arXiv subjects

Sherry Sarkar

Publications and source records attributed to Sherry Sarkar.

12 recordsLinked to original sources

arXiv2026

Semi-Streaming Algorithms for Submodular Maximization under Random Arrival Order

We study random order semi-streaming algorithms for submodular maximization under a wide range of combinatorial constraint classes, including matroids, matroid $p$-parity, $p$-exchange systems and $p$-systems. For most of these classes of constraints, our results are the first improvement over what is known to be achievable for adversarial order. For matroids, matching and $p$-matchoids, previous random order results were known, and we improve over some of these as well. In the case of matroids, our improved results show a separation between adversarial and random order semi-streaming algorithms, and exponentially improve the number of passes necessary for getting $1 - 1/e - \varepsilon$ approximation for maximizing a monotone submodular function subject to a matroid constraint. We also prove a new hardness result showing a similar separation for $p$-systems. Our results are based on two new technical tools. One tool provides a general way to translate offline algorithms for many classes of constraints into random order semi-streaming algorithms. The other tool is a semi-streaming variant of a recently proposed offline algorithm for matroid constraints.

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arXiv2026

Online Steiner Forest with Recourse

In the online Steiner forest problem we are given a graph $G$, and a sequence of terminal pairs $(u_i,v_i)$ which arrive in an online fashion. We are asked to maintain a low-cost subgraph in which each $u_i$ is connected to $v_i$ for all the pairs that have arrived so far. If we are not allowed to delete edges from our solution, then the best possible competitive ratio is $\Theta(\log n)$. In this work, we initiate the study of low-recourse algorithms for online Steiner forest. We give an algorithm that maintains a constant-competitive solution and has an amortized recourse of $O(\log n)$, i.e., inserts and deletes $O(\log n)$ edges per demand on average.

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arXiv2026

Improved Algorithms for Fair Matroid Submodular Maximization

Submodular maximization subject to matroid constraints is a central problem with many applications in machine learning. As algorithms are increasingly used in decision-making over datapoints with sensitive attributes such as gender or race, it is becoming crucial to enforce fairness to avoid bias and discrimination. Recent work has addressed the challenge of developing efficient approximation algorithms for fair matroid submodular maximization. However, the best algorithms known so far are only guaranteed to satisfy a relaxed version of the fairness constraints that loses a factor 2, i.e., the problem may ask for $\ell$ elements with a given attribute, but the algorithm is only guaranteed to find $\lfloor \ell/2 \rfloor$. In particular, there is no provable guarantee when $\ell=1$, which corresponds to a key special case of perfect matching constraints. In this work, we achieve a new trade-off via an algorithm that gets arbitrarily close to full fairness. Namely, for any constant $\varepsilon>0$, we give a constant-factor approximation to fair monotone matroid submodular maximization that in expectation loses only a factor $(1-\varepsilon)$ in the lower-bound fairness constraint. Our empirical evaluation on a standard suite of real-world datasets -- including clustering, recommendation, and coverage tasks -- demonstrates the practical effectiveness of our methods.

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arXiv2025

Sum-Of-Squares To Approximate Knapsack

These notes give a self-contained exposition of Karlin, Mathieu and Nguyen's tight estimate of the integrality gap of the sum-of-squares semidefinite program for solving the knapsack problem. They are based on a sequence of three lectures in CMU course on Advanced Approximation Algorithms in Fall'21 that used the KMN result to introduce the Sum-of-Squares method for algorithm design. The treatment in these notes uses the pseudo-distribution view of solutions to the sum-of-squares SDPs and only rely on a few basic, reusable results about pseudo-distributions.

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arXiv2024

The Online Submodular Assignment Problem

Online resource allocation is a rich and varied field. One of the most well-known problems in this area is online bipartite matching, introduced in 1990 by Karp, Vazirani, and Vazirani [KVV90]. Since then, many variants have been studied, including AdWords, the generalized assignment problem (GAP), and online submodular welfare maximization. In this paper, we introduce a generalization of GAP which we call the submodular assignment problem (SAP). This generalization captures many online assignment problems, including all classical online bipartite matching problems as well as broader online combinatorial optimization problems such as online arboricity, flow scheduling, and laminar restricted allocations. We present a fractional algorithm for online SAP that is (1-1/e)-competitive. Additionally, we study several integral special cases of the problem. In particular, we provide a (1-1/e-epsilon)-competitive integral algorithm under a small-bids assumption, and a (1-1/e)-competitive integral algorithm for online submodular welfare maximization where the utility functions are given by rank functions of matroids. The key new ingredient for our results is the construction and structural analysis of a "water level" vector for polymatroids, which allows us to generalize the classic water-filling paradigm used in online matching problems. This construction reveals connections to submodular utility allocation markets and principal partition sequences of matroids.

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arXiv2024

The Secretary Problem with Predicted Additive Gap

The secretary problem is one of the fundamental problems in online decision making; a tight competitive ratio for this problem of $1/\mathrm{e} \approx 0.368$ has been known since the 1960s. Much more recently, the study of algorithms with predictions was introduced: The algorithm is equipped with a (possibly erroneous) additional piece of information upfront which can be used to improve the algorithm's performance. Complementing previous work on secretary problems with prior knowledge, we tackle the following question: What is the weakest piece of information that allows us to break the $1/\mathrm{e}$ barrier? To this end, we introduce the secretary problem with predicted additive gap. As in the classical problem, weights are fixed by an adversary and elements appear in random order. In contrast to previous variants of predictions, our algorithm only has access to a much weaker piece of information: an \emph{additive gap} $c$. This gap is the difference between the highest and $k$-th highest weight in the sequence. Unlike previous pieces of advice, knowing an exact additive gap does not make the problem trivial. Our contribution is twofold. First, we show that for any index $k$ and any gap $c$, we can obtain a competitive ratio of $0.4$ when knowing the exact gap (even if we do not know $k$), hence beating the prevalent bound for the classical problem by a constant. Second, a slightly modified version of our algorithm allows to prove standard robustness-consistency properties as well as improved guarantees when knowing a range for the error of the prediction.

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arXiv2024

The Online Submodular Assignment Problem

Online resource allocation is a rich and varied field. One of the most well-known problems in this area is online bipartite matching, introduced in 1990 by Karp, Vazirani, and Vazirani [KVV90]. Since then, many variants have been studied, including AdWords, the generalized assignment problem (GAP), and online submodular welfare maximization. In this paper, we introduce a generalization of GAP which we call the submodular assignment problem (SAP). This generalization captures many online assignment problems, including all classical online bipartite matching problems as well as broader online combinatorial optimization problems such as online arboricity, flow scheduling, and laminar restricted allocations. We present a fractional algorithm for online SAP that is $(1-\frac{1}{e})$-competitive. Additionally, we study several integral special cases of the problem. In particular, we provide a $(1-\frac{1}{e}-\epsilon)$-competitive integral algorithm under a small-bids assumption, and a $(1-\frac{1}{e})$-competitive integral algorithm for online submodular welfare maximization where the utility functions are given by rank functions of matroids. The key new ingredient for our results is the construction and structural analysis of a "water level" vector for polymatroids, which allows us to generalize the classic water-filling paradigm used in online matching problems. This construction reveals connections to submodular utility allocation markets and principal partition sequences of matroids.

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arXiv2023

Maintaining Matroid Intersections Online

Maintaining a maximum bipartite matching online while minimizing recourse/augmentations is a well studied problem, motivated by content delivery, job scheduling, and hashing. A breakthrough result of Bernstein, Holm, and Rotenberg (\emph{SODA 2018}) resolved this problem up to a logarithmic factors. However, we may need a richer class of combinatorial constraints (e.g., matroid constraints) to model other problems in scheduling and resource allocation. We consider the problem of maintaining a maximum independent set of an arbitrary matroid $\mathcal{M}$ and a partition matroid $\mathcal{P}$ in the online setting. Specifically, at each timestep $t$ one part $P_t$ of the partition matroid (i.e., a subset of elements) is revealed: we must now select at most one of these newly-revealed elements, but can exchange some of the previously selected elements for new ones from previous parts, to maintain a maximum independent set on the elements seen thus far. The goal is to minimize the number of augmentations/changes done by our algorithm. If $\mathcal{M}$ is also a partition matroid, we recover the problem of maintaining a maximum bipartite matching online with recourse as a special case. In our work, we allow arbitrary matroids $\mathcal{M}$, and so we can model broader classes of problems. Our main result is an $O(n \log^2 n)$-competitive algorithm, where $n$ is the rank of the largest common base; this matches the current best quantitative bound for the bipartite matching special case. Our result builds substantively on the breakthrough result of Bernstein, Holm, and Rotenberg for maintaining bipartite matchings: a key contribution of our work is to make connections to market equilibria and prices, and our use of properties of these equilibria in submodular utility allocation markets to prove our bound on the number of augmentations.

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arXiv2021

Matroid-Based TSP Rounding for Half-Integral Solutions

We show how to round any half-integral solution to the subtour-elimination relaxation for the TSP, while losing a less-than-1.5 factor. Such a rounding algorithm was recently given by Karlin, Klein, and Oveis Gharan based on sampling from max-entropy distributions. We build on an approach of Haddadan and Newman to show how sampling from the matroid intersection polytope, and a new use of max-entropy sampling, can give better guarantees.

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arXiv2020

Tolerance for colorful Tverberg partitions

Tverberg's theorem bounds the number of points $\mathbb{R}^d$ needed for the existence of a partition into $r$ parts whose convex hulls intersect. If the points are colored with $N$ colors, we seek partitions where each part has at most one point of each color. In this manuscript, we bound the number of color classes needed for the existence of partitions where the convex hulls of the parts intersect even after any set of $t$ colors is removed. We prove asymptotically optimal bounds for $t$ when $r \le d+1$, improve known bounds when $r>d+1$, and give a geometric characterization for the configurations of points for which $t=N-o(N)$.

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arXiv2020

Quantitative combinatorial geometry for concave functions

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and $H$-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single $H$-convex set. We obtain colorful and fractional variants of all our Helly-type theorems.

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arXiv2018

Stability for Take-Away Games

In this paper, we study a family of take-away games called $α$-tag, parametrized by a real number $α\ge 1$. We show that for any given $α$, there is a half-open interval $I_α$ containing $α$ such that the set of losing positions for $α$-tag is the same as the set of losing positions for $β$-tag if and only if $β\in I_α$. We then end with some results and conjectures on the nature of these intervals.

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