Alternating paths algorithm

Algorithmic problem: The graph [math]\displaystyle{ G }[/math] is biparite. Type of algorithm: loop Auxillary data:

Abstract view

Invariant: Before and after each iteration, [math]\displaystyle{ M }[/math] is a matching. Variant: [math]\displaystyle{ |M| }[/math] increases by [math]\displaystyle{ 1 }[/math]. Break condition: There is no more augmenting alternating path.

Induction basis

Abstract view: is initialized to be some matching, for example, . Implementation: Obvious. Proof: Nothing to show.

Induction step

Abstract view: If there is an augmenting alternating path, use it to increase ; otherwise, terminate the algorithm and return . Implementation: Call the algorithm Find augmenting alternating path. If this call fails, terminate the algorithm and return . Otherwise, let denote the set of all edges on the path delivered by that call. Let be the symmetric difference of and Correctness:

Complexity

Statement: Proof:

Further information