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 $Π$ admits a solution-preserving dynamic programming algorithm that produces tables of size at most $α(k,n)$ when processing a tree decomposition of width at most $k$ of an $n$-vertex graph $G$, then the polytope $P_Π(G)$ defined as the convex-hull of solutions of $Π$ in $G$ has extension complexity at most $O(α(k,n)\cdot n)$. 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.

翻译：顶点子集问题是一类图上的组合优化问题，其目标在于寻找满足预定条件的顶点子集。解决顶点子集问题的两种主要方法包括基于树状结构（如树分解或团分解）的动态规划，以及线性规划。本文通过建立这两种方法之间的紧密联系，证明了若顶点子集问题Π存在一个解保持动态规划算法，该算法在处理宽度至多为k的n顶点图G的树分解时，生成的表格大小至多为α(k,n)，则由G中Π的解的凸包定义的多面体P_Π(G)的扩展复杂度至多为O(α(k,n)·n)。此外，在指数时间假设下，该上界是最优的。一方面，我们的结果表明，针对组合问题的ETH最优解保持动态规划算法，能够为这些问题的实例相关的解多面体生成最优规模的参数化扩展形式化。另一方面，扩展形式化理论领域中获得的无条件下界，为解保持动态规划算法的表格复杂度提供了无条件下界。