arXiv subjects

David Wajc

Publications and source records attributed to David Wajc.

35 records · Page 2Linked to original sources

arXiv2021

Beating the Folklore Algorithm for Dynamic Matching

The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence $2$-approximate) matching in $O(n)$ worst-case update time in $n$-node graphs. We present the first deterministic algorithm which outperforms the folklore algorithm in terms of {\em both} approximation ratio and worst-case update time. Specifically, we give a $(2-Ω(1))$-approximate algorithm with $O(m^{3/8})=O(n^{3/4})$ worst-case update time in $n$-node, $m$-edge graphs. For sufficiently small constant $ε>0$, no deterministic $(2+ε)$-approximate algorithm with worst-case update time $O(n^{0.99})$ was known. Our second result is the first deterministic $(2+ε)$-approximate weighted matching algorithm with $O_ε(1)\cdot O(\sqrt[4]{m}) = O_ε(1)\cdot O(\sqrt{n})$ worst-case update time. Our main technical contributions are threefold: first, we characterize the tight cases for \emph{kernels}, which are the well-studied matching sparsifiers underlying much of the $(2+ε)$-approximate dynamic matching literature. This characterization, together with multiple ideas -- old and new -- underlies our result for breaking the approximation barrier of $2$. Our second technical contribution is the first example of a dynamic matching algorithm whose running time is improved due to improving the \emph{recourse} of other dynamic matching algorithms. Finally, we show how to use dynamic bipartite matching algorithms as black-box subroutines for dynamic matching in general graphs without incurring the natural $\frac{3}{2}$ factor in the approximation ratio which such approaches naturally incur.

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arXiv2021

Online Stochastic Max-Weight Bipartite Matching: Beyond Prophet Inequalities

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? We study the above questions for the online stochastic maximum-weight matching problem under vertex arrivals. For this problem, a number of $1/2$-competitive algorithms are known against optimum offline. This is the best possible ratio for this problem, as it generalizes the original single-item prophet inequality problem. We present a polynomial-time algorithm which approximates the optimal online algorithm within a factor of $0.51$ -- beating the best-possible prophet inequality. In contrast, we show that it is PSPACE-hard to approximate this problem within some constant $α< 1$.

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arXiv2021

The Stationary Prophet Inequality Problem

We study a continuous and infinite time horizon counterpart to the classic prophet inequality, which we term the stationary prophet inequality problem. Here, copies of a good arrive and perish according to Poisson point processes. Buyers arrive similarly and make take-it-or-leave-it offers for unsold items. The objective is to maximize the (infinite) time average revenue of the seller. Our main results are pricing-based policies which (i) achieve a $1/2$-approximation of the optimal offline policy, which is best possible, and (ii) achieve a better than $(1-1/e)$-approximation of the optimal online policy. Result (i) improves upon bounds implied by recent work of Collina et al. (WINE'20), and is the first optimal prophet inequality for a stationary problem. Result (ii) improves upon a $1-1/e$ bound implied by recent work of Aouad and Saritaç (EC'20), and shows that this prevalent bound in online algorithms is not optimal for this problem.

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arXiv2021

The Greedy Algorithm is \emph{not} Optimal for On-Line Edge Coloring

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 \emph{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 $(1/2+c)\cdot x_e$, for a constant $c>0.027$.

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arXiv2021

Near-Optimal Schedules for Simultaneous Multicasts

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 $O(C+D) + o(\log n)$.

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arXiv2021

Streaming Submodular Matching Meets the Primal-Dual Method

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. $\bullet$ $3+2\sqrt{2}\approx 5.828$ for monotone MSM, improving the previous best ratio of $7.75$. $\bullet$ $4+3\sqrt{2}\approx 7.464$ for non-monotone MSM, improving the previous best ratio of $9.899$. $\bullet$ $3+ε$ for maximum weight b-matching, improving the previous best ratio of $4+ε$. On the lower bounds front, we improve on the previous best lower bound of $\frac{e}{e-1}\approx 1.582$ for MSM, and show ETH-based lower bounds of $\approx 1.914$ for polytime monotone MSM streaming algorithms. Our most substantial contributions are our algorithmic techniques. We show that the (randomized) primal-dual method, which originated in the study of maximum weight matching (MWM), is also useful in the context of MSM. To our knowledge, this is the first use of primal-dual based analysis for streaming submodular optimization. We also show how to reinterpret previous algorithms for MSM in our framework; hence, we hope our work is a step towards unifying old and new techniques for streaming submodular maximization, and that it paves the way for further new results.

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arXiv2020

Online Edge Coloring Algorithms via the Nibble Method

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 \emph{one-sided vertex arrivals} by Cohen et al.~[FOCS'19]. In a similar vein, we study edge coloring under widely-studied relaxations of the online model. Our main result is in the \emph{random-order} online model. For this model, known results fall short of the Bar-Noy et al.~conjecture, either in the degree bound [Aggarwal et al.~FOCS'03], or number of colors used [Bahmani et al.~SODA'10]. We achieve the best of both worlds, thus resolving the Bar-Noy et al.~conjecture in the affirmative for this model. Our second result is in the adversarial online (and dynamic) model with \emph{recourse}. A recent algorithm of Duan et al.~[SODA'19] yields a $(1+ε)Δ$-edge-coloring with poly$(\log n/ε)$ recourse. We achieve the same with poly$(1/ε)$ recourse, thus removing all dependence on $n$. Underlying our results is one common offline algorithm, which we show how to implement in 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 proven successful for distributed edge coloring. We display its usefulness in the context of online algorithms.

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arXiv2020

Rounding Dynamic Matchings Against an Adaptive Adversary

We present a new dynamic matching sparsification scheme. From this scheme we derive a framework for dynamically rounding fractional matchings against \emph{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 \emph{oblivious} adversary). In particular, for any constant $ε>0$, our framework yields $(2+ε)$-approximate algorithms with constant update time or polylog worst-case update time, as well as $(2-δ)$-approximate algorithms in bipartite graphs with arbitrarily-small polynomial update time, with all these algorithms' guarantees holding against adaptive adversaries. All these results achieve \emph{polynomially} better update time to approximation tradeoffs than previously known to be achievable against adaptive adversaries.

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arXiv2020

Network Coding Gaps for Completion Times of Multiple Unicasts

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.

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arXiv2019

Tight Bounds for Online Edge Coloring

Vizing's celebrated theorem asserts that any graph of maximum degree $Δ$ admits an edge coloring using at most $Δ+1$ colors. In contrast, Bar-Noy, Naor and Motwani showed over a quarter century 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 $Δ(1+o(1))$ 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 \emph{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 $\big(\frac{e}{e-1} - Ω(1) \big) Δ$-edge-coloring algorithm exists. We then provide an optimal, $\big(\frac{e}{e-1}+o(1)\big)Δ$-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.

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arXiv2019

Stochastic Online Metric Matching

We study the minimum-cost metric perfect matching problem under online i.i.d arrivals. We are given a fixed 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.

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arXiv2019

Online Matching with General Arrivals

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 $\frac{1}{2}$ 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 $\frac{1}{2}$ competitive. 2. For general vertex arrivals, randomization helps --- there exists a randomized $(\frac{1}{2}+Ω(1))$-competitive online matching algorithm.

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arXiv2018

Simplified and Space-Optimal Semi-Streaming for $(2+ε)$-Approximate Matching

In a recent breakthrough, Paz and Schwartzman (SODA'17) presented a single-pass ($2+ε$)-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 $ε>0$. 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 $Ω(n\log n)$ bits. Our analyses rely on a simple use of the primal-dual method and a simple accounting method.

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arXiv2018

Round- and Message-Optimal Distributed Graph Algorithms

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 $\tilde{O}(D+\sqrt n)$ round complexity and $\tilde{O}(m)$ message complexity. Furthermore, our algorithms require an optimal $\tilde{O}(D)$ rounds and $\tilde{O}(n)$ messages on planar, genus-bounded, treewidth-bounded and pathwidth-bounded graphs.

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arXiv2018

Fully-Dynamic Bin Packing with Limited Repacking

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 \emph{while repacking items sparingly} between updates. Formally, each item $i$ has a \emph{movement cost} $c_i\geq 0$, and we want to use $α\cdot OPT$ bins and incur a movement cost $γ\cdot c_i$, either in the worst case, or in an amortized sense, for $α, γ$ as small as possible. We call $γ$ the \emph{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.

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arXiv2018

Dynamic Matching: Reducing Integral Algorithms to Approximately-Maximal Fractional Algorithms

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 $(2+ε)$-approximate integral matching algorithm with small polylog worst-case update time. For the $(2+ε)$-approximation regime only a \emph{fractional} fully-dynamic $(2+ε)$-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.

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arXiv2015

Near-Optimum Online Ad Allocation for Targeted Advertising

Motivated by Internet targeted advertising, we address several ad allocation problems. Prior work has established these problems admit no randomized online algorithm better than $(1-\frac{1}{e})$-competitive (\cite{karp1990optimal,mehta2007adwords}), 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.~\cite{buchbinder2007online}, 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 $1-\frac{d-1}{k+d-1}$, tending to \emph{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 $1-(1-\frac{1}{d})^k>1-\frac{1}{e^{k/d}}$, or \emph{exponentially} better loss as a function of $k/d$, and strictly better than $1-\frac{1}{e}$ whenever $k\geq d$. 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.

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