Cost Preserving Dependent Rounding for Allocation Problems

TL;DR

This paper introduces a dependent randomized rounding scheme that preserves cost constraints and concentration properties, enabling improved approximation algorithms for complex allocation problems like the Budgeted Santa Claus Problem.

Contribution

The paper develops a novel dependent rounding algorithm that guarantees cost preservation and concentration, extending its application to advanced allocation problems.

Findings

Guarantees cost of rounded solution does not exceed fractional solution

Provides an O(log n)-approximation for the Budgeted Santa Claus Problem

Works for assignment problems with complex restrictions

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.