Ejection chains, reference structures and alternating path methods for traveling salesman problems

Ejection chain procedures are based on the notion of generating compound sequences of moves, leading from one solution to another, by linked steps in which changes in selected elements cause other elements to be ''ejected from'' their current state, position or value assignment. This paper introduces new ejection chain strategies designed to generate neighborhoods of compound moves with attractive properties for traveling salesman problems. These procedures derive from the principle of creating a reference structure that coordinates the options for the moves generated. We focus on ejection chain processes related to alternating paths, and introduce three reference structures, of progressively greater scope, that produce both classical and non-standard alternating path trajectories. Theorems and examples show that various rules for exploiting these structures can generate moves not available to customary neighborhood search approaches. We also provide a reference structure that makes it possible to generate a collection of alternating paths that precisely expresses the symmetric difference between two tours. In addition to providing new results related to generalized alternating paths, in directed and undirected graphs, we lay a foundation for achieving a combinatorial leverage effect, where an investment of polynomial effort yields solutions dominating exponential numbers of alternatives. These consequences are explored in a sequel.