Separating a superclass of comb inequalities in planar graphs

Letchford, A. N. (2000) Separating a superclass of comb inequalities in planar graphs. Mathematics of Operations Research, 25 (3). pp. 443-454. ISSN 0364-765X

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Many classes of valid and facet-inducing inequalities are known for the family of polytopes associated with the Symmetric Travelling Salesman Problem (STSP), including subtour elimination, 2-matching and comb inequalities. For a given class of inequalities, an exact separation algorithm is a procedure which, given an LP relaxation vector x*, finds one or more inequalities in the class which are violated by x*, or proves that none exist. Such algorithms are at the core of the highly successful branch-and-cut algorithms for the STSP. However, whereas polynomial time exact separation algorithms are known for subtour elimination and 2-matching inequalities, the complexity of comb separation is unknown. A partial answer to the comb problem is provided in this paper. We define a generalization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges with x*_e > 0 is planar. The separation algorithm runs in O(n^3) time, where n is the number of vertices in the graph.

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Mathematics of Operations Research
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11 Jul 2011 17:56
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21 Nov 2022 20:56