Optimal Extended Formulations from Optimal Dynamic Programming Algorithms

TL;DR

This paper establishes a fundamental link between dynamic programming algorithms and linear programming formulations for vertex subset problems, showing that optimal algorithms lead to optimal extended formulations, with bounds proven to be tight under ETH.

Contribution

It demonstrates that solution-preserving dynamic programming algorithms directly translate into small extended formulations for the associated polytopes, establishing a tight connection between algorithmic and polyhedral complexity.

Findings

Optimal DP algorithms imply small extended formulations.

Lower bounds on DP table size match polyhedral complexity bounds.

Results are tight under the exponential time hypothesis (ETH).

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 $P_{\Pi}(G)$ defined as the convex-hull of solutions of $\Pi$ in $G$ has extension complexity at most $O(\alpha(k,n)\cdot n)$. Additionally, this upper bound is optimal under the exponential time hypothesis (ETH). On 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. On 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.