Cost Preserving Dependent Rounding for Allocation Problems

2025-02-12

AI Summary

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.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We present a dependent randomized rounding scheme, which rounds fractional solutions to integral solutions satisfying certain hard constraints on the output while preserving Chernoff-like concentration properties. In contrast to previous dependent rounding schemes, our algorithm guarantees that the cost of the rounded integral solution does not exceed that of the fractional solution. Our algorithm works for a class of assignment problems with restrictions similar to those of prior works. In a non-trivial combination of our general result with a classical approach from Shmoys and Tardos [Math. Programm.'93] and more recent linear programming techniques developed for the restricted assignment variant by Bansal, Sviridenko [STOC'06] and Davies, Rothvoss, Zhang [SODA'20], we derive a O(log n)-approximation algorithm for the Budgeted Santa Claus Problem. In this new variant, the goal is to allocate resources with different values to players, maximizing the minimum value a player receives, and satisfying a budget constraint on player-resource allocation costs.

Problem

• Cost-preserving dependent rounding
• Budget-constrained resource allocation
• Maximizing minimum player value

Innovation

• Dependent randomized rounding scheme
• Cost-preserving integral solutions
• O(log n)-approximation algorithm