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（VSPs）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VSPs的两种主要方法是基于树状结构（如树分解或团分解）的动态规划以及线性规划。在本工作中，我们通过证明以下结论，在这两种方法之间建立了紧密联系：若一个顶点子集问题$Π$存在一个保持解性质的动态规划算法，该算法在处理一个$n$顶点图$G$的宽度至多为$k$的树分解时，产生的表格大小至多为$α(k,n)$，则定义为$G$中$Π$的解的凸包的$P_Π(G)$多面体，其扩展复杂度至多为$O(α(k,n)\cdot n)$。此外，在指数时间假设（ETH）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最优保持解性质的动态规划算法，会为这些问题实例相关的解多面体产生最优尺寸的参数化扩展形式。另一方面，在扩展形式理论领域获得的无条件下界，则转化为保持解性质的动态规划算法在表格复杂度上的无条件下界。