Uncapacitated flow-based extended formulations

• 成果类型: Article
• 署名作者: Fiorini, Samuel; Pashkovich, Kanstantsin
• 署名单位: Universite Libre de Bruxelles; University of Padua
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-015-0862-9
• 发表日期: 2015
• 页码: 117-131
• 关键词: integrality gaps; sherali-adams; Combinatorial Optimization; relaxations

摘要

An extended formulation of a polytope is a linear description of this polytope using extra variables besides the variables in which the polytope is defined. The interest of extended formulations is due to the fact that many interesting polytopes have extended formulations with a lot fewer inequalities than any linear description in the original space. This motivates the development of methods for, on the one hand, constructing extended formulations and, on the other hand, proving lower bounds on the sizes of extended formulations. Network flows are a central paradigm in discrete optimization, and are widely used to design extended formulations. We prove exponential lower bounds on the sizes of uncapacitated flow-based extended formulations of several polytopes, such as the (bipartite and non-bipartite) perfect matching polytope and TSP polytope. We also give new examples of flow-based extended formulations, e.g., for 0/1-polytopes defined from regular languages.