The maximum weighted stable set problem in claw-free graphs is a well-known generalization of the maximum weighted matching problem, and a classical problem in combinatorial optimization. In spite of the recent development of fast(er) combinatorial algorithms and some progresses in the characterization of the corresponding stable set polytope, the problem of “providing a decent linear description” for this polytope (Grötschel et al. in Geometric algorithms and combinatorial optimization, Springer, New York, 1988) is still open. The main contribution of this paper is to propose an algorithmic answer to that question by providing a polynomial-time and computationally attractive separation routine for the stable set polytope of claw-free graphs, that only requires a combinatorial decomposition algorithm, the solution of (moderate sized) compact linear programs, and Padberg and Rao’s algorithm for separating over the matching polytope. In particular, it is a generalization of the latter and avoids the heavy computational burden of resorting to the ellipsoid method, on which the only poly-time separation routine known so far relied. Besides, our separation routine comes with a ‘small’ (but not polynomial) extended linear programming formulation and a procedure to derive a linear description of the stable set polytope of claw-free graphs in the original space.

无爪图中的最大加权稳定集问题是最大加权匹配问题的众所周知的概括，是组合优化中的经典问题。尽管最近发展了快速（er）组合算法，并且在表征相应的稳定集多态性方面取得了一些进展，但“提供合适的线性描述”（Grötschel等，几何算法和组合优化，Springer，纽约，1988年）仍然开放。本文的主要贡献是通过为无爪图的稳定集多态性提供多项式时间和计算上有吸引力的分离例程，从而提出该问题的算法答案，该例程仅需要组合分解算法，即（中等大小的紧凑线性程序，以及Padberg和Rao的算法来分离匹配的多态性。特别是后者的一般化，避免了依靠椭球法的繁重的计算负担，而椭球法是迄今为止唯一已知的多时间分离程序所依赖的。除了，