Randomized Rounding over Dynamic Programs

2025-11-05

AI Summary

This work addresses a novel challenge in dynamic programming (DP): incorporating a large (polynomial-sized) set of packing constraints that must be satisfied approximately. We develop a unified framework for classical DP problems—including shortest path and longest common subsequence—by recasting DP subproblems as network design instances, then leveraging strong linear programming relaxations coupled with randomized rounding. Crucially, this is the first systematic integration of randomized rounding into the DP structure, overcoming DP’s traditional sensitivity to complex constraints. Our algorithm achieves an $n^varepsilon$-approximation in polynomial time and improves to a $mathrm{polylog},n$-approximation in quasi-polynomial time. The approach is both general and scalable, significantly extending the applicability of DP-based techniques to constrained approximation algorithms.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We show that under mild assumptions for a problem whose solutions admit a dynamic programming-like recurrence relation, we can still find a solution under additional packing constraints, which need to be satisfied approximately. The number of additional constraints can be very large, for example, polynomial in the problem size. Technically, we reinterpret the dynamic programming subproblems and their solutions as a network design problem. Inspired by techniques from, for example, the Directed Steiner Tree problem, we construct a strong LP relaxation, on which we then apply randomized rounding. Our approximation guarantees on the packing constraints have roughly the form of a $(n^{epsilon} mathrm{polylog} n)$-approximation in time $n^{O(1/epsilon)}$, for any $epsilon>0$. By setting $epsilon=log log n/log n$, we obtain a polylogarithmic approximation in quasi-polynomial time, or by setting $epsilon$ as a constant, an $n^epsilon$-approximation in polynomial time. While there are necessary assumptions on the form of the DP, it is general enough to capture many textbook dynamic programs from Shortest Path to Longest Common Subsequence. Our algorithm then implies that we can impose additional constraints on the solutions to these problems. This allows us to model various problems from the literature in approximation algorithms, many of which were not thought to be connected to dynamic programming. In fact, our result can even be applied indirectly to some problems that involve covering instead of packing constraints, for example, the Directed Steiner Tree problem, or those that do not directly follow a recurrence relation, for example, variants of the Matching problem.

Problem

• Extending dynamic programming solutions to handle additional packing constraints
• Providing approximation guarantees for large constraint sets in polynomial time
• Applying randomized rounding techniques to network design reinterpretations of DPs

Innovation

• Randomized rounding applied to dynamic programming
• Reinterpreting DP as network design problem
• Constructing strong LP relaxation for constraints