Almost Optimal Streaming Algorithms for Coverage Problems

Maximum coverage and minimum set cover problems --collectively called coverage problems-- have been studied extensively in streaming models. However, previous research not only achieve sub-optimal approximation factors and space complexities, but also study a restricted set arrival model which makes an explicit or implicit assumption on oracle access to the sets, ignoring the complexity of reading and storing the whole set at once. In this paper, we address the above shortcomings, and present algorithms with improved approximation factor and improved space complexity, and prove that our results are almost tight. Moreover, unlike most of previous work, our results hold on a more general edge arrival model. More specifically, we present (almost) optimal approximation algorithms for maximum coverage and minimum set cover problems in the streaming model with an (almost) optimal space complexity of $\tilde{O}(n)$, i.e., the space is {\em independent of the size of the sets or the size of the ground set of elements}. These results not only improve over the best known algorithms for the set arrival model, but also are the first such algorithms for the more powerful {\em edge arrival} model. In order to achieve the above results, we introduce a new general sketching technique for coverage functions: This sketching scheme can be applied to convert an $\alpha$-approximation algorithm for a coverage problem to a (1-\eps)\alpha-approximation algorithm for the same problem in streaming, or RAM models. We show the significance of our sketching technique by ruling out the possibility of solving coverage problems via accessing (as a black box) a (1 \pm \eps)-approximate oracle (e.g., a sketch function) that estimates the coverage function on any subfamily of the sets

Streaming Algorithms for Connectivity Augmentation

We study the $k$-connectivity augmentation problem ($k$-CAP) in the single-pass streaming model. Given a $(k-1)$-edge connected graph $G=(V,E)$ that is stored in memory, and a stream of weighted edges $L$ with weights in $\{0,1,\dots,W\}$, the goal is to choose a minimum weight subset $L'\subseteq L$ such that $G'=(V,E\cup L')$ is $k$-edge connected. We give a $(2+\epsilon)$-approximation algorithm for this problem which requires to store $O(\epsilon^{-1} n\log n)$ words. Moreover, we show our result is tight: Any algorithm with better than $2$-approximation for the problem requires $\Omega(n^2)$ bits of space even when $k=2$. This establishes a gap between the optimal approximation factor one can obtain in the streaming vs the offline setting for $k$-CAP. We further consider a natural generalization to the fully streaming model where both $E$ and $L$ arrive in the stream in an arbitrary order. We show that this problem has a space lower bound that matches the best possible size of a spanner of the same approximation ratio. Following this, we give improved results for spanners on weighted graphs: We show a streaming algorithm that finds a $(2t-1+\epsilon)$-approximate weighted spanner of size at most $O(\epsilon^{-1} n^{1+1/t}\log n)$ for integer $t$, whereas the best prior streaming algorithm for spanner on weighted graphs had size depending on $\log W$. Using our spanner result, we provide an optimal $O(t)$-approximation for $k$-CAP in the fully streaming model with $O(nk + n^{1+1/t})$ words of space. Finally we apply our results to network design problems such as Steiner tree augmentation problem (STAP), $k$-edge connected spanning subgraph ($k$-ECSS), and the general Survivable Network Design problem (SNDP). In particular, we show a single-pass $O(t\log k)$-approximation for SNDP using $O(kn^{1+1/t})$ words of space, where $k$ is the maximum connectivity requirement

Robust Communication Complexity of Matching: EDCS Achieves 5/6 Approximation

We study the robust communication complexity of maximum matching. Edges of an arbitrary $n$-vertex graph $G$ are randomly partitioned between Alice and Bob independently and uniformly. Alice has to send a single message to Bob such that Bob can find an (approximate) maximum matching of the whole graph $G$. We specifically study the best approximation ratio achievable via protocols where Alice communicates only $\widetilde{O}(n)$ bits to Bob. There has been a growing interest on the robust communication model due to its connections to the random-order streaming model. An algorithm of Assadi and Behnezhad [ICALP'21] implies a $(2/3+\epsilon_0 \sim .667)$-approximation for a small constant $0 < \epsilon_0 < 10^{-18}$, which remains the best-known approximation for general graphs. For bipartite graphs, Assadi and Behnezhad [Random'21] improved the approximation to .716 albeit with a computationally inefficient (i.e., exponential time) protocol. In this paper, we study a natural and efficient protocol implied by a random-order streaming algorithm of Bernstein [ICALP'20] which is based on edge-degree constrained subgraphs (EDCS) [Bernstein and Stein; ICALP'15]. The result of Bernstein immediately implies that this protocol achieves an (almost) $(2/3 \sim .666)$-approximation in the robust communication model. We present a new analysis, proving that it achieves a much better (almost) $(5/6 \sim .833)$-approximation. This significantly improves previous approximations both for general and bipartite graphs. We also prove that our analysis of Bernstein's protocol is tight

AFQN: approximate Qn estimation in data streams

We present afqn (Approximate Fast Qn), a novel algorithm for approximate computation of the Qn scale estimator in a streaming setting, in the sliding window model. It is well-known that computing the Qn estimator exactly may be too costly for some applications, and the problem is a fortiori exacerbated in the streaming setting, in which the time available to process incoming data stream items is short. In this paper we show how to efficiently and accurately approximate the Qn estimator. As an application, we show the use of afqn for fast detection of outliers in data streams. In particular, the outliers are detected in the sliding window model, with a simple check based on the Qn scale estimator. Extensive experimental results on synthetic and real datasets confirm the validity of our approach by showing up to three times faster updates per second. Our contributions are the following ones: (i) to the best of our knowledge, we present the first approximation algorithm for online computation of the Qn scale estimator in a streaming setting and in the sliding window model; (ii) we show how to take advantage of our UDDSketch algorithm for quantile estimation in order to quickly compute the Qn scale estimator; (iii) as an example of a possible application of the Qn scale estimator, we discuss how to detect outliers in an input data stream

Submodular Secretary Problem with Shortlists under General Constraints

In submodular k-secretary problem, the goal is to select k items in a randomly ordered input so as to maximize the expected value of a given monotone submodular function on the set of selected items. In this paper, we introduce a relaxation of this problem, which we refer to as submodular k-secretary problem with shortlists. In the proposed problem setting, the algorithm is allowed to choose more than k items as part of a shortlist. Then, after seeing the entire input, the algorithm can choose a subset of size k from the bigger set of items in the shortlist. We are interested in understanding to what extent this relaxation can improve the achievable competitive ratio for the submodular k-secretary problem. In particular, using an O(k) shortlist, can an online algorithm achieve a competitive ratio close to the best achievable online approximation factor for this problem? We answer this question affirmatively by giving a polynomial time algorithm that achieves a 1 - 1/e - epsilon -O(k^{-1}) competitive ratio for any constant epsilon>0, using a shortlist of size eta {epsilon}(k)=O(k). Also, for the special case of m-submodular functions, we demonstrate an algorithm that achieves a 1 - epsilon competitive ratio for any constant epsilon > 0, using an O(1) shortlist. Finally, we show that our algorithm can be implemented in the streaming setting using a memory buffer of size eta{epsilon}(k)=O(k) to achieve a 1 - 1/e - epsilon - O(k^{-1}) approximation for submodular function maximization in the random order streaming model. This substantially improves upon the previously best known approximation factor of 1/2 + 8*10^{-14} [Norouzi-Fard et al. 2018] that used a memory buffer of size O(k log k). We further generalize our results to the case of matroid constraints. We design an algorithm that achieves a 1/2(1 - 1/e^2 - epsilon - O(1/k)) competitive ratio for any constant epsilon>0, using a shortlist of size O(k). This is especially surprising considering that the best known competitive ratio for the matroid secretary problem is O(log log k). An important application of our algorithm is for the random order streaming of submodular functions. We show that our algorithm can be implemented in the streaming setting using O(k) memory. It achieves a 1/2 (1 - 1/e^2 - epsilon - O(1/k)) approximation. The previously best known approximation ratio for streaming submodular maximization under matroid constraint is 0.25 (adversarial order) due to [Feldman et al.], [Chekuri et al.], and [Chakrabarti et al.]. Moreover, we generalize our results to the case of p-matchoid constraints and give a frac{1}{p+1}(1 - 1/e^{p+1} - epsilon - O(1/k)) approximation using O(k) memory, which asymptotically approaches the best known offline guarantee frac{1}{p+1} [Nemhauser et al.]. Finally we empirically evaluate our results on real world data sets such as YouTube video and Twitter stream