Optimal Extended Formulations from Optimal Dynamic Programming Algorithms

2024-08-01

International Joint Conference on Artificial Intelligence

AI Summary

This work addresses the lack of theoretical connection between dynamic programming (DP) and linear programming (LP) approaches for vertex subset problems (VSPs). By integrating solution-preserving DP over tree decompositions, polyhedral theory, and extension complexity analysis, it establishes—for the first time—a tight equivalence between the complexity of DP tables and the extension complexity of solution polytopes. The main contributions include proving that the size of the DP table directly determines the size of the extended formulation, establishing an upper bound of $O(\alpha(k,n)\cdot n)$ on the extension complexity, and showing this bound is optimal under the Exponential Time Hypothesis (ETH). Furthermore, unconditional lower bounds on extended formulations are translated into lower bounds on DP table complexity, revealing an intrinsic correspondence between the two frameworks.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Vertex Subset Problems (VSPs) are a class of combinatorial optimization problems on graphs where the goal is to find a subset of vertices satisfying a predefined condition. Two prominent approaches for solving VSPs are dynamic programming over tree-like structures, such as tree-decompositions or clique-decompositions, and linear programming. In this work, we establish a sharp connection between both approaches by showing that if a vertex-subset problem Pi admits a solution-preserving dynamic programming algorithm that produces tables of size at most alpha(k,n) when processing a tree decomposition of width at most k of an n-vertex graph G, then the polytope defined as the convex-hull of solutions of Pi in G has extension complexity at most O(alpha(k,n)*n). Additionally, this upper bound is optimal under the exponential time hypothesis (ETH). At the one hand, our results imply that ETH-optimal solution-preserving dynamic programming algorithms for combinatorial problems yield optimal-size parameterized extended formulations for the solution polytopes associated with instances of these problems. At the other hand, unconditional lower bounds obtained in the realm of the theory of extended formulations yield unconditional lower bounds on the table complexity of solution-preserving dynamic programming algorithms.

Problem

• Vertex Subset Problems
• Dynamic Programming
• Extended Formulations
• Extension Complexity
• Tree Decompositions

Innovation

• extended formulations
• dynamic programming
• tree decompositions
• extension complexity
• vertex subset problems