exclu-
1.3 Organization
We first introduce the cover error in Section 2 and then give our results for online-time routing problems and online-list network design problems in Section resp. Section Finally, we present empirical experiments Section 5 Details on extensions, further experiments and missing proofs
can be found in the full version [6 of this paper, which is also part of the supplemental material.
2 The cover error
Given an input prediction Ñt and the actual input R. we design an error measure that covers every erroneously predicted item, i.e., all unexpected requests It' ft and all absent predicted requests R) R. As concrete example think of OLTSP where Saarning-augmentt certain degree) the predicted requests in Åt and thus follows an optimal tour on ft. After serving predicted request (ft, f). it may serve some unexpected actual requests Rt' C R \ ft that have already been released and are relatively close to (f+ f) (both time- and location-wise). In our terminology. think of (λ, f) covering R', and observe that the cost for this cover shall naturally be equal to the optimal cost for serving R' when starting in 0 at time f. Conversely, the predicted requests that have not shown up and are nevertheless visited, Åt R, should be covered by actual requests, to make up for the extra cost incurred by these superftuous visits.
algorithm trusts (at least to
We can arguably expect that any well-performing algorithm should be as least as good as serving all unexpected requests R R A and all absent predicted requests ft \ R in such partitions which can be covered by, respectively, predicted and actual requests in the cheapest possible way.

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12
10
10
10
12
R \ ft
R
R
R
(a) Instance metric space (b) Min-cost k-hyperedge cover (c) (R) R) At ft)
Figure 1: Example for metric instance and input prediction with min-cost k-hyperedge cover of the set of unxxpected requests R' R. The actual requests are filled green and the predicted requests are encircled red. The labels show which points in the metric space correspond to which nodes in the bipartite graphs.
We now embed this intuition into a precise definition. Let A and B be two sets of (possibly different) size and let 2 1. We define a bipartite hypergraph G₁ = (A B, H) where is the set of all endpoint in B and at most & endpoints in A. A k-hyperedge
hyperedges which have exactly cover of by B is a set of hyperedges H'' C H in Gę such that every vertex A is incident to least hyperedge in H'' If every hyperedge h H of G has associated cost 7(h), minimum-cost k-hyperedge cover 't' a k-hyperedge cover which minimizes the total hyperedge cost Σ (h'). We denote the value of min-cost k-hyperedge cover of A by B by I (A, B). Finally, the cover error, denoted by A& R. ft). is given by
As (t) To (R, R) + (R,
Notice that allow arbitrary large hyperedges (A: oo) to cover predicted requests h. We emphasize that all results also hold for symmetric error definition I (R, R) (RR, R), because (A, B) 2 Ti+1 (A, B), for any i. Nevertheless we use this asymmetric definition to obtain a stronger bound when covering R. Intuitively, this is possible because all predicted requests are known advance (as opposed the actual requests, which arrive online).
We simply write Aę if Å and R are clear from the context. Since error measure shall give value zero if R R, require that the cost of every hyperedge {a} U {6} for some A and b B is equal to zero if a = b. Then, all vertices in A can be covered trivially by B, and we conclude that (A B, B) (A, B). Figure depicts example of k-hyperedge cover.
remains to specify the cost 7(A', b) of a hyperedge A' (b}. Although we will give precise definitions separately for every concrete problem, all definitions follow certain paradigm. That is, the cost (A', b) shall be equal to the value of an optimal solution for the subinstance induced by A' with respect to b. This anchoring requirement the single detail which has to be specified for concrete problem. Note that this matches our intuition discussed above for OLTSP.
Comparison to other error measures We compare the cover еттоr to previously proposed error measures for the (undirected) online Steiner tree problem. Xu and Moseley [47) define a prediction error 77 = max RR. [ R), the number of erroneous requests, and prove that their algorithm is (log(min) (A, D), where D is the value of min-cost perfect matching between two equally sized subsets of and ft, and is the total number of unmatched points in Rt and Å. They prove for their algorithms a multiplicative error dependency w.r.t. log(min( [[R) A}) and an additive error dependency w.r.t. D.
(.m.)))/cmmmtttiiv Azar et al. [12) introduce the metric error with outliers X =
We give a family of instances with n actual requests and an input prediction for which the algorithms of Azar et al. [12] and Xu and Moseley [47) perform arguably well, but their error measures and analyses yield bound of (log n) OPT ((s), which could be achieved even without predictions. For some > 0, the instance is composed of one terminal request
requests in an e-ball around the Steiner point T2, no request is exactly on Ø2. Both 21 and æ2 are predicted. We can 1. For the metric error with
and
immediately observe that the number of erroncous requests 47 is
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outliers [1]. note that any perfect matching is composed of at most two matches, therefore A 2 n n-2
and (c)). On the other hand, our cover enror is bounded by A S A₁ On <) for any because 22 covers all requests in the 6-ball around it. Then, Theorem Saarrem@concin conclldes that the algorithm of Azar et al. [12] indeed constant competitive for this instance when e -
Online metric TSP with predictions
Let M = (X,d) be metric space, consisting of set of points X, with origin X and metric d. In the Online Metric Traveling Salesperson Problem (OLTSP), a set of unknown requests R is released online over time. A request (x. r) is composed of point x E X and release date Rgo. ie., the time at which the request becomes known and can be served. The task of an algorithm is to route server, which initially in the origin and moves unit speed, through all requests back the origin. The objective is to minimize the makespan, i.e., the total time required for this task.
The OLTSP with predictions, is an OLTSP in which we are given additionally an a priori prediction R on the set of requests. We assume that the server receives a signal when it is back at the origin after serving all the actual requests. Unlike in the classic OLTSP, this is important, as otherwise an algorithm might continue considering predicted requests, thus, ruling out any robustness.
To specify our cover error, we define the cost of a hyperedge R' ( r')} the extra cost of serving erroneous (unexpected or predicted absent) requests R'' w.r.t. a request x' (actual or predicted);
(ISP(R' r')) optimal makespan for serving instance R' from origin x' and initial time r'.
To get some intuition, consider X Rgo and algorithm that does not move before time t if there requests. It will receive an adversarial request (t, and hence encounters ratio of _/2 [18] J. To overcome this when having (almost) accurate predictions, the server has to move towards (predicted) requests before they actually arrive. However, this pre-moving technique brings several challenges. an algorithm moves the server without interruption to predicted request at the beginning of the instance, adversary would immediately spawn the single actual request at the origin, giving an unbounded robustness. the server would directly move back, one can similarly argue that the consistency is at least 3. Therefore, the key is to define a proper waiting strategy before moving towards predicted requests. We show that we can execute an arbitrary online algorithm while delaying pre-moving to gain information about the instance. This is very delicate, since too much delay clearly weakens the consistency, but too little delay gives weak robustness. We prove the following result.
Theorem 5. Let a E (0, 1/2) and let A be a (1 + a) coonsistttt deterministic learning augmented algorithm for OLTSP. Then, A can be 3-robust only for 2 1 1. This holds even the half-line.
Our final algorithm uses a hyperparameter a > 0 to configure the waiting duration and thereby achieves tight asymptotic consisteney-robustnees tradeoff. Intuitively, we can express our confidence in the prediction using a and get customized guarantees.
3.1
A general framework for OLTSP with predictions
Our strategy involves an initial delay phase in which we follow arbitrary online algorithm, up to some predetermined time depending on the cost Ô of an optimal tour f of the predicted requests f. After that, we start following 2, adjusting whenever the actual requests deviate from the predictions. We call this greedy strategy PREDICTREPLAN (PREDREPLAN for short), due to the analogy with the Let p()) be the server's location time L. classic REPLAN heuristic 7
Algorithm 1 PREDREPLAN
Follow f. Whenever unexpected request r) is released, recompute and follow fastest tour from p(r) to the origin serving all unserved predicted requests as well as all the unserved unexpected requests. If the server receives end signal in the origin, terminate.
While this algorithm might move towards predicted requests which are known to be absent to make the analysis clearer, practical implementation ignores these and thereby only improve its performance.

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We formally define the class of algorithms DELAYTRUST. that is parameterized by our trust parame- ter a O, which scales the delay. Let A be any p-competitive online algorithm for OLTSP.
Algorithm 2 DELAYTRUST
Follow A long for time t it holds S aC' - d(p(())
ii:
Move the server to the origin
ii: Follow the PREDREPLAN strategy until the end
We refer to the execution of each line as a phase. We now prove the main Theorem I for OLTSP. by showing for DELAY TRUST separately an error-dependent bound in Lemma 6 and a robustness bound in Lemma Given an OLTSP instance, we denote by C the makespan an optimal tour T serving all actual requests.
Lemma 6. DELAYTRUST has competitive ratio of at most (1 a) (1 for any 2 0.
Proof. We first bound Ò. Fix a min-cost oo-hyperedge cover of ft by R and an optimal tour T for R. For every hyperedge ft' {(x, } in the cover, we extend T by adding the optimal offline OLTSP tour for R' which starts at x at the time t at which T serves x. Note that, since r I t, the makespan of this subtour is bounded by the cost of Rt' U r)). Since every predicted request is covered by at least one hyperedge, the constructed tour serves At and we conclude that Ö S C* + Do (R, R).
We now bound the makespan of the tour of the algorithm. If the algorithm terminates in Phases (i)
R)) I (1 (1+00 a.) (C* Az). or (ii), its makespan is at most aC S a (C* + Too R))
Otherwise, the algorithm reaches Phase (iii.. There it first computes at most E serving all unserved predicted requests. The makespan only increases when unexpected requests arrive. To this end, fix min-cost 1- 1-hyperedge cover of R by R and a hyperedge {(x', r')} U { of this cover. We upper bound the additional cost due to (a', r²) by the cost of an excursion from the algorithm's 's current tour serving (x',r") The algorithm might find a faster tour serve all unserved requests and henceforth uses that. We distinguish two cases depending on the algorithm's 's remaining tour before request (a', r') arrived. If ft is not part of this tour, consider excursion which immediately deviates from p(r') to serve (x', r') and then returns to p(r'). By the triangle inequality, the length of this excursion bounded by twice the distance between p(r') and â. plus the cost for optimally serving (x', from & when starting at time f. Due to our assumption, (œ, f) must have already been served at some time with r' > t 2 Thus, the algorithm's 's server is at most r' - f units away from a at time r', and the total time incurred for this excursion is bounded by
optimal tour f of length
2 (r' - f) + 7
TSP (f(x', r'),,r)),,,) (2, (2) < y TSP (f(a', ,r')), (f).
Note that the inequality due to the fact that (a', r') can only served after its release date.
In the other case, the algorithm's 's server will visit â at some later point in time, especially at least once after time P. We thus wait until the algorithm reaches â at some time t f, and then serve (x', (x',r") using at most
(((x', ,r)), (f, f)) additional time. See Figure 2 for an illustration of both cases.
Since every actual request is covered by one hyperedge, conclude that Phase (iii) takes time at most Ò 2.3.T T,, ft). Adding the time for Phases (i) and (ii) gives a makespan of at most
(1+0))+3-Ti(R R) & (1+0) (c*
(R R)) - (1+a) (C" + O
Lemma 7. DELAYTRUST has a competitive ratio of at most 1 + p + &. for any > and any p-competitive algorithm used in Phase (i).
Proof. If the algorithm terminates during Phase (i) or (ii), the competitive ratio is p. We are guaranteed to finish in one of these two phases if pC S aC.
If we terminate within Phase (iii,, then Ć < LC Once the last request has arrived at some time Plast S C* our tour stays fixed. We distinguish two cases. If the last request arrives before the end of Phase (ii), then the cost of our tour comprises of the cost for finishing Phase (ii), which is at
a
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f3
8
Y ((7), (23.
(a) The server (square) plans (dashed) and follows (solid) pre- dicted tour. Two hyperedges are completely released: [7] is covered by 3, which is already served, and {6} is covered by 1, which will he served in the future. Note that neither 1, 8 nor 9 have heen released yet. Also notice that PREDREPLAN might find faster tour. Figure 2
R
(b) Min-cost 1-hyperedge cover of R
most aC, and the cost of PREDREPL.AN for following the predicted tour, including all unexpected yet unserved requests, which is at most & + C The total cost is thus bounded from above by
aC+C+0" < (1+ (rC++==(++000
C" (1+
C
In the second case, the last request arrives in Phase (iii). In this case the cost after Tiast is the cost of following the predicted tour, adapted for incorporating the unexpected, yet unserved, requests. This bounded above by the cost of returning to the origin, following the predicted tour T, and finally following the optimal tour, T" Note that the cost of returning to the origin is at most Flast aC* Hence, we complete the proof by upper bounding the makespan, for any P 2 1, by
Tiast + Plast - aC -CC < (1 (1-a) a)
c- c²= - p-pt
C < (1 + 11+p+=0 P Cr
C" 01+
3.2 Extensions and improvements
An improved algorithm for OLTSP with predictions A best possible online algorithm for OLTSP SMARTSTART, which is 2-competitive (7). Using this in Phase (i) of DELAYTRUST, Theorem I yields robustness factor of at most 3 2. We exploit SMARTSTART'S waiting strategy to serve yet unserved requests and expedite Phase (iii) avoiding unnecessary waiting time, obtaining an algorithm, SMARTTRUST, with improved robustness factor 2 + 2 stated in Theorem
a
Algorithms with polynomial running time Algorithms DELAYTRUST and SMARTTRUST require the computation of optimal TSP tours on subinstances. NP-hardness of TSP prohibits polynomial algo-
running time, unless P=NP. We provide performance guarantees for our rithm framework when using polynomial-time v-approximation algorithms for solving TSP, which guarantee to find TSP tour within a factor v of the optimum.
We use modified efficient TREDREPLAY strategy which uses v-approximate solution instead of an optimal solution and further ensures that errors due such approximations do not add up too much compared to our eтror budget A₁. Adjusting the proof of Theorem Dyilld the following result.
Theorem 8. Given a v-approximation algorithm for metric TSP, the competitive ratio of the polyno- mial time DELAYTRUST using polynomial time p-competitive online algorithm in Phase (i) is, for +
any a > 0, bounded by min {(1+v) + a) --2 (p (1 v1 2)))
Online metric Dial-a-Ridh with predictions The Online Metric Dial-a-Ride Problem (OLDARP) is generalization of OLTSP where each request (a: x² ,r) has starting location x and destina- tion rd To serve request, the server must first visit x at some time not earlier than r, and then a We assume that the server can carry at most one request at the time and cannot store it after pickup.
We show that slight modifications of DELAYTRUST and SMARTTRUST yield Theorems and for this generalized setting. We define the cost function 7. DDRP for the cover error:
y DaRP (R', (x", rd r)) = min TSP (R', (x",r))) r))),TTTT TSP (R', (xd r d(x", ))) + D.
where the maximum transportation distance in Rn R.. Intuitively, an excursion can start from after time from z after time d(æ", _zd) to serve R', whatever is the shorter of the two.
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An improved algorithm for OLTSP on the half-line metric When restricting the metric space to X Rzo, the best possible online algorithm is 3 g-competitive (18). We design a learning- augmented algorithm tailored to this metric space and a minimalistic prediction, namely, a single value & predicting the optimal makespan C". We prove the following error-dependent performance bound, which gives an almost tight consistency-robustees tradeoff W.r.t Theorem [5]
Theorem 9. There is learning augnented algorithum for the half-line metric that has for every a E (0, 1/2] a competitive ratio of at most min { + a) (1 2: 3.
Online network design problems with predictions
This section sketches the applicability of our new error measure for the online-list problems Steiner Tree, Steiner Forest and (capacitated) facility location. To prove new dror-dependent performance bounds as stated in Theorem 3 and Theorem and analyze it w.r.t. our cover error measure. The key is an appropriate, problem-spectne the cost of a hyperedge and the corresponding analysis. Recall that the cost of hyperedge R' U {x'}
revisit the algorithms proposed by Azar et al. [12]
definition of
should (intuitively) be equal to the value an optimal solution for serving a set of unexpected (predicted absent) requests R' w.r.t. predicted (actual) request x'. We define the cost functions:
Steiner tree: ( xx) = cost of an optimal Steiner tree for terminals R' with root x'.
Steiner forest:
SF (R', x') = cost of an optimal Steiner forest for terminal pairs R' when connecting via x' ()) is free.
Facility location: (R, x²) = cost opening facility x' and assigning clients R' to it.
Our main technical contribution for online network design problems are the proofs of Theorem 3 and Theorem 4 We now give some intuition on how these proofs work by considering the online Steiner tree problem, and defer details and results for this and the other problems to the full version (16).
On a high level, the algorithm by Azar et al. 12 for the online Steiner tree problem does the following. Each new request (terminal) is connected to the current solution greedily by buying edges on a shortest path to a vertex of the current tree. When the Greedy cost increased sufficiently (with thresholds following a doubling-strateeyy, the algorithm spends a certain budget (depending on the spent Greedy cost) on connecting as many future predicted requests as possible to the current solution.
The proof of Theorem 3 splits the execution of this algorithm into two parts, where the first part considers the time until all predicted requests are satisfied, and the second part the remaining exccution. We then use sub-result provided by Azar et al. [I2] which roughly bounds the total cost of the first part by the optimal solutions of R and Rł and the total cost of the second part by the cost of the algorithm for serving specific subsequences of the request sequence. To further bound the first part, we use the structure of a min-cost oo hyypeede cover of to prove an upper bound of at most (1)) OPT + O(1) Do c (ft, R). For the second part, we consider the total cost the Greedy algorithm incurs for a hyperedge of min-cost k-hyperedge cover of R, and conclude by the bounded hyperedge size and Greedy properties that this is at most Olog k) times the hyperedge cost, yielding total bound of O(1) OPT + Oloog k) T& [& (R, ft). Here we especially use the fact that in our chosen partition of the algorithm's 's execution, any predicted terminal 2 which covers actual requests
must have already been served in the first part.
5 Experiments
We performed various empirical experiments on real-world OLTSP instances that demonstrate the benefits of using our algorithms over classic online algorithms. We consider the road network of Manhattan [1]41] and compose 100 instances of 10 requests each based on taxi pickup requests from a dataset offered by the NYC Taxi & Limousine Commission We compare SMARTTRUST with the classic online algorithms REPLAN [7). IGNORE 24 44] and SMARTSTART (7): all algorithms use efficient TSP heuristics. We report for every experiment and instance the empirical competitive ratio, ie. the average ratio between the algorithms performance and the approximated value of the optimal makespan, as well as error bars that denote the 95% confidence interval over all instances.
data. page downloaded 02/05/22

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Smarthust (aw0.)
Smarthrus (ow0.0)
Smarthust (cm5 0J
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2.0
Smarthue (p-0.:
Sealen
Smenthust (sw0.3)
Smarthus (e-0
SmartStart
Smarthust (sw0.5)
Replan
Smarthuut 1e-0.5:
Smartiust (o=1.0)
SmartStart
1.-
2.6
1.6
14-
1.2
12-
10-
10-
10"
10
102
30a
30a
10'
12
10
10
Noise parameter
Numbar of correclly predicted requests.
(a) Noise only in request locations
(b) Partial instance predicted correctly
Figure 3: Experimental results for two different prediction settings (100 instances with 10 requests each)
We sketch here two relevant experiments and defer further details to the full version [6 The first experiment considers synthetic predictions with Gaussian noise a only in the request locations, i.e., the release dates are predicted correctly. The results (Figure (3a) show that SMARTTRUST with = 0.1 dominates classic algorithms even for arbitrarily bad predictions. only part of the actual instance is predicted, which is an interesting and practice-relevant variant. Again, the results (Figure 3b) show that for small values a, SMARTTRUST outperforms all classic algorithms.
second experiment
Concluding remarks
The universal cover error can applied arbitrary problems with uncertain inputs. As seems to be the first eтror measure that captures arrival times, it seems very natural to investigate, in particular, other online-time problems such as, e.g., scheduling problems. Further, would be interesting to identify more compact or smaller predictions. While we predict a full input instance much less information might sufficient to gain high-quality solutions. This can be only partial information algorithmic actions, such
about the input instance predictions request sequences. In the latter case, we can directly apply our framework after approximating the predicted tour by some time-stamped discrete points and using those as input prediction.
optimal tour instead of
Acknowledgments and Disclosure of Funding
The work in this paper is supported in part by: the Netherlands Organisation for Scientific Research (NWO) through project OCENW. Www.W.GROT.2D.. P.RROOT.201..00 2019,01 "Optimization for and with Machine Learning (OPTIMAL)" and Gravitation grant NETWORKS SAEKKS--22.000 002.003; the German Science Foundation (DFG) under contract 146371743 -
TRR 89 Invasive Computing: the ERC Advanced Grant 788893 AMDROMA "Algorithmic and Mechanism Design Research in Online Markets": the MIUR PRIN project ALGADIMAR "Algorithms, Games, and Digital Markets"; and the MUR FSE REACT BU PON R&T 2014-2020.
The map data used in experiments is copyrighted by OpenStreetMap contributors and available from https:/wwww.pennstretmm..ppp
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
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(b) Did you describe the limitations of your work? [Yes]
(c) Did you discuss any potential negative societal impacts of your work? [N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full of assumptions of all theoretical results? [Yes]
(b) Did you include complete proofs all theoretical results? [Yes] See supplemental material for most of the full proofs.
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main exper- imental results (either in the supplemental material or URL.? [Yes] See source code folder in supplemental material; instructions can be found in README.md.
(b) Did you specify all the training details (e.g.. data splits, hyperparameters, how they were chosen)? [Yes] See Appendix in the supplemental material
(c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)? [Yes] See Appendix D in the supplemental material
(d) Did you include the total amount of compute and the type of resources used (e.g.. type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix D in the supplemental material
4. If you using existing assets (e.g., code, data, models) curating/releasing assets...
(a) If your work uses existing assets, did you cite the creators? [Yes] OpenStreetMap contributors cited
(b) Did you mention the license of the assets? [N/A]
(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]
(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A]
(e) Did you discuss whether the data you using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given participants and screenshots, applicable? [N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
13


Reviews:
[{"review_id": "kIi1vYUzj80", "reviewer": "Reviewer_UJPw", "summary": "This paper introduces a new notion of error for algorithms with predictions in the context of online metric graph problems. The predictions considered are the set of all requests, e.g., the offline problem instance. At a high level, the \"cover error\" introduced in this paper measures the optimal solution cost on the set of requests that appear in the prediction but not in the real instance and vice versa. This is formalized as a min-cost bipartite hypergraph cover problem to account for these two types of error via \"detours\" from the true instance. The authors give instantiations of this general error measure to online TSP, online Steiner tree/forest, and online facility location and develop algorithms that use predictions and whose performance is parameterized by the cover error.", "questions": "Could the authors please elaborate on the weakness discussed above? In some sense, it seems like this paper brings the error to the algorithm as opposed to the more natural idea of designing an algorithm to work better with some standalone notion of error.", "limitations": "I think the discussion of limitations is adequate.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n- The cover error has several benefits, which the authors do a good job highlighting. In particular, it accounts for the asymmetry in the impacts of the errors in which we predict requests that aren't there and errors in which we miss real requests in our predictions.\n- The authors develop algorithms parameterized by the cover error for various online graph problems, achieving bounds that can significantly improve on the worst-case if the cover error is small.\n- I think that this error measure and the resulting algorithms will be of interest to those working on algorithms with predictions.\n\nWeaknesses\n- Naturally, more complex error measures can more precisely capture the performance of algorithms with predictions. I worry that in some sense the definition of this error measure almost makes the fact that there exist good algorithm if this error is small a tautology. More basic error measures may be loose in certain respects, but they tell us something clear about the relation between the prediction and instance which we may or may not be able to leverage algorithmically: they have some meaning somewhat separate from the algorithmic task. It seems that saying the cover error is basically an algorithmic fact: there are a small set of errors which we can cover by work which we must do anyway.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "hrSvAKdpZxm", "reviewer": "Reviewer_Jom7", "summary": "This paper gives a new error measure for online algorithms with predictions for metric problems of two kinds: (a) online TSP and Dial-a Ride, and (b) online Steiner tree and forest. The two categories differ in their notion of online: the first category has requests released over (continuous) time, while the second category just has an online sequence of requests but no actual notion of time. The error measure is innovative and tries to address some shortcomings of previous measures. In particular, the paper compares to two papers from AAAI 22 and SODA 22. The first paper uses set differences to characterize error and the non-errorneous part of the prediction needs to exactly match the actual input locations. The second paper relaxes this notion by allowing the common set between predictions and actual input to not match exactly and uses a minimum cost matching between these points to quantify the quality of the match. At a high level, the deficiency being addressed in this paper is that this minimum cost matching (or the stricter requirement that the common part matches exactly) forces the cardinality of the common parts of the predicted and actual sets to be equal. So, for example, if there are many real input locations that are all close to a single predicted location, the previous error measures have large error while one can argue that the prediction is actually a good one in a qualitative sense. To address this, the new error measure being introduced in this paper (roughly speaking) replaces the edges in the matching with hyperedges with one point on one side (predicted or actual) and multiple points on the other side. The cost of the hyperedge is defined in a natural way based on the problem at hand. \n\nUsing this new measure, the paper gives the following results: (a) for online TSP and Dial-a-Ride, it gives online algorithms that degrade gracefully with the new notion of error, and (b) for Steiner tree and forest, it shows that the algorithm from the previous SODA 22 paper gracefully degrades with this notion of error. Although the algorithm in (b) is the one from the previous paper, the analysis with respect to the new error is new.", "questions": "1. Can you say more on the asymmetry of your error measure (using $\\Lambda_\\infty$ and $\\Lambda_k$ in the two directions)? Is this necessary, i.e., does something go wrong if you defined the error measure symmetrically? This asymmetry actually shows up in other places as well (e.g., in the Almanza et al. work on online facility location with predictions where the bounds apply to the case when predictions contain all the actual requests but not vice-versa). So, I suspect this is an important (and necessary) asymmetry. The paper should include a discussion about this.\n\n2. Theorems 1 and 2 should be replaced by informal statements if possible. At this point, $\\Lambda_1$ hasn't even been defined formally, and as such, it is difficult to interpret the formal bounds of these theorems without knowing this definition precisely. I believe $C^*$ wasn't defined before the theorems either (but this is easier to fix by just adding the definition).", "limitations": "There is no explicit discussion of limitations. This can be included, perhaps in the conclusions section.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1. I think the online algorithms with ML predictions (or more generally data-driven algorithms) area is a very important one. This paper is exactly in the direction that is particularly important for this area, which is to explore new paradigms and frameworks that are natural and have interesting technical challenges. As the paper correctly claims, there is no unanimity on what the correct measure of error is (as against fairly universal acceptance of the notions of consistency and robustness) and this is a significant handicap because each problem requires its own specialized techniques to showing smoothness bounds with error. So, I really like what the paper sets out to do, which is to give a reasonable notion of error that applies to many (at least metric) problems, and then give somewhat general techniques for obtaining smooth algorithms with respect to this error measure (at least techniques that extend beyond single problems).\n\n2. The algorithms for online TSP and Dial-a-Ride with predictions are new. These are important problems in their own right, so initiating the study of online algorithms with predictions for these problems is an important step irrespective of the particular error measure being used here.\n\nWeaknesses:\n\n1. I did not completely understand why the paper eliminates the difference sets in the error measure that existed in the previous definitions. For instance, if the predictions are completely bogus, then it seems the actual location of the predictions should not matter. In particular, one should be able to recover robustness guarantees from error dependent bounds, but that does not seem to be the case here. I suspect it is possible to keep the current definition but only apply it to a common part (chosen in an adversarial manner) and allow difference sets outside the common part, thereby recovering robustness.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "YLeq30aaXyB", "reviewer": "Reviewer_xjD5", "summary": "The paper studies learning-based online graph problems, focused on online TSP and its variants, online Steiner tree and forest, and online facility location. In each problem, there is a sequence of node requests arriving online, and an algorithm must move or select certain edges to serve these requests. Moving or buying edges come with costs, so the goal is to minimize the total cost, measured against the offline OPT. \n\nClassic algorithms are focused on the worst-case competitive analysis and generally do not consider real world data distributions. In the algorithms with prediction model, the algorithm is instead provided with a predicted sequence of requests. We’d like to show that if the prediction is good, a learning-augmented algorithm can improve upon the classic worst-case guarantee. Otherwise, the algorithm should be robust and retain a worst-case competitive ratio.\n\nThe main contribution of the paper is a new error metric of the learned predictions (based on hypergraph edge cover). The paper claims that this metric bypasses some difficult cases of prior related work, and provides algorithms that work well given good predictions under this metric. Empirical evaluations demonstrate that the algorithms achieve good performance on real data.\n", "questions": "\nAlgorithm 2 requires “predicted optimal cost C hat” as its input. I’d like to clarify that this is the OPT value of the predicted request sequence Rhat. Moreover, we know that optimally solving TSP is NP-hard. Can we solve C hat approximately up to a small constant? Based on the proof of lemma 6, I think we can actually provide a C hat that is, say, 3/2, approximation of the optimal Chat. This would worsen the final bound by only a fixed constant, though. Is that correct? \n", "limitations": " \nOverall, I believe the paper is technically novel and has potential to make an impact. However, as it currently stands, it likely requires significant revision to improve its presentation. I have listed some concrete suggestions above.\n\n\n", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "\nStrengths:\n-----\n\nThe paper gives a novel notion of error metric of learned predictions in online graph problems.  Given the “dummy request example” (line 189 below), I think this new notion is reasonably motivated.  In particular, this example demonstrates a difficult case where the prediction is conceptually good, but not captured by the error metric of recent work, including Azar et al. [Online graph algorithms with predictions. SODA 22]. \n\nThe technical claims in the full paper are sound (but I have not checked the proof details in the supplement). \n\nThe paper also gives experimental evaluations. \n\n\nWeaknesses:\n------\n\nThe paper is not well written, particularly the early sections. Some statements appear vague and only get specified (much) later, and sentences hard to parse. Below, I list a few for the author(s) to fix. I find they seriously obstruct my reading. \n\n* Line 90: “Since we can use cost functions based…”  This sentence seems very vague to me. Is it a feature that prior work such as [12] doesn’t have? My understanding is that the matching-based error of [12] is also rather general and independent of the problem (but can be applied to all these node-arrival online graph problems). \n* Line 92: “Further, it allows to integrate both actual and predicted release dates for online-time problems.” How? Up to this point, the cover error has not been fully specified, so I couldn’t tell why this is true. \n* Line 97: “Although previously studied…” Regarding this dummy requests example, I suggest it’s written concretely right here in the intro. (I get that it’s defined later in line 189 and below)\n* C* is undefined in Theorem 1.\n* Line 131: “These bounds hold simultaneously for [...] the bounds provided in [12]” — What does this mean? (I get that it holds for any k, but not the second part.) Is it suggesting that these bounds recover some of the results in [12]?\n\nFinally I suggest that the definition of the cover error be given earlier in the intro section. \n\nMinor errors: \n-----\nLine 77: “In all our problems the cost of a hyperedge [...]”   In my first reading, this sentence feels long and a bit hard to parse. (What does “anchor” mean here?) \n\nLine 80: “E.g….” This is not a complete sentence.\n\nLine 85: “unavoidably” -> “unavoidable”\n\nLine 89: I suggest using an itemize here to list these points\n\nLine 90: “problem-independent” -> “problem-independently”\n\nFigure 1: maybe state that the numbers on the nodes are just indices, and do not correspond to the order of the request sequence. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}, {"review_id": "K-9uUDhUYzi", "reviewer": "Reviewer_6xM7", "summary": "The paper considers online algorithms for classic graph problems when predictions regarding the requests are provided to the algorithm. The key contribution of the paper is to define a new notion of error to measure the quality of predictions provided for such problems. The new error attempts to capture the intuition that if following the predictions leads to substantially worse performance than the optimal solution, then such predictions should suffer from large errors. To make this notion precise, the paper defines the error as the size of the minimum cost edge cover in an associated hypergraph where the edge costs capture the amount of excess cost required to satisfy incorrectly predicted requests.\n\nFor the online traveling salesman problem (and its dial-a-ride generalization), the paper provides a framework to “combine” a known robust online algorithm with an offline TSP algorithm on the predicted requests to obtain consistent and robust algorithms. As in many prior work, the algorithm is parameterized by an alpha that represents the trust of the decision maker in the predictions and simultaneously provides a competitive ratio of (1+alpha) if the predictions are correct, and also O(\\rho/\\alpha) if the predictions are incorrect where \\rho is the competitiveness of the base algorithm. \n\nUsing the new error measure, the paper also reanalyzes the algorithm by Azar et al for online steiner tree / forest with predictions and show that it obtains a total cost of at most O(1) OPT + f(k, error) [as opposed to the logarithmic dependence on the error in the multiplier for OPT in Azar et al.] yielding tighter guarantees for certain predictions.\n", "questions": "1. In the definition of Cover Error in section 2, it is not mentioned how the hyperedges are constructed. Does the hypergraph G_k have all possible hyperedges of the specified size?\n2. Related to the point above, since the hyperedges (and their costs) are not defined yet, the statement that \\Gamma_i((A,B) >= \\Gamma_{i+1}(A,B) [on line 179] is not well justified.\n", "limitations": "None", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is well motivated and the new error formulation is pretty intuitive for graph problems.\nOverall I quite like the paper and find the contributions to be non-trivial and interesting. The paper is well-written and reads well. The experimental section is also well written and the experiments are pretty comprehensive for a primarily theoretical paper.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept