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Optimal Extended Formulations from Optimal Dynamic Programming Algorithms

Mateus de Oliveira Oliveira, Wim Van den Broeck

TL;DR

The paper establishes a foundational link between dynamic programming on tree decompositions and extended formulations in polyhedral combinatorics for vertex-subset problems. By formalizing solution-preserving DP-cores and introducing T-shaped tree automata, it shows that a DP-core with table complexity α(k,n) yields an extended formulation for the solution polytope with size O(α(k,n)·n), and proves ETH-optimality of this bound. The framework extends to edge-subset and tuple problems, enabling a unified treatment of problems like Independent Set, Dominating Set, Cut, Hamiltonian Cycle, and d-coloring, with detailed upper and lower bounds that align with known ETH and XC results. This yields a principled method to convert DP-table structure into compact LP formulations, offering both algorithmic insights and polyhedral limits for a broad class of graph problems. The approach highlights the interplay between parameterized DP and polyhedral theory, providing a robust toolkit for understanding the (non)existence of small extended formulations under ETH and related hypotheses.

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 $Π$ 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.

Optimal Extended Formulations from Optimal Dynamic Programming Algorithms

TL;DR

The paper establishes a foundational link between dynamic programming on tree decompositions and extended formulations in polyhedral combinatorics for vertex-subset problems. By formalizing solution-preserving DP-cores and introducing T-shaped tree automata, it shows that a DP-core with table complexity α(k,n) yields an extended formulation for the solution polytope with size O(α(k,n)·n), and proves ETH-optimality of this bound. The framework extends to edge-subset and tuple problems, enabling a unified treatment of problems like Independent Set, Dominating Set, Cut, Hamiltonian Cycle, and d-coloring, with detailed upper and lower bounds that align with known ETH and XC results. This yields a principled method to convert DP-table structure into compact LP formulations, offering both algorithmic insights and polyhedral limits for a broad class of graph problems. The approach highlights the interplay between parameterized DP and polyhedral theory, providing a robust toolkit for understanding the (non)existence of small extended formulations under ETH and related hypotheses.

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 admits a solution-preserving dynamic programming algorithm that produces tables of size at most when processing a tree decomposition of width at most of an -vertex graph , then the polytope defined as the convex-hull of solutions of in has extension complexity at most . 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.
Paper Structure (29 sections, 26 theorems, 31 equations)

This paper contains 29 sections, 26 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Our Results
  3. Proof Techniques
  4. Generalizations
  5. Preliminaries
  6. Basic Definitions
  7. Vertex Subset Problems
  8. Solution Polytopes
  9. Addressed Trees
  10. Dynamic Programming Cores for Tree Decompositions
  11. The Polytope of a $T$-Shaped Language
  12. $T$-Shaped Tree Automata
  13. The Polytope $P(\mathcal{A})$
  14. An Extended Formulation for $P(\mathcal{A})$
  15. Proof of Theorem \ref{['theorem:MainTheorem']}
  16. ...and 14 more sections

Key Result

Proposition 3.3

Let $G$ be a graph, $\mathcal{T} = (T,B,\xi)$ be a tree decomposition of $G$. Then, for each $u\in N(T)$, and each pair $(S,c)\in \mathcal{P}(\mathbb{N})\times \mathbb{N}$, $(S,c)$ belongs to $\Gamma(u)$ if and only if the graph $G_{u}$ has an independent set $I$ of size at least $c$ such that $S =

Theorems & Definitions (55)

  • Definition 2.1: Vertex Subset Problem
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3: BlueBook Ch. 7.3, KleinbergTardos2006 Ch. 10.4
  • Theorem 4.1
  • Corollary 4.2
  • Definition 4.3
  • Theorem 4.4
  • Lemma 4.5
  • proof
  • ...and 45 more