randomized rounding

This paper studies the budgeted Santa Claus problem: allocating heterogeneous resources to players under a total cost constraint to maximize the minimum player value. The core challenge lies in rounding a fractional solution of the linear programming relaxation to an integral one that satisfies hard constraints—namely, each player receives at most one resource and the total cost remains within budget—without loss in objective value. To this end, the paper introduces, for the first time, a *cost-preserving dependent rounding* algorithm: it guarantees that the rounded solution’s total cost never exceeds that of the fractional solution, while preserving Chernoff-type concentration bounds. This overcomes a fundamental limitation of prior dependent rounding schemes, which cannot control cost inflation. As a result, the paper achieves the first $O(log n)$-approximation algorithm for the budgeted Santa Claus problem, significantly improving both theoretical guarantees and practical applicability for budget-constrained resource allocation.

This paper studies online resource allocation under adversarial conditions with reusable resources, focusing on two leasing models: kRental-Fixed (fixed lease duration) and kRental-Variable (variable lease duration), where a decision-maker must allocate k identical reusable units to dynamically arriving lease requests. We propose a price-based fractional allocation framework, augmented by novel lossless online rounding and bounded-correlation rounding techniques—introducing controlled dependence among unit assignments while preserving per-unit independent processing, thereby overcoming performance bottlenecks of traditional independent rounding. Theoretically, we achieve the optimal randomized competitive ratio for kRental-Fixed and an asymptotically optimal competitive ratio for kRental-Variable. To our knowledge, this is the first work to establish tight theoretical bounds for reusable leasing problems, significantly enhancing both the efficiency and robustness of online decision-making.

In resource-constrained policy experiments, balancing strict budget constraints with high estimation precision remains challenging. Method: This paper proposes a causal inference framework based on dependent randomized rounding: it converts continuous treatment assignment probabilities into binary intervention decisions while strictly respecting budget constraints, and introduces negative dependence among assignments to preserve marginal probabilities and substantially reduce estimator variance. Contribution/Results: This is the first systematic application of dependent rounding to budget-aware experimental design. We theoretically establish that the method strictly tightens the variance upper bounds of both inverse probability weighting (IPW) and generalized linear model estimators—surpassing the fundamental precision limits of independent rounding. Empirical results demonstrate an average 32% reduction in mean squared error under fixed budgets, establishing a new paradigm for high-precision, cost-efficient policy evaluation.

This paper addresses resource allocation in two-sided markets under multidimensional fairness criteria—including proportionality, envy-freeness, and stability. We propose a tunable, parameterized framework for approximately feasible allocations and establish, for the first time, its theoretical existence. Our analysis characterizes the fundamental trade-offs among market capacity, demand profiles, and fairness requirements. To compute such allocations, we design an iterative rounding algorithm grounded in fractional assignment and derive novel sufficient conditions for controlling feasibility deviation. The framework unifies and strengthens guarantees for three canonical problems: school choice, stable matching with couples, and political seat apportionment. It significantly improves the existence guarantees of approximately feasible solutions across multiple fairness axioms, while preserving policy adaptability and theoretical rigor.

This work presents the first experimental study that simultaneously approximates solutions to both the Max-Cut problem and the Weighted Fractional Cut Cover problem. Building upon a primal-dual framework, the approach integrates semidefinite programming (SDP) relaxation with randomized hyperplane rounding, while replacing conventional theoretical algorithms with a linear programming (LP) solver to substantially enhance empirical performance. Using only ⌈128 ln m⌉ random samples, the method consistently achieves an approximation ratio close to the Goemans–Williamson bound of 0.878 for the vast majority of instances, demonstrating robustness and reproducibility. Notably, the LP-based variant frequently outperforms theoretical expectations, thereby validating the efficacy and practicality of the proposed hybrid strategy.

This work addresses the problem of efficiently and exactly solving discounted Markov decision processes (DMDPs) for the optimal value function and policy. We propose a novel reduction framework that decomposes the exact solution into two subproblems: policy evaluation and computation of an approximately optimal value function. Leveraging state-of-the-art techniques in approximate dynamic programming, we design both deterministic and randomized algorithms tailored to these subtasks. Our approach achieves significantly improved computational efficiency, yielding the fastest known exact DMDP solver to date. The resulting algorithms demonstrate clear advantages over existing methods, both theoretically—through tighter complexity bounds—and empirically—via superior practical performance.

This work addresses the long-standing open problem of designing polynomial-time constant-factor truthful mechanisms for budget-feasible procurement under subadditive valuations. Building upon a compensation-based framework, the authors introduce a novel mechanism that integrates marginal contribution payments with self-bounding function approximation. A key technical contribution is a new smoothing lemma showing that any subadditive function can be approximated within a factor of 2 by a self-bounding function. Leveraging potential analysis and the demand query model, the proposed mechanism achieves an approximation ratio of $2e + 1$ (improving to $2e$ in large markets) for subadditive valuations—significantly surpassing the previous best bound of 33—and also yields improved guarantees for submodular and XOS settings. Furthermore, the paper establishes the existence of a $2e$-approximate core in multi-winner elections.

This work addresses the issue of load imbalance on left nodes and degraded solution quality caused by independent randomized rounding in assignment graphs. To overcome this, the authors propose a dependent rounding mechanism based on the Dirichlet distribution: each left node generates Dirichlet-distributed random variables over its incident edges, and right nodes select edges accordingly, naturally inducing strong negative correlations. The method is structurally simple, avoids complex tie-breaking strategies, and achieves stronger negative correlation than existing approaches. It yields improved approximation guarantees—0.68 for oblivious online dependent rounding (surpassing the previous 0.652) and 1.387 for unrelated-machine scheduling to minimize weighted completion time (improving upon the prior 1.398)—demonstrating significantly enhanced rounding performance for fractional solutions in combinatorial optimization.

This work addresses the challenge of identifying a universally acceptable randomized allocation—termed a “lottery”—among stakeholders in high-stakes AI deployments. The authors propose an adaptive query framework that efficiently determines the existence of such a feasible lottery by iteratively presenting candidate allocations to stakeholders and collecting binary feedback. Their key contributions include deterministic and randomized algorithms that substantially reduce the number of required queries, as well as a learning-augmented mechanism that, when predictions are accurate, dramatically lowers query complexity while preserving worst-case theoretical guarantees. They also establish an information-theoretic lower bound showing that query complexity must scale linearly with the number of stakeholders and logarithmically with the desired accuracy—bounds that cannot be improved upon. Empirical evaluations demonstrate that the proposed approach significantly outperforms exhaustive enumeration strategies.

This work addresses the k-clustering problem with the constraint of opening exactly k facilities, focusing on approximation algorithms where the cost is defined as the p-th power of distances. Building upon the standard LP relaxation, the authors propose an iterative randomized rounding algorithm that, in expectation, satisfies the cardinality constraint while controlling the expected cost. The key contribution is the first LP-rounding-based (3^p + 1)/2–LMP (Lagrangian Multiplier Preserving) approximation algorithm for general L_p^p costs, unifying the treatment across different values of p. Using probabilistic rounding techniques, this LMP guarantee is converted into a true (1+ε)-approximation. Notably, the framework recovers the optimal (2+ε)-approximation for k-median, improves the metric k-means approximation ratio from 5.83 to 5+ε, and achieves a (4+ε)-approximation in Euclidean space matching the current best bound.