{"type":"ScholarlyArticle","title":"Optimal Extended Formulations from Optimal Dynamic Programming Algorithms","authors":[{"name":"Mateus de Oliveira Oliveira","slug":"mateus-de-oliveira-oliveira","given":"Mateus","family":"de Oliveira Oliveira","affiliation":"Department of Computer and Systems Sciences , Stockholm University , Sweden; Department of Informatics , University of Bergen , Norway"},{"name":"Wim Van den Broeck","slug":"wim-van-den-broeck","given":"Wim","family":"Van den Broeck","affiliation":"Department of Informatics , University of Bergen , Norway"}],"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).\r\n\r\nAt 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.","doi":"10.24963/ijcai.2024/208","url":"https://ijcai.org/proceedings/2024/208","pdf":"https://ijcai.org/proceedings/2024/0208.pdf","markdown":"https://ijcai.org/proceedings/2024/208.md","bibtex":"https://ijcai.org/proceedings/2024/bibtex/208","volume":{"slug":"2024","title":"Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2024","url":"https://ijcai.org/proceedings/2024/","year":2024,"location":"Jeju, Korea","dates":"3-9 August 2024","editors":"Kate Larson","isbn":"978-1-956792-04-1","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2024"},"section":"Constraint Satisfaction and Optimization","track":"Main Track","number":208,"pages":{"first":1881,"last":1888},"keywords":["Constraint Satisfaction and Optimization: General","Constraint Satisfaction and Optimization: CSO: Mixed discrete and continuous optimization","Constraint Satisfaction and Optimization: CSO: Constraint optimization problems"],"videos":[],"references":[{"position":1,"raw":"[Aboulker et al., 2019] Pierre Aboulker, Samuel Fiorini, Tony Huynh, Marco Macchia, and Johanna Seif. 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