We consider the classic online problem of scheduling on a single machine to minimize total flow time. In STOC 2021, the concept of robustness to distortion in processing times was introduced: for every distortion factor $\mu$, an $O(\mu^2)$-competitive algorithm $\operatorname{ALG}_{\mu}$ which handles distortions up to $\mu$ was presented. However, using that result requires one to know the distortion of the input in advance, which is impractical. We present the first \emph{distortion-oblivious} algorithms: algorithms which are competitive for \emph{every} input of \emph{every} distortion, and thus do not require knowledge of the distortion in advance. Moreover, the competitive ratios of our algorithms are $\tilde{O}(\mu)$, which is a quadratic improvement over the algorithm from STOC 2021, and is nearly optimal (we show a randomized lower bound of $\Omega(\mu)$ on competitiveness).
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We propose a deterministic Kaczmarz algorithm for solving linear systems $A\x=\b$. Different from previous Kaczmarz algorithms, we use reflections in each step of the iteration. This generates a series of points distributed with patterns on a sphere centered at a solution. Firstly, we prove that taking the average of $O(\eta/\epsilon)$ points leads to an effective approximation of the solution up to relative error $\epsilon$, where $\eta$ is a parameter depending on $A$ and can be bounded above by the square of the condition number. We also show how to select these points efficiently. From the numerical tests, our Kaczmarz algorithm usually converges more quickly than the (block) randomized Kaczmarz algorithms. Secondly, when the linear system is consistent, the Kaczmarz algorithm returns the solution that has the minimal distance to the initial vector. This gives a method to solve the least-norm problem. Finally, we prove that our Kaczmarz algorithm indeed solves the linear system $A^TW^{-1}A \x = A^TW^{-1} \b$, where $W$ is the low-triangular matrix such that $W+W^T=2AA^T$. The relationship between this linear system and the original one is studied.
In 1961, Gomory and Hu showed that the max-flow values of all $n\choose 2$ pairs of vertices in an undirected graph can be computed using only $n-1$ calls to any max-flow algorithm. Even assuming a linear-time max-flow algorithm, this yields a running time of $O(n^3)$ for this problem. We break this 60-year old barrier by giving an $\tilde{O}(n^{23/8})$-time algorithm for the Gomory-Hu tree problem. Our result is unconditional, i.e., it does not rely on a linear-time max-flow algorithm.
A Level Ancestory query LA($u$, $d$) asks for the the ancestor of the node $u$ at a depth $d$. We present a simple solution, which pre-processes the tree in $O(n)$ time with $O(n)$ extra space, and answers the queries in $O(\log\ {n})$ time. Though other optimal algorithms exist, this is a simple enough solution that could be taught and implemented easily.
We study revenue maximization in multi-item multi-bidder auctions under the natural item-independence assumption - a classical problem in Multi-Dimensional Bayesian Mechanism Design. One of the biggest challenges in this area is developing algorithms to compute (approximately) optimal mechanisms that are not brute-force in the size of the bidder type space, which is usually exponential in the number of items in multi-item auctions. Unfortunately, such algorithms were only known for basic settings of our problem when bidders have unit-demand [CHMS10,CMS15] or additive valuations [Yao15]. In this paper, we significantly improve the previous results and design the first algorithm that runs in time polynomial in the number of items and the number of bidders to compute mechanisms that are $O(1)$-approximations to the optimal revenue when bidders have XOS valuations, resolving the open problem raised in [CM16,CZ17]. Moreover, the computed mechanism has a simple structure: It is either a posted price mechanism or a two-part tariff mechanism. As a corollary of our result, we show how to compute an approximately optimal and simple mechanism efficiently using only sample access to the bidders' value distributions. Our algorithm builds on two innovations that allow us to search over the space of mechanisms efficiently: (i) a new type of succinct representation of mechanisms - the marginal reduced forms, and (ii) a novel Lift-and-Round procedure that concavifies the problem.
We study the problem of \emph{dynamic regret minimization} in $K$-armed Dueling Bandits under non-stationary or time varying preferences. This is an online learning setup where the agent chooses a pair of items at each round and observes only a relative binary `win-loss' feedback for this pair, sampled from an underlying preference matrix at that round. We first study the problem of static-regret minimization for adversarial preference sequences and design an efficient algorithm with $O(\sqrt{KT})$ high probability regret. We next use similar algorithmic ideas to propose an efficient and provably optimal algorithm for dynamic-regret minimization under two notions of non-stationarities. In particular, we establish $\tO(\sqrt{SKT})$ and $\tO({V_T^{1/3}K^{1/3}T^{2/3}})$ dynamic-regret guarantees, $S$ being the total number of `effective-switches' in the underlying preference relations and $V_T$ being a measure of `continuous-variation' non-stationarity. The complexity of these problems have not been studied prior to this work despite the practicability of non-stationary environments in real world systems. We justify the optimality of our algorithms by proving matching lower bound guarantees under both the above-mentioned notions of non-stationarities. Finally, we corroborate our results with extensive simulations and compare the efficacy of our algorithms over state-of-the-art baselines.
We consider the problem of controlling a Linear Quadratic Regulator (LQR) system over a finite horizon $T$ with fixed and known cost matrices $Q,R$, but unknown and non-stationary dynamics $\{A_t, B_t\}$. The sequence of dynamics matrices can be arbitrary, but with a total variation, $V_T$, assumed to be $o(T)$ and unknown to the controller. Under the assumption that a sequence of stabilizing, but potentially sub-optimal controllers is available for all $t$, we present an algorithm that achieves the optimal dynamic regret of $\tilde{\mathcal{O}}\left(V_T^{2/5}T^{3/5}\right)$. With piece-wise constant dynamics, our algorithm achieves the optimal regret of $\tilde{\mathcal{O}}(\sqrt{ST})$ where $S$ is the number of switches. The crux of our algorithm is an adaptive non-stationarity detection strategy, which builds on an approach recently developed for contextual Multi-armed Bandit problems. We also argue that non-adaptive forgetting (e.g., restarting or using sliding window learning with a static window size) may not be regret optimal for the LQR problem, even when the window size is optimally tuned with the knowledge of $V_T$. The main technical challenge in the analysis of our algorithm is to prove that the ordinary least squares (OLS) estimator has a small bias when the parameter to be estimated is non-stationary. Our analysis also highlights that the key motif driving the regret is that the LQR problem is in spirit a bandit problem with linear feedback and locally quadratic cost. This motif is more universal than the LQR problem itself, and therefore we believe our results should find wider application.
A $(1+\epsilon)$-approximate distance oracle of an edge-weighted graph is a data structure that returns an approximate shortest path distance between any two query vertices up to a $(1+\epsilon)$ factor. Thorup (FOCS 2001, JACM 2004) and Klein (SODA 2002) independently constructed a $(1+\epsilon)$-approximate distance oracle with $O(n\log n)$ space, measured in number of words, and $O(1)$ query time when $G$ is an undirected planar graph with $n$ vertices and $\epsilon$ is a fixed constant. Many follow-up works gave $(1+\epsilon)$-approximate distance oracles with various trade-offs between space and query time. However, improving $O(n\log n)$ space bound without sacrificing query time remains an open problem for almost two decades. In this work, we resolve this problem affirmatively by constructing a $(1+\epsilon)$-approximate distance oracle with optimal $O(n)$ space and $O(1)$ query time for undirected planar graphs and fixed $\epsilon$. We also make substantial progress for planar digraphs with non-negative edge weights. For fixed $\epsilon > 0$, we give a $(1+\epsilon)$-approximate distance oracle with space $o(n\log(Nn))$ and $O(\log\log(Nn)$ query time; here $N$ is the ratio between the largest and smallest positive edge weight. This improves Thorup's (FOCS 2001, JACM 2004) $O(n\log(Nn)\log n)$ space bound by more than a logarithmic factor while matching the query time of his structure. This is the first improvement for planar digraphs in two decades, both in the weighted and unweighted setting.
We study streaming algorithms for two fundamental geometric problems: computing the cost of a Minimum Spanning Tree (MST) of an $n$-point set $X \subset \{1,2,\dots,\Delta\}^d$, and computing the Earth Mover Distance (EMD) between two multi-sets $A,B \subset \{1,2,\dots,\Delta\}^d$ of size $n$. We consider the turnstile model, where points can be added and removed. We give a one-pass streaming algorithm for MST and a two-pass streaming algorithm for EMD, both achieving an approximation factor of $\tilde{O}(\log n)$ and using polylog$(n,d,\Delta)$-space only. Furthermore, our algorithm for EMD can be compressed to a single pass with a small additive error. Previously, the best known sublinear-space streaming algorithms for either problem achieved an approximation of $O(\min\{ \log n , \log (\Delta d)\} \log n)$ [Andoni-Indyk-Krauthgamer '08, Backurs-Dong-Indyk-Razenshteyn-Wagner '20]. For MST, we also prove that any constant space streaming algorithm can only achieve an approximation of $\Omega(\log n)$, analogous to the $\Omega(\log n)$ lower bound for EMD of [Andoni-Indyk-Krauthgamer '08]. Our algorithms are based on an improved analysis of a recursive space partitioning method known generically as the Quadtree. Specifically, we show that the Quadtree achieves an $\tilde{O}(\log n)$ approximation for both EMD and MST, improving on the $O(\min\{ \log n , \log (\Delta d)\} \log n)$ approximation of [Andoni-Indyk-Krauthgamer '08, Backurs-Dong-Indyk-Razenshteyn-Wagner '20].
In the minimum $k$-cut problem, we want to find the minimum number of edges whose deletion breaks the input graph into at least $k$ connected components. The classic algorithm of Karger and Stein runs in $\tilde O(n^{2k-2})$ time, and recent, exciting developments have improved the running time to $O(n^k)$. For general, weighted graphs, this is tight assuming popular hardness conjectures. In this work, we show that perhaps surprisingly, $O(n^k)$ is not the right answer for simple, unweighted graphs. We design an algorithm that runs in time $O(n^{(1-\epsilon)k})$ where $\epsilon>0$ is an absolute constant, breaking the natural $n^k$ barrier. This establishes a separation of the two problems in the unweighted and weighted cases.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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