A retailer is purchasing goods in bundles from suppliers and then selling these goods in bundles to customers; her goal is to maximize profit, which is the revenue obtained from selling goods minus the cost of purchasing those goods. In this paper, we study this general trading problem from the retailer's perspective, where both suppliers and customers arrive online. The retailer has inventory constraints on the number of goods from each type that she can store, and she must decide upon arrival of each supplier/customer which goods to buy/sell in order to maximize profit. We design an algorithm with logarithmic competitive ratio compared to an optimal offline solution. We achieve this via an exponential-weight-update dynamic pricing scheme, and our analysis dual fits the retailer's profit with respect to a linear programming formulation upper bounding the optimal offline profit. We prove (almost) matching lower bounds, and we also extend our result to an incentive compatible mechanism. Prior to our work, algorithms for trading bundles were known only for the special case of selling an initial inventory.

In fully-dynamic consistent clustering, we are given a finite metric space $(M,d)$, and a set $F\subseteq M$ of possible locations for opening centers. Data points arrive and depart, and the goal is to maintain an approximately optimal clustering solution at all times while minimizing the recourse, the total number of additions/deletions of centers over time. Specifically, we study fully dynamic versions of the classical $k$-center, facility location, and $k$-median problems. We design algorithms that, given a parameter $\beta\geq 1$, maintain an $O(\beta)$-approximate solution at all times, and whose total recourse is bounded by $O(\log |F| \log \Delta) \cdot OPT_{rec}^{\beta}$. Here $OPT_{rec}^{\beta}$ is the minimal recourse of an offline algorithm that maintains a $\beta$-approximate solution at all times, and $\Delta$ is the metric aspect ratio. We obtain our results via a reduction to the recently proposed Positive Body Chasing framework of [Bhattacharya Buchbinder Levin Saranurak, FOCS 2023], which we show gives fractional solutions to our clustering problems online. Our contribution is to round these fractional solutions while preserving the approximation and recourse guarantees. We complement our positive results with logarithmic lower bounds which show that our bounds are nearly tight.

In the stochastic set cover problem (Grandoni et al., FOCS ‘08), we are given a collection S of m sets over a universe U of size N, and a distribution D over elements of U. The algorithm draws n elements one-by-one from D and must buy a set to cover each element on arrival; the goal is to minimize the total cost of sets bought during this process. A universal algorithm a priori maps each element u ∈ U to a set S(u) such that if U ⊆ U is formed by drawing n times from distribution D, then the algorithm commits to outputting S(U). Grandoni et al. gave an O(log mN)-competitive universal algorithm for this stochastic set cover problem.

We study the problem of chasing positive bodies in $\ell_{1}$: given a sequence of bodies $K_{t}=\left\{x^{t} \in \mathbb{R}_{+}^{n} \mid C^{t} x^{t} \geq 1, P^{t} x^{t} \leq 1\right\}$ revealed online, where $C^{t}$ and $P^{t}$ are nonnegative matrices, the goal is to (approximately) maintain a point $x_{t} \in K_{t}$ such that $\sum_{t}\left\|x_{t}-x_{t-1}\right\|_{1}$ is minimized. This captures the fully-dynamic low-recourse variant of any problem that can be expressed as a mixed packing-covering linear program and thus also the fractional version of many central problems in dynamic algorithms such as set cover, load balancing, hyperedge orientation, minimum spanning tree, and matching.We give an $O(\log d)$-competitive algorithm for this problem, where d is the maximum row sparsity of any matrix $C^{t}$. This bypasses and improves exponentially over the lower bound of $\sqrt{n}$ known for general convex bodies. Our algorithm is based on iterated information projections, and, in contrast to general convex body chasing algorithms, is entirely memoryless.We also show how to round our solution dynamically to obtain the first fully dynamic algorithms with competitive recourse for all the stated problems above; i.e. their recourse is less than the recourse of every other algorithm on every update sequence, up to polylogarithmic factors. This is a significantly stronger notion than the notion of absolute recourse in the dynamic algorithms literature.

Motivated by the design of real system storage hierarchies, we study the block-aware caching problem, a generalization of classic caching in which fetching (or evicting) pages from the same block incurs the same cost as fetching (or evicting) just one page from the block. Given a cache of size k, and a sequence of requests from n pages partitioned into given blocks of size β ≤ k, the goal is to minimize the total cost of fetching to (or evicting from) cache. This problem captures generalized caching as a special case, which is already NP-hard offline. We show the following suite of results:

We give a polynomial-time algorithm for Online-SetCover with a competitive ratio of$O(\log mn)$when the elements are revealed in random order, matching the best possible offline bound of$O(\log n)$when the number of sets$m$is polynomial in the number of elements$n$, and circumventing the$\Omega(\log m \log n)$lower bound known in adversarial order. We also extend the result to solving pure covering IPs when constraints arrive in random order. The algorithm is a multiplicative-weights-based round-and-solve approach we call LearnOrCover. We maintain a coarse fractional solution that is neither feasible nor monotone increasing, but can nevertheless be rounded online to achieve the claimed guarantee (in the random order model). This gives a new offline algorithm for Setcover that performs a single pass through the elements, which may be of independent interest.

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 primaldual algorithms achieving the following approximation ratios. for monotone MSM, improving the previous best ratio of 7.75. for non-monotone MSM, improving the previous best ratio of 9.899. for maximum weight b-matching, improving the previous best ratio of 4 + ∊. On the lower bounds front, we improve on the previous best lower bound of for MSM, and show ETH-based lower bounds of ≈ 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.

In submodular covering problems, we are given a monotone, nonnegative submodular function f:2N→ R+and wish to find the min-cost set S ⊆ N such that f(S)=f(N). When f is a coverage function, this captures Setcover as a special case. We introduce a general framework for solving such problems in a fully-dynamic setting where the function f changes over time, and only a bounded number of updates to the solution (a.k.a. recourse) is allowed. For concreteness, suppose a nonnegative monotone submodular integer-valued function gt is added or removed from an active set G(t)at each time t. If f(t)=Σ(g∈G(t)g) is the sum of all active functions, we wish to maintain a competitive solution to Submodularcover for f(t)as this active set changes, and with low recourse. For example, if each gt is the (weighted) rank function of a matroid, we would be dynamically maintaining a low-cost common spanning set for a changing collection of matroids. We give an algorithm that maintains an O(log(fmax/fmin)) - competitive solution, where fmax, fminare the largest/smallest marginals of f(t). The algorithm guarantees a total recourse of O(log(cmax/cmin)·Σt≤Tgt(N)), where cmax, cminare the largest/smallest costs of elements in N. This competitive ratio is best possible even in the offline setting, and the recourse bound is optimal up to the logarithmic factor. For monotone sub-modular functions that also have positive mixed third derivatives, we show an optimal recourse bound of O(Σt≤Tgt(N)). This structured class includes set-coverage functions, so our algorithm matches the known O(log n)-competitiveness and O(1) recourse guarantees for fully-dynamic Setcover. Our work simultaneously simplifies and unifies previous results, as well as generalizes to a significantly larger class of covering problems. Our key technique is a new potential function inspired by Tsallis entropy. We also extensively use the idea of Mutual Coverage, which generalizes the classic notion of mutual information.

Say that we are given samples from a distribution $ψ$ over an $n$-dimensional space. We expect or desire $ψ$ to behave like a product distribution (or a $k$-wise independent distribution over its marginals for small $k$). We propose the problem of enumerating/list-decoding all large subcubes where the distribution $ψ$ deviates markedly from what we expect; we refer to such subcubes as skewed subcubes. Skewed subcubes are certificates of dependencies between small subsets of variables in $ψ$. We motivate this problem by showing that it arises naturally in the context of algorithmic fairness and anomaly detection. In this work we focus on the special but important case where the space is the Boolean hypercube, and the expected marginals are uniform. We show that the obvious definition of skewed subcubes can lead to intractable list sizes, and propose a better definition of a minimal skewed subcube, which are subcubes whose skew cannot be attributed to a larger subcube that contains it. Our main technical contribution is a list-size bound for this definition and an algorithm to efficiently find all such subcubes. Both the bound and the algorithm rely on Fourier-analytic techniques, especially the powerful hypercontractive inequality. On the lower bounds side, we show that finding skewed subcubes is as hard as the sparse noisy parity problem, and hence our algorithms cannot be improved on substantially without a breakthrough on this problem which is believed to be intractable. Motivated by this, we study alternate models allowing query access to $ψ$ where finding skewed subcubes might be easier.

In the submodular cover problem, we are given a monotone submodular function f: 2N → ℝ+, and we want to pick the min-cost set S such that f (S) = f (N). This captures the set cover problem when f is a coverage function. Motivated by problems in network monitoring and resource allocation, we consider the submodular cover problem in an online setting. As a concrete example, suppose at each time t, a nonnegative monotone submodular function gt is given to us. We define as the sum of all functions seen so far. We need to maintain a submodular cover of these submodular functions f(1), f(2), … f(T) in an online fashion; i.e., we cannot revoke previous choices. Formally, at each time t we produce a set St ⊆ N such that f(t)(St) = f(t)(N)—i.e., this set St is a cover—such that St–1 ⊆ St, so previously decisions to pick elements cannot be revoked. (We actually allow more general sequences {f(t)} of submodular functions, but this sum-of-simpler-submodular-functions case is useful for concreteness.) We give polylogarithmic competitive algorithms for this online submodular cover problem. The competitive ratio on an input sequence of length T is O(ln n ln(T · fmax/fmin)), where fmax and fmin are the largest and smallest marginals for functions f(t), and |N| = n. For the special case of online set cover, our competitive ratio matches that of Alon et al. [AAA+09], which are best possible for polynomial-time online algorithms unless NP ⊆ BPP [Kor04]. Since existing offline algorithms for submodular cover are based on greedy approaches which seem difficult to implement online, the technical challenge is to (approximately) solve the exponential-sized linear programming relaxation for submodular cover, and to round it, both in the online setting. Moreover, to get our competitiveness bounds, we define a (seemingly new) generalization of mutual information to general submodular functions, which we call mutual coverage; we hope this will be useful in other contexts.

We study robust subspace estimation in the streaming and distributed settings. Given a set of n data points {ai}{i=1}^n in R^d and an integer k, we wish to find a linear subspace S of dimension k for which sumi M(dist(S, ai)) is minimized, where dist(S,x) := min_{y in S} |x-y|_2, and M() is some loss function. When M is the identity function, S gives a subspace that is more robust to outliers than that provided by the truncated SVD. Though the problem is NP-hard, it is approximable within a (1+epsilon) factor in polynomial time when k and epsilon are constant. We give the first sublinear approximation algorithm for this problem in the turnstile streaming and arbitrary partition distributed models, achieving the same time guarantees as in the offline case. Our algorithm is the first based entirely on oblivious dimensionality reduction, and significantly simplifies prior methods for this problem, which held in neither the streaming nor distributed models.

We present an approach for answering questions that span multiple sentences and exhibit sophisticated cross-sentence anaphoric phenomena, evaluating on a rich source of such questions -the math portion of the Scholastic Aptitude Test (SAT).By using a tree transducer cascade as its basic architecture, our system (called EU-CLID) propagates uncertainty from multiple sources (e.g.coreference resolution or verb interpretation) until it can be confidently resolved.Experiments show the first-ever results (43% recall and 91% precision) on SAT algebra word problems.We also apply EUCLID to the public Dolphin algebra question set, and improve the state-of-the-art F 1 -score from 73.9% to 77.0%.