Abstract

The rich literature on online Bayesian selection problems has long focused on so-called prophet inequalities, which compare the gain of an online algorithm to that of a “prophet” who knows the future. An equally-natural, though significantly less well-studied benchmark is the optimum online algorithm, which may be omnipotent (i.e., computationally-unbounded), but not omniscient. What is the computational complexity of the optimum online? How well can a polynomial-time algorithm approximate it?

Motivated by applications in ride hailing, we study the above questions for the online stochastic maximum-weight matching problem under vertex arrivals. This problem was recently introduced by Ezra, Feldman, Gravin and Tang (EC’20), who gave a 1/2-competitive algorithm for it. This is the best possible ratio, as this problem is a generalization of the original single-item prophet inequality.

We present a polynomial-time algorithm which approximates optimal online within a factor of 0.51—beating the best-possible prophet inequality. At the core of our result are a new linear program formulation, an algorithm that tries to match the arriving vertices in two attempts, and an analysis that bounds the correlation resulting from the second attempts. In contrast, we show that it is PSPACE-hard to approximate this problem within some constant α<1.

Abstract

Nearly three decades ago, Bar-Noy, Motwani and Naor showed that no online edge-coloring algorithm can edge color a graph optimally. Indeed, their work, titled "the greedy algorithm is optimal for on-line edge coloring", shows that the competitive ratio of 2 of the naïve greedy algorithm is best possible online. However, their lower bound required bounded-degree graphs, of maximum degree Δ = O(log n), which prompted them to conjecture that better bounds are possible for higher-degree graphs. While progress has been made towards resolving this conjecture for restricted inputs and arrivals or for random arrival orders, an answer for fully general adversarial arrivals remained elusive.

We resolve this thirty-year-old conjecture in the affirmative, presenting a (1.9+o(1))-competitive online edge coloring algorithm for general graphs of degree Δ=ω(log n) under vertex arrivals. At the core of our results, and of possible independent interest, is a new online algorithm which rounds a fractional bipartite matching x online under vertex arrivals, guaranteeing that each edge e is matched with probability (1/2+c) x_e, for a constant c>0.027.

Abstract

We study the store-and-forward packet routing problem for simultaneous multicasts, in which multiple packets have to be forwarded along given trees as fast as possible.

This is a natural generalization of the seminal work of Leighton, Maggs and Rao, which solved this problem for unicasts, i.e., the case where all trees are paths. They showed the existence of asymptotically optimal O(C + D)-length schedules, where the congestion C is the maximum number of packets sent over an edge and the dilation D is the maximum depth of a tree. This improves over the trivial O(CD) length schedules.

We prove a lower bound for multicasts, which shows that there do not always exist schedules of non-trivial length, o(CD). On the positive side, we construct O(C+D+log^2 n)-length schedules in any n-node network. These schedules are near-optimal, since our lower bound shows that this length cannot be improved to O(C+D) + o(log n).

Abstract

Many recent distributed optimization algorithms achieve existentially-optimal running times which cannot be improved on some pathological worst-case topologies. Still, most networks of interest allow for exponentially faster algorithms. This motivates two questions:

• What network topology parameters determine the complexity of distributed optimization?
• Are there universally-optimal algorithms that are as fast as possible on every topology?

Abstract

We study streaming submodular maximization subject to matching/b-matching constraints (MSM/MSbM), and present improved upper and lower bounds for these problems. On the upper bounds front, we give primal-dual algorithms achieving the following approximation ratios.

• 3+2√2≈5.828 for monotone MSM, improving the previous best ratio of 7.75.
• 4+2√3≈7.464 for non-monotone MSM, improving the previous best ratio of 9.899.
• 3+ϵ for maximum weight b-matching, improving the previous best ratio of 4+ϵ.

Abstract

Nearly thirty years ago, Bar-Noy, Motwani and Naor [IPL'92] conjectured that an online (1+o(1))Δ-edge-coloring algorithm exists for n-node graphs of maximum degree Δ=ω(log n). This conjecture remains open in general, though it was recently proven for bipartite graphs under one-sided vertex arrivals by Cohen et al. [FOCS'19]. In a similar vein, we study online edge coloring under widely-studied relaxations of the online model.

Our main result is in the random-order online model. For this model, Aggarwal et al. [FOCS'03] gave a (1+o(1))Δ-coloring algorithm for multigraphs with Δ=ω(n^2), and Bahmani et al. [SODA'10] gave a 1.26Δ-coloring algorithm for Δ=ω(log n). Whether the number of colors of Aggarwal et al. and the degree bound of Bahmani et al. could be achieved simultaneously (thus resolving the Bar-Noy et al. conjecture under random-order arrivals) remained open. In this work, we answer this open question in the affirmative; we present an online algorithm that, for some constant γ < 1, uses Δ+O(Δ^γ log^1-γ n) = (1+o(1))Δ colors, w.h.p.

Our second result concerns the adversarial online (and even dynamic) model with recourse. Here, the algorithm is allowed to recolor a small number of edges per edge arrival/departure (this number is referred to as recourse). For this problem, a recent algorithm of Duan et al. [SODA'19] yields a (1+ε)Δ-edge-coloring with poly(log n/ε) recourse. In this work, we given an algorithm that achieves the same with poly(1/ε) recourse. In particular, we remove the dependence on n.

Underlying our results is one common offline algorithm, which we show how to implement efficiently for these two online models. Our algorithm, based on the Rödl Nibble Method, is an adaptation of the distributed algorithm of Dubhashi et al. [TCS'98]. The Nibble Method has been highly successful in the distributed edge coloring literature, and we display its usefulness in the context of online algorithms.

Abstract

We study network coding gaps for the problem of makespan minimization of multiple unicasts. In this problem distinct packets at different nodes in a network need to be delivered to a destination specific to each packet, as fast as possible. The network coding gap specifies how much coding packets together in a network can help compared to the more natural approach of routing.

While makespan minimization using routing has been intensely studied for the multiple unicasts problem, no bounds on network coding gaps for this problem are known. We develop new techniques which allow us to upper bound the network coding gap for the makespan of k unicasts, proving this gap is at most polylogarithmic in k. Complementing this result, we show there exist instances of k unicasts for which this coding gap is polylogarithmic in k. Our results also hold for average completion time, and more generally any ell_p norm of completion times.

Abstract

We present a new dynamic matching sparsification scheme. From this scheme we derive a framework for dynamically rounding fractional matchings against adaptive adversaries. Plugging in known dynamic fractional matching algorithms into our framework, we obtain numerous randomized dynamic matching algorithms which work against adaptive adversaries (the first such algorithms, as all previous randomized algorithms for this problem assumed an oblivious adversary). In particular, for any constant ɛ>0, our framework yields (2+ɛ)-approximate algorithms with constant update time or polylog worst-case update time, as well as (2-ɛ)-approximate algorithms in bipartite graphs with arbitrarily-small polynomial update time, with all these algorithms' guarantees holding against adaptive adversaries. All these results achieve polynomially better update time to approximation tradeoffs than previously known to be achievable against adaptive adversaries.

Abstract

The online matching problem was introduced by Karp, Vazirani and Vazirani nearly three decades ago. In that seminal work, they studied this problem in bipartite graphs with vertices arriving only on one side, and presented optimal deterministic and randomized algorithms for this setting. In comparison, more general arrival models, such as edge arrivals and general vertex arrivals, have proven more challenging, and positive results are known only for various relaxations of the problem. In particular, even the basic question of whether randomization allows one to beat the trivially-optimal deterministic competitive ratio of 1/2 for either of these models was open. In this paper, we resolve this question for both these natural arrival models, and show the following.

1. For edge arrivals, randomization does not help | no randomized algorithm is better than 1/2 competitive.
2. For general vertex arrivals, randomization helps | there exists a randomized (1/2+ Ω(1))-competitive online matching algorithm.

Abstract

Vizing’s celebrated theorem asserts that any graph of maximum degree
admits an edge coloring using at most Δ+1 colors. In contrast, Bar-Noy, Motwani and Naor showed over a quarter century ago that the trivial greedy algorithm, which uses 2Δ-1 colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to low-degree graphs, with Δ = O(log n), and they conjectured the existence of online algorithms using
(1+o(1))·Δ colors for Δ = ω(log n). Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS’03 and Bahmani et al., SODA’10).

We resolve the above conjecture for adversarial vertex arrivals in bipartite graphs, for which we present a (1+o(1))·Δ-edge-coloring algorithm for Δ = ω(log n) known a priori. Surprisingly, if Δ is not known ahead of time, we show that no online (e/(e-1)-Ω(1))·Δ-edge-coloring algorithm exists. We then provide an optimal, (e/(e-1)+o(1))·Δ
-edge-coloring algorithm for unknown Δ = ω(log n). Key to our results, and of possible independent interest, is a novel fractional relaxation for edge coloring, for which we present optimal fractional online algorithms and a near-lossless online rounding scheme, yielding our optimal randomized algorithms.

Abstract

We study the minimum-cost metric perfect matching problem under online i.i.d arrivals. We are given a xed metric with a server at each of the points, and then requests arrive online, each drawn independently from a known probability distribution over the points. Each request has to be matched to a free server, with cost equal to the distance. The goal is to minimize the expected total cost of the matching.

Such stochastic arrival models have been widely studied for the maximization variants of the online matching problem; however, the only known result for the minimization problem is a tight O(log n)-competitiveness for the random-order arrival model. This is in contrast with the adversarial model, where an optimal competitive ratio of O(log n) has long been conjectured and remains a tantalizing open question.

In this paper, we show improved results in the i.i.d arrival model. We show how the i.i.d model can be used to give substantially better algorithms: our main result is an O((log log log n)2)- competitive algorithm in this model. Along the way we give a 9-competitive algorithm for the line and tree metrics. Both results imply a strict separation between the i.i.d model and the adversarial and random order models, both for general metrics and these much-studied metrics.

Abstract

In a recent breakthrough, Paz and Schwartzman (SODA'17) presented a single-pass (2+ɛ)-approximation algorithm for the maximum weight matching problem in the semi-streaming model. Their algorithm uses O(n log^2 n) bits of space, for any constant ɛ>0.

We present two simplified and more intuitive analyses, for essentially the same algorithm, which also improve the space complexity to the optimal bound of O(n log n) bits --- this is optimal as the output matching requires \Omega(n log n) bits. Our analyses rely on a simple use of the primal dual method and a simple accounting method.

Abstract

Distributed graph algorithms that separately optimize for either the number of rounds used or the total number of messages sent have been studied extensively. However, algorithms simultaneously efficient with respect to both measures have been elusive. For example, only very recently was it shown that for Minimum Spanning Tree (MST), an optimal message and round complexity is achievable (up to polylog terms) by a single algorithm in the CONGEST model of communication.

In this paper we provide algorithms that are simultaneously round- and message-optimal for a number of well-studied distributed optimization problems. Our main result is such a distributed algorithm for the fundamental primitive of computing simple functions over each part of a graph partition. From this algorithm we derive round- and message-optimal algorithms for multiple problems, including MST, Approximate Min-Cut and Approximate Single Source Shortest Paths, among others. On general graphs all of our algorithms achieve worst-case optimal ˜O(D+sqrt{n}) round complexity and ˜O(m) message complexity. Furthermore, our algorithms require an optimal ˜O(D) rounds and ˜O(n) messages on planar, genus-bounded, treewidth-bounded and pathwidth-bounded graphs.

Abstract

We study the classic Bin Packing problem in a fully-dynamic setting, where new items can arrive and old items may depart. We want algorithms with low asymptotic competitive ratio while repacking items sparingly between updates. Formally, each item i has a movement cost c_i≥0, and we want to use α·OPT bins and incur a movement cost γ·c_i, either in the worst case, or in an amortized sense, for α, γ as small as possible. We call γ the recourse of the algorithm. This is motivated by cloud storage applications, where fully-dynamic Bin Packing models the problem of data backup to minimize the number of disks used, as well as communication incurred in moving file backups between disks. Since the set of files changes over time, we could recompute a solution periodically from scratch, but this would give a high number of disk rewrites, incurring a high energy cost and possible wear and tear of the disks. In this work, we present optimal tradeoffs between number of bins used and number of items repacked, as well as natural extensions of the latter measure.

Abstract

We present a simple randomized reduction from fully-dynamic integral matching algorithms to fully-dynamic “approximately-maximal” fractional matching algorithms. Applying this reduction to the recent fractional matching algorithm of Bhattacharya, Henzinger, and Nanongkai (SODA 2017), we obtain a novel result for the integral problem. Specifically, our main result is a randomized fully-dynamic (2+ɛ)-approximate integral matching algorithm with small polylog worst-case update time. For the (2+ɛ)-approximation regime only a fractional fullydynamic (2+ɛ)-matching algorithm with worst-case polylog update time was previously known, due to Bhattacharya et al. (SODA 2017). Our algorithm is the first algorithm that maintains approximate matchings with worst-case update time better than polynomial, for any constant approximation ratio. As a consequence, we also obtain the first constant-approximate worst-case polylogarithmic update time maximum weight matching algorithm.

Abstract

In this paper we study the classic online matching problem, introduced in the seminal work of Karp, Vazirani and Vazirani (STOC 1990), in regular graphs. For such graphs, an optimal deterministic algorithm as well as efficient algorithms under stochastic input assumptions were known. In this work, we present a novel randomized algorithm with competitive ratio tending to one on this family of graphs, under adversarial arrival order. Our main contribution is a novel algorithm which achieves competitive ratio 1-O(sqrt(log d)/ sqrt(d)) in expectation on d-regular graphs. In contrast, we show that all previously-studied online algorithms have competitive ratio strictly bounded away from one. Moreover, we show the convergence rate of our algorithm’s competitive ratio to one is nearly tight, as no algorithm achieves competitive ratio better than 1-O(1/sqrt(d)). Finally, we show that our algorithm yields a similar competitive ratio with high probability, as well as guaranteeing each vertex a probability of being matched tending to one.

Abstract

Abstract

We present a faster distributed broadcasting primitive for the classical radio network model.

The most basic distributed radio network broadcasting primitive - called Decay - dates back to PODC'87 result of Bar-Yehuda, Goldreich, and Itai. In any radio network with some informed source nodes, running Decay for O(d log n + log^2 n) rounds informs all nodes at most d hops away from a source with high probability. Since 1987 this primitive has been the most important building block for implementing many other functionalities in radio networks. The only improvements to this decades-old algorithm are slight variations due to [Czumaj, Rytter; FOCS'03] and [Kowalski and Pelc, PODC'03] which achieve the same functionality in O(d log(n/d) + log^2 n) rounds. To obtain a speedup from this, d and thus also the network diameter need to be near linear, i.e., larger than n^(1-eps).

Our new distributed primitive spreads messages for d hops in O(d (log n log log n)/(log d) + log^{O(1)} n) rounds with high probability. This improves over Decay for any super-polylogarithmic d = log^{omega(1)} n and achieves near-optimal O(d log log n) running time for d = n^eps. This also makes progress on an open question of Peleg.

Abstract

Motivated by Internet targeted advertising, we address several ad allocation problems. Prior work has established these problems admit no randomized online algorithm better than (1-1/e)-competitive ([Karp et al. 1990; Mehta et al. 2007]), yet simple heuristics have been observed to perform much better in practice. We explain this phenomenon by studying a generalization of the bounded-degree inputs considered by [Buchbinder et al. 2007), graphs which we call (k,d)-bounded. In such graphs the maximal degree on the online side is at most d and the minimal degree on the offline side is at least k. We prove that for such graphs, these problems' natural greedy algorithms attain competitive ratio 1-(d-1)/(k+d-1), tending to one as d/k tends to zero. We prove this bound is tight for these algorithms.

Next, we develop deterministic primal-dual algorithms for the above problems achieving competitive ratio 1-(1-1/d)k>1-1/e^{k/d}, or exponentially better loss as a function of k/d, and strictly better than 1-1/e whenever k ≥ d. We complement our lower bounds with matching upper bounds for the vertex- weighted problem. Finally, we use our deterministic algorithms to prove by dual-fitting that simple randomized algorithms achieve the same bounds in expectation. Our algorithms and analysis differ from previous ad allocation algorithms, which largely scale bids based on the spent fraction of their bidder's budget, whereas we scale bids according to the number of times the bidder could have spent as much as her current bid. Our algorithms differ from previous online primal-dual algorithms, as they do not maintain dual feasibility, but only primal-to-dual ratio, and only attain dual feasibility upon termination. We believe our techniques could find applications to other well- behaved online packing problems.

Abstract

The majority of Web email is known to be generated by machines even when one excludes spam. Many machine-generated email messages such as invoices or travel itineraries are critical to users. Recent research studies establish that causality relations between certain types of machine- generated email messages exist and can be mined. These relations exhibit a link between a given message to a past message that gave rise to its creation. For example, a shipment notification message can often be linked to a past online purchase message. Instead of studying how an incoming message can be linked to the past, we propose here to focus on predicting future email arrival as implied by causality relations. Such a prediction method has several potential applications, ranging from improved ad targeting in up sell scenarios to reducing false positives in spam detection.

We introduce a novel approach for predicting which types of machine-generated email messages, represented by so-called "email templates", a user should receive in future time windows. Our prediction approach relies on (1) statistically inferring causality relations between email templates, (2) building a generative model that explains the inbox of each user using those causality relations, and (3) combining those results to predict which email templates are likely to appear in future time frames. We present preliminary experimental results and some data insights obtained by analyzing several million inboxes of Yahoo Mail users, who voluntarily opted-in for such research.

Abstract

In many computational and economic models of multi-agent interaction, each participant repeatedly "best-responds" to the others' actions. Game theory research on the prominent "best-response dynamics" model typically relies on the premise that the interaction between agents is somehow synchronized. However, in many real-life settings, e.g., internet protocols and large-scale markets, the interaction between participants is asynchronous. We tackle the following important questions: (1) When are best-response dynamics guaranteed to converge to an equilibrium even under asynchrony? (2) What is the (computational and communication) complexity of verifying guaranteed convergence? We show that, in general, verifying guaranteed convergence is intractable. In fact, our main negative result establishes that this task is undecidable. We exhibit, in contrast, positive results for several environments of interest, including complete, computationally-tractable, characterizations of convergent systems. We discuss the algorithmic implications of our results, which extend beyond best-response dynamics to applications such as asynchronous Boolean circuits.