A Variational-Calculus Approach to Online Algorithm Design and Analysis
Pan Xu
TL;DR
This paper introduces a variational-calculus (VC) framework to analyze the asymptotic behavior of factor-revealing and policy-revealing LPs that arise in online algorithm design. By reformulating the LP limit as a variational problem and applying Euler–Lagrange techniques and Lagrange multipliers, the authors derive analytical solutions and tight bounds, often via simple ODE-based reasoning. They demonstrate the approach through three case studies—BALANCE, RANKING, and the classical Secretary problem—establishing the same $1/\mathsf{e}$ bound as known results and providing unique optimal solutions where applicable. The VC method offers a general, potentially broadly applicable tool for analyzing online algorithms beyond instance-specific constructions, with promising extensions to higher-dimensional and multi-constraint settings.
Abstract
Factor-revealing linear programs (LPs) and policy-revealing LPs arise in various contexts of algorithm design and analysis. They are commonly used techniques for analyzing the performance of approximation and online algorithms, especially when direct performance evaluation is challenging. The main idea is to characterize the worst-case performance as a family of LPs parameterized by an integer $n \ge 1$, representing the size of the input instance. To obtain the best possible bounds on the target ratio (e.g., approximation or competitive ratios), we often need to determine the optimal objective value (and the corresponding optimal solution) of a family of LPs as $n \to \infty$. One common method, called the Primal-Dual approach, involves examining the constraint structure in the primal and dual programs, then developing feasible analytical solutions to both that achieve equal or nearly equal objective values. Another approach, known as \emph{strongly factor-revealing LPs}, similarly requires careful investigation of the constraint structure in the primal program. In summary, both methods rely on \emph{instance-specific techniques}, which is difficult to generalize from one instance to another. In this paper, we introduce a general variational-calculus approach that enables us to analytically study the optimal value and solution to a family of LPs as their size approaches infinity. The main idea is to first reformulate the LP in the limit, as its size grows infinitely large, as a variational-calculus instance and then apply existing methods, such as the Euler-Lagrange equation and Lagrange multipliers, to solve it. We demonstrate the power of our approach through three case studies of online optimization problems and anticipate broader applications of this method.