arXiv subjects

Sahil Singla

Publications and source records attributed to Sahil Singla.

At least 19 recordsLinked to original sources

arXiv2026

As Easy as Rank One: Optimal Algorithms for Matroid Secretary and Single-Sample Matroid Prophet

We resolve the matroid secretary conjecture in its strong form and the single-sample matroid prophet inequality problem, achieving competitive ratios $e$ and $2$, respectively. Both ratios are optimal, even for rank-one matroids. For matroid secretary, our ordinal algorithm accepts the elements of the canonical optimum independently, each with probability $1/e$. It needs only the number of elements in advance and uses an expected polynomial number of independence queries on arrived elements. For single-sample matroid prophet inequalities, our algorithm uses one sample per independent value distribution and $O(n^2)$ independence queries on a known $n$-element matroid. It works for every arrival order fixed independently of the samples, values, and algorithmic randomness. The selected elements and their values have exactly the joint distribution obtained by independently retaining each element--value pair in the canonical optimum of a fresh product draw with probability $1/2$. Our main algorithms maintain a reference state whose greedy solution contains every accepted element. Its updates, made even after rejections, preserve exact distributional invariants. A short reversible-update argument gives a $4$-probability-competitive secretary algorithm, already resolving the original conjecture; a related invariant gives the optimal single-sample prophet inequality. For secretary, we characterize the distributions achievable by ordinal algorithms through a finite polytope. A continuous-time formulation then lets us design an accepted-set distribution achieving the optimal ratio $e$ and realize it efficiently. We also obtain guarantees for matroid intersections.

cs.DS↗

arXiv2026

Improved Combinatorial Auctions with and without Samples

We give new mechanisms for combinatorial auctions in the offline, secretary, and prophet-with-samples models. Specifically, we obtain the following guarantees for social welfare maximization with $m$ items and $n$ XOS buyers: * In the prophet-with-samples version, buyers arrive in adversarial order and draw their values from independent distributions. We give a universally truthful, polynomial-time $(2+\eps)$-approximation mechanism using $O(\log\!\log m)$ samples from each distribution. The approximation factor is tight even for a single item and without the incentive constraints, due to the classic prophet inequality. Compared to the previous state-of-the-art with the same guarantees [Dütting--Kesselheim--Lucier--Reiffenhäuser--Singla (FOCS 2024)], our running time is exponentially faster, and our sample complexity is doubly-exponentially lower, resolving two explicit open problems by Dütting et al. * Using the same techniques, we also obtain improved mechanisms in the secretary and offline versions, where the values are worst-case, and buyers are processed in either random order (secretary) or without any timing restrictions (offline). After a long line of works, the state of the art in both models is an $O((\log\!\log m)^2)$-approximation due to [Assadi--Kesselheim--Singla (SODA 2021)], who explicitly ask whether it can be improved to $O(\log\!\log m)$. We indeed obtain an $O(\log\!\log m)$-approximate universally truthful mechanism for the secretary and offline models.

cs.GT↗

arXiv2026

Efficient Algorithms for Online Subadditive Combinatorial Allocations

For the online combinatorial allocation problem with subadditive valuations, Correa and Cristi (STOC 2023) proved the existence of a $6$-competitive online algorithm, improving on the previous best $O(\log\!\log m)$-competitive online algorithm due to Dütting, Kesselheim, and Lucier (FOCS 2020), where $m$ is the number of items. However, Correa and Cristi's result is existential, and it was left open whether a constant competitive ratio is attainable via an efficient online algorithm that uses a polynomial number of demand oracle queries. In this work, we answer this affirmatively, giving an expected-polynomial-time $(6 + ε)$-competitive online algorithm for any constant $ε> 0$. Our techniques also recover, in the offline setting, the $(2 + ε)$-approximation result of Feige (STOC, 2006). Finally, when the buyers' valuations are drawn from identical distributions, we exploit symmetry to obtain an improved $(60/11+ε)$-competitive algorithm. Starting with the natural configuration LP relaxation for the problem, our main technical contribution is a recursive Bundle Score Generator (BSG) that resolves item conflicts by assigning correlated scores to the items requested by each buyer. Unlike the Random Score Generator whose existence is shown by Correa and Cristi, our BSG is efficiently computable in expected polynomial time. Moreover, it satisfies a stochastic dominance property that is sufficient to recover both Feige's offline result and Correa and Cristi's online result.

cs.DS↗

arXiv2026

The Matroid Secretary Conjecture is True

We resolve the matroid secretary conjecture, giving an online algorithm that accepts each element of the offline optimum with probability at least $1/4$. The algorithm only needs the number of elements in advance and independence-oracle access to subsets of already-arrived elements; it does not need to know the matroid upfront.

cs.DS↗

arXiv2026

The Cube-Root Phenomenon in Online Carpooling

We consider the online carpooling problem, where edges arrive online and must be oriented immediately while keeping the discrepancy between the indegree and outdegree at each vertex small. We prove that the natural Greedy algorithm incurs discrepancy $O(\min\{T^{1/3},n\})$ after $T$ arrivals. This resolves a question of Ajtai et al., who showed that any deterministic algorithm must incur $Ω(\min\{T^{1/3},n\})$ discrepancy, and gave an algorithm with $O(\min\{T^{1/2},n\})$ discrepancy. We also show a similar square-root to cube-root improvement in the stochastic setting, where $O(n)$ edges are sampled independently from an underlying $n$-vertex graph $G$. Formally, we show an $O((\log n)^{1/3})$ bound for random arrivals from any $Δ$-regular graph $G$. When $Δ= Ω((\log n)^3)$, we show the more refined bound of $O((\log n/\log Δ)^{1/3}+\log\log n)$ on the discrepancy. We show that the cube-root term in the previous bound is essential, while the $\log\log n$ term is already known to be necessary for random arrivals from complete graphs. The previous upper bounds here were $O((\log n)^{1/2})$, which follow from the breakthrough works on online discrepancy due to Kulkarni, Reis, and Rothvoss, and Aden-Ali. Our techniques for proving such cube-root-type bounds may be of independent interest, as the standard quadratic-potential and subgaussian analyses underlying the previous general bounds appear inherently unable to go below square-root-type guarantees.

cs.DS↗

arXiv2026

Online Allocation using Few Samples

We study online allocation problems where $n$ requests over $m$ resources arrive in an adversarial order and must be served immediately and irrevocably. This framework captures both Online Resource Allocation, where the goal is to maximize value subject to resource budgets, and Online Load Balancing, where the goal is to minimize the makespan. We seek $(1\pmε)$-competitive algorithms in the large-budget or large-makespan regime. We consider a sampling model that generalizes the following two well-studied sampling models. In the Single-Sample Prophet Inequality ($\mathsf{SSPI}$) model, request $t$ is drawn from an unknown distribution $\mathcal{D}_t$, and the algorithm is given one independent sample from each $\mathcal{D}_t$ before the online phase. In the $p$-$\mathsf{Sample}$ model, the requests are adversarial, but a uniformly random $p$-fraction is revealed upfront as training data. Although near-optimal algorithms are known in the easier random-order model ($\mathsf{RO}$), where the requests arrive in a uniformly random order, prior algorithms for $\mathsf{SSPI}$ and $p$-$\mathsf{Sample}$ were problem-specific and incurred substantially worse dependencies on $ε$, $m$, and $n$. Our main contribution is a general framework that converts $\mathsf{RO}$ algorithms into algorithms for the $p$-$\mathsf{Preview}$ model, a model that generalizes both $\mathsf{SSPI}$ and $p$-$\mathsf{Sample}$. As consequences, we obtain near-optimal bounds for Online Resource Allocation, generalized Online Load Balancing, and online mixed packing-covering problems in these adversarial-order sampling models, significantly improving the bounds of [Ghuge, Singla, Wang (STOC'25)] and [Gupta and Molinaro (SODA'26)].

cs.DS↗

arXiv2026

Adversarial Online Classification with a Preview

Worst-case online classification is governed by sequential complexity, such as Littlestone dimension, and can be impossible even for statistically simple classes, such as thresholds of VC dimension one. We study a preview model in which an oblivious adversary fixes an entire labeled sequence of length $T$, a uniformly random subset of size $pT$ is revealed before prediction begins, and the remaining $(1-p)T$ examples are then presented in their original adversarial order. Against the best full-sequence hypothesis evaluated on the unrevealed examples, we characterize the dependence on the preview rate $p$: for binary classes of VC dimension $d$, the optimal excess loss is $Θ(d/p+\sqrt{dT})$, up to the trivial cap at $T$; for multiclass classes we obtain the corresponding $\widetilde O(d_{\rm DS}/p+\sqrt{d_{\rm Nat}T})$ bound with no dependence on the number of labels. Thus a random preview can replace worst-case sequential complexity by classical statistical dimensions without randomizing the online order. To achieve the sharp binary bound, our ChainedPrediction algorithm uses an online analogue of chaining, implemented as a multiscale aggregation algorithm rather than only as an analytic argument.

cs.LG↗

arXiv2026

Online Algorithms via Minimax and Posterior Matching

Competitive analysis is central to the study of online algorithms, but upper bounds are often highly problem-specific. We develop a more unifying methodology via the minimax viewpoint. Guided by Yao's principle, we reduce worst-case competitive analysis to Bayesian online design under an arbitrary correlated prior over arrival sequences. For such a prior, let $X^*$ be the hindsight-optimal fractional solution for the realized instance, and let $X^{(t)}=\mathbb E[X^*\mid \mathcal F_t]$ be its posterior process. Our guiding rule is posterior matching: at each time $t$, choose the feasible online action that tracks the current posterior $X^{(t)}$ as closely as the online constraints permit. We show that this single principle yields optimal or near-optimal guarantees for several classical online fractional problems, including set cover, load balancing, matching and more general resource-allocation problems, recovering or improving state-of-the-art bounds in these settings with norm/concave objectives. Via known rounding reductions, it also yields randomized integral guarantees for weighted paging, MTS on star metrics, and ski-rental. At a technical level, our analysis reduces competitive guarantees to key probabilistic inequalities for the vector martingales generated by the posterior of the offline optimum. The resulting framework gives a reusable route from Bayesian online design under arbitrary correlated priors to information-theoretic worst-case competitive guarantees.

cs.LG↗

arXiv2026

Philosopher and Prophet Inequalities for Divisible Items

We study online welfare maximization with divisible resources. A sequence of $n$ players arrive one by one; upon arrival, each player draws a valuation function over $m$ divisible items from a known distribution, reveals this valuation, and must be allocated an irrevocable fractional bundle subject to unit supply constraints. While online welfare maximization has been extensively studied for indivisible items and combinatorial valuations, much less is known when the resources are divisible and players have multi-dimensional concave valuations. We give approximation algorithms for monotone concave valuations satisfying diminishing returns. Our main result is a $2/3$-approximation to the optimal online policy, also known as the philosopher benchmark. The algorithm is guided by a low-dimensional concave relaxation of the online benchmark and rounds it via a new single-item capped online contention resolution scheme. This Capped-OCRS problem allocates to each realized type no more than its prescribed fractional bundle while preserving a $2/3$-fraction of that bundle in expectation. Its analysis uses a submartingale potential for the remaining side, we show that computing the optimal online policy is #P-hard even for a single divisible item. We also obtain a tight prophet inequality against the offline hindsight optimum. We show that a fixed-price auction with one linear per-unit price for each original divisible item achieves a $1/2$-approximation to the offline/prophet benchmark. The prices are obtained by aggregating Aumann--Shapley supporting prices, a continuous analogue of supporting prices for submodular/XOS set functions, and yield simple item prices rather than copy-dependent prices arising from discretization. The factor $1/2$ for the prophet benchmark is information-theoretically tight even for one item with linear valuations.

cs.DS↗

arXiv2026

Online Graph Balancing and the Power of Two Choices

In the classic online graph balancing problem, edges arrive sequentially and must be oriented immediately upon arrival, to minimize the maximum in-degree. For adversarial arrivals, the natural greedy algorithm is $O(\log n)$-competitive, and this bound is the best possible for any algorithm, even with randomization. We study this problem in the i.i.d. model where a base graph $G$ is known in advance and each arrival is an independent uniformly random edge of $G$. This model generalizes the standard power-of-two choices setting, corresponding to $G = K_n$, where the greedy algorithm achieves an $O(\log\!\log n)$ guarantee. We ask whether a similar bound is possible for arbitrary base graphs. While the greedy algorithm is optimal for adversarial arrivals and also for i.i.d. arrivals from regular base graphs (such as $G = K_n$), we show that it can perform poorly in general: there exist mildly irregular graphs $G$ for which greedy is $\widetildeΩ(\log n)$-competitive under i.i.d. arrivals. In sharp contrast, our main result is an $O(\log\!\log n)$-competitive online algorithm for every base graph $G$; this is optimal up to constant factors, since an $Ω(\log\!\log n)$ lower bound already holds even for the complete graph $G = K_n$. The key new idea is a notion of log-skewness for graphs, which captures the irregular substructures in $G$ that force the offline optimum to be large. Moreover, we show that any base graph can be decomposed into ``skew-biregular'' pieces at only $O(\log\!\log n)$ scales of log-skewness, and use this to design a decomposition-based variant of greedy that is $O(\log\!\log n)$-competitive.

cs.DS↗

arXiv2026

Secretary, Prophet, and Stochastic Probing via Big-Decisions-First

We revisit three fundamental problems in algorithms under uncertainty: the Secretary Problem, Prophet Inequality, and Stochastic Probing, each subject to general downward-closed constraints. When elements have binary values, all three problems admit a tight $\tildeΘ(\log n)$-factor approximation guarantee. For general (non-binary) values, however, the best known algorithms lose an additional $\log n$ factor when discretizing to binary values, leaving a quadratic gap of $\tildeΘ(\log n)$ vs. $\tildeΘ(\log^2 n)$. We resolve this quadratic gap for all three problems, showing $\tildeΩ(\log^2 n)$-hardness for two of them and an $O(\log n)$-approximation algorithm for the third. While the technical details differ across settings, and between algorithmic and hardness proofs, all our results stem from a single core observation, which we call the Big-Decisions-First Principle: Under uncertainty, it is better to resolve high-stakes (large-value) decisions early.

cs.DS↗

arXiv2025

Visualisation for the CIS benchmark scanning results

In this paper, we introduce GraphSecure, a web application that provides advanced analysis and visualisation of security scanning results. GraphSecure enables users to initiate scans for their AWS account, validate them against specific Center for Internet Security (CIS) Benchmarks and return results, showcase those returned results in the form of statistical charts and warn the users about their account status.

cs.CR↗

arXiv2025

Online Combinatorial Optimization with Graphical Dependencies

Most existing work in online stochastic combinatorial optimization assumes that inputs are drawn from independent distributions -- a strong assumption that often fails in practice. At the other extreme, arbitrary correlations are equivalent to worst-case inputs via Yao's minimax principle, making good algorithms often impossible. This motivates the study of intermediate models that capture mild correlations while still permitting non-trivial algorithms. In this paper, we study online combinatorial optimization under Markov Random Fields (MRFs), a well-established graphical model for structured dependencies. MRFs parameterize correlation strength via the maximum weighted degree $Δ$, smoothly interpolating between independence ($Δ= 0$) and full correlation ($Δ\to \infty$). While naïvely this yields $e^{O(Δ)}$-competitive algorithms and $Ω(Δ)$ hardness, we ask: when can we design tight $Θ(Δ)$-competitive algorithms? We present general techniques achieving $O(Δ)$-competitive algorithms for both minimization and maximization problems under MRF-distributed inputs. For minimization problems with coverage constraints (e.g., Facility Location and Steiner Tree), we reduce to the well-studied $p$-sample model. For maximization problems (e.g., matchings and combinatorial auctions with XOS buyers), we extend the "balanced prices" framework for online allocation problems to MRFs.

cs.DS↗

arXiv2025

Integral Online Algorithms for Set Cover and Load Balancing with Convex Objectives

Online Set Cover and Load Balancing are central problems in online optimization, and there is a long line of work on developing algorithms for these problems with convex objectives. Although we know optimal online algorithms with $\ell_p$-norm objectives, recent developments for general norms and convex objectives that rely on the online primal-dual framework apply only to fractional settings due to large integrality gaps. Our work focuses on directly designing integral online algorithms for Set Cover and Load Balancing with convex objectives, bypassing the convex-relaxation and the primal-dual technique. Some of the main implications are: 1. For Online Set Cover, we can extend the results of Azar et. al. (2016) for convex objectives and of Kesselheim, Molinaro, and Singla (2024) for symmetric norms from fractional to integral settings. 2. Our results for convex objectives and symmetric norms even apply to the online generalized scheduling problem, which generalizes both Set Cover and Load Balancing. Previous works could only handle the offline version of this problem with norm objectives (Deng, Li, and Rabani 2023). 3. Our methods easily extend to settings with disjoint-composition of norms. This allows us to recover or improve the norm-composition results of Nagarajan and Shen (2020), and Kesselheim, Molinaro, and Singla (2024), and to extend our results to a large class of norms beyond symmetric. Our approach is to first reduce these problems to online packing problems, and then to design good approximation algorithms for the latter. To solve these packing problems, we use two key ideas. First, we decouple the global packing problem into a series of local packing problems on different machines. Next, we choose random activation thresholds for machines such that conditional on a machine being activated, the expected number of jobs it covers is high compared to its cost.

cs.DS↗

arXiv2025

Efficient Approximation Schemes for Stochastic Probing and Selection-Stopping Problems

In this paper, we propose a general framework to design {efficient} polynomial time approximation schemes (EPTAS) for fundamental stochastic combinatorial optimization problems. Given an error parameter $ε>0$, such algorithmic schemes attain a $(1-ε)$-approximation in $t(ε)\cdot poly(|{\cal I}|)$ time, where $t(\cdot)$ is a function that depends only on $ε$ and $|{\cal I}|$ denotes the input length. Technically speaking, our approach relies on presenting tailor-made reductions to a newly-introduced multi-dimensional Santa Claus problem. Even though the single-dimensional version of this problem is already known to be APX-Hard, we prove that an EPTAS can be designed for a constant number of machines and dimensions, which hold for each of our applications. To demonstrate the versatility of our framework, we first study selection-stopping settings to derive an EPTAS for the Free-Order Prophets problem [Agrawal et al., EC~'20] and for its cost-driven generalization, Pandora's Box with Commitment [Fu et al., ICALP~'18]. These results constitute the first approximation schemes in the non-adaptive setting and improve on known \emph{inefficient} polynomial time approximation schemes (PTAS) for their adaptive variants. Next, turning our attention to stochastic probing problems, we obtain an EPTAS for the adaptive ProbeMax problem as well as for its non-adaptive counterpart; in both cases, state-of-the-art approximability results have been inefficient PTASes [Chen et al., NIPS~'16; Fu et al., ICALP~'18].

cs.DS↗

arXiv2025

Improved and Oracle-Efficient Online $\ell_1$-Multicalibration

We study \emph{online multicalibration}, a framework for ensuring calibrated predictions across multiple groups in adversarial settings, across $T$ rounds. Although online calibration is typically studied in the $\ell_1$ norm, prior approaches to online multicalibration have taken the indirect approach of obtaining rates in other norms (such as $\ell_2$ and $\ell_{\infty}$) and then transferred these guarantees to $\ell_1$ at additional loss. In contrast, we propose a direct method that achieves improved and oracle-efficient rates of $\widetilde{\mathcal{O}}(T^{-1/3})$ and $\widetilde{\mathcal{O}}(T^{-1/4})$ respectively, for online $\ell_1$-multicalibration. Our key insight is a novel reduction of online $\ell_1$-multicalibration to an online learning problem with product-based rewards, which we refer to as \emph{online linear-product optimization} ($\mathtt{OLPO}$). To obtain the improved rate of $\widetilde{\mathcal{O}}(T^{-1/3})$, we introduce a linearization of $\mathtt{OLPO}$ and design a no-regret algorithm for this linearized problem. Although this method guarantees the desired sublinear rate (nearly matching the best rate for online calibration), it is computationally expensive when the group family $\mathcal{H}$ is large or infinite, since it enumerates all possible groups. To address scalability, we propose a second approach to $\mathtt{OLPO}$ that makes only a polynomial number of calls to an offline optimization (\emph{multicalibration evaluation}) oracle, resulting in \emph{oracle-efficient} online $\ell_1$-multicalibration with a rate of $\widetilde{\mathcal{O}}(T^{-1/4})$. Our framework also extends to certain infinite families of groups (e.g., all linear functions on the context space) by exploiting a $1$-Lipschitz property of the $\ell_1$-multicalibration error with respect to $\mathcal{H}$.

cs.LG↗

arXiv2025

Single-Sample and Robust Online Resource Allocation

Online Resource Allocation problem is a central problem in many areas of Computer Science, Operations Research, and Economics. In this problem, we sequentially receive $n$ stochastic requests for $m$ kinds of shared resources, where each request can be satisfied in multiple ways, consuming different amounts of resources and generating different values. The goal is to achieve a $(1-ε)$-approximation to the hindsight optimum, where $ε>0$ is a small constant, assuming each resource has a large budget. In this paper, we investigate the learnability and robustness of online resource allocation. Our primary contribution is a novel Exponential Pricing algorithm with the following properties: 1. It requires only a \emph{single sample} from each of the $n$ request distributions to achieve a $(1-ε)$-approximation for online resource allocation with large budgets. Such an algorithm was previously unknown, even with access to polynomially many samples, as prior work either assumed full distributional knowledge or was limited to i.i.d.\,or random-order arrivals. 2. It is robust to corruptions in the outliers model and the value augmentation model. Specifically, it maintains its $(1 - ε)$-approximation guarantee under both these robustness models, resolving the open question posed in Argue, Gupta, Molinaro, and Singla (SODA'22). 3. It operates as a simple item-pricing algorithm that ensures incentive compatibility. The intuition behind our Exponential Pricing algorithm is that the price of a resource should adjust exponentially as it is overused or underused. It differs from conventional approaches that use an online learning algorithm for item pricing. This departure guarantees that the algorithm will never run out of any resource, but loses the usual no-regret properties of online learning algorithms, necessitating a new analytical approach.

cs.DS↗

arXiv2025

Focus-N-Fix: Region-Aware Fine-Tuning for Text-to-Image Generation

Text-to-image (T2I) generation has made significant advances in recent years, but challenges still remain in the generation of perceptual artifacts, misalignment with complex prompts, and safety. The prevailing approach to address these issues involves collecting human feedback on generated images, training reward models to estimate human feedback, and then fine-tuning T2I models based on the reward models to align them with human preferences. However, while existing reward fine-tuning methods can produce images with higher rewards, they may change model behavior in unexpected ways. For example, fine-tuning for one quality aspect (e.g., safety) may degrade other aspects (e.g., prompt alignment), or may lead to reward hacking (e.g., finding a way to increase rewards without having the intended effect). In this paper, we propose Focus-N-Fix, a region-aware fine-tuning method that trains models to correct only previously problematic image regions. The resulting fine-tuned model generates images with the same high-level structure as the original model but shows significant improvements in regions where the original model was deficient in safety (over-sexualization and violence), plausibility, or other criteria. Our experiments demonstrate that Focus-N-Fix improves these localized quality aspects with little or no degradation to others and typically imperceptible changes in the rest of the image. Disclaimer: This paper contains images that may be overly sexual, violent, offensive, or harmful.

cs.CV↗