A polyhedral study of the generalized vertex packing problem

• 成果类型: Article
• 署名作者: Sherali, HD; Smith, JC
• 署名单位: Virginia Polytechnic Institute & State University; University of Arizona
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-004-0504-0
• 发表日期: 2006
• 页码: 367-390
• 关键词: one programming-problems; relaxations; hierarchy; polytopes; graphs

摘要

The traditional vertex packing problem defined on an undirected graph identifies the largest weighted independent set of nodes, that is, a set of nodes whose induced subgraph contains no edges. In this paper, we examine a generalized vertex packing problem (GVP-k) in which k violations to the independent set restriction are permitted, whereby k edges may exist within the subgraph induced by the chosen set of nodes. A particular context in which such problems arise is in the national airspace planning model of Sherali, Smith, and Trani (2000), where a set of flight-plans need to be composed for various flights subject to conflict, workload, and equity considerations. The GVP-k structure arises in modeling the air-traffic control sector workload restrictions, which stipulate that for each sector and during each discretized time-slot, the number of aircraft conflicts that would need to be resolved should not exceed k, for some k >= N1. We derive several classes of facetial valid inequalities for GVP-k for certain specially structured subgraphs, identifying polynomial-sized convex hull representations for some of these cases. Related constraint generation routines are also developed, and some computational results are provided to demonstrate the efficacy of utilizing the proposed valid inequalities in solving GVP-k for different values of k.