When uncertainty meets costly information gathering, a fundamental question emerges: which data points should we probe to unlock near-optimal solutions? Sparsification of stochastic packing problems addresses this trade-off. The existing notions of sparsification measure the level of sparsity, called degree, as the ratio of queried items to the optimal solution size. While effective for matching and matroid-type problems with uniform structures, this cardinality-based approach fails for knapsack-type constraints where feasible sets exhibit dramatic structural variation. We introduce a polyhedral sparsification framework that measures the degree as the smallest scalar needed to embed the query set within a scaled feasibility polytope, naturally capturing redundancy without relying on cardinality. Our main contribution establishes that knapsack, multiple knapsack, and generalized assignment problems admit (1-ε)-approximate sparsifiers with degree polynomial in 1/p and 1/ε - where p denotes the independent activation probability of each element - remarkably independent of problem dimensions. The key insight involves grouping items with similar weights and deploying a charging argument: when our query set misses an optimal item, we either substitute it directly with a queried item from the same group or leverage that group’s excess contribution to compensate for the loss. This reveals an intriguing complexity-theoretic separation - while the multiple knapsack problem lacks an FPTAS and generalized assignment is APX-hard, their sparsification counterparts admit efficient (1-ε)-approximation algorithms that identify polynomial degree query sets. Finally, we raise an open question: can such sparsification extend to general integer linear programs with degree independent of problem dimensions?

Much of learning theory is concerned with the design and analysis of probably approximately correct (PAC) learners. The closely related transductive model of learning has recently seen more scrutiny, with its learners often used as precursors to PAC learners. Our goal in this work is to understand and quantify the exact relationship between these two models. First, we observe that modest extensions of existing results show the models to be essentially equivalent for realizable learning for most natural loss functions, up to low order terms in the error and sample complexity. The situation for agnostic learning appears less straightforward, with sample complexities potentially separated by a $\frac{1}{\epsilon}$ factor. This is therefore where our main contributions lie. Our results are two-fold: For agnostic learning with bounded losses (including, for example, multiclass classification), we show that PAC learning reduces to transductive learning at the cost of low-order terms in the error and sample complexity. This is via an adaptation of the reduction of Aden-Ali et al. (2023a) to the agnostic setting. For agnostic binary classification, we show the converse: transductive learning is essentially no more difficult than PAC learning. Together with our first result this implies that the PAC and transductive models are essentially equivalent for agnostic binary classification. This is our most technical result, and involves two key steps: (a) A symmetrization argument on the agnostic one-inclusion graph (OIG) of Long (1998) to derive the worst-case agnostic transductive instance, and (b) expressing the error of the agnostic OIG algorithm for this instance in terms of the empirical Rademacher complexity of the class. We leave as an intriguing open question whether our second result can be extended beyond binary classification to show the transductive and PAC models equivalent more broadly.

We introduce a novel definition for a small set R of k points being "representative" of a larger set in a metric space. Given a set V (e.g., documents or voters) to represent, and a set C of possible representatives, our criterion requires that for any subset S comprising a theta fraction of V, the average distance of S to their best theta*k points in R should not be more than a factor gamma compared to their average distance to the best theta*k points among all of C. This definition is a strengthening of proportional fairness and core fairness, but - different from those notions - requires that large cohesive clusters be represented proportionally to their size. Since there are instances for which - unless gamma is polynomially large - no solutions exist, we study this notion in a resource augmentation framework, implicitly stating the constraints for a set R of size k as though its size were only k/alpha, for alpha > 1. Furthermore, motivated by the application to elections, we mostly focus on the "ordinal" model, where the algorithm does not learn the actual distances; instead, it learns only for each point v in V and each candidate pairs c, c' which of c, c' is closer to v. Our main result is that the Expanding Approvals Rule (EAR) of Aziz and Lee is (alpha, gamma) representative with gamma

Motivated by the growing interest in correlation-robust stochastic optimization, we investigate stochastic selection problems beyond independence. Specifically, we consider the instructive case of pairwise-independent priors and matroid constraints. We obtain essentially-optimal bounds for contention resolution and prophet inequalities. The impetus for our work comes from the recent work of Caragiannis et. al. [WINE 2022], who derived a constant factor approximation for the single-choice prophet inequality with pairwise-independent priors.