Version 5.0 The LEDA User Manual

Heuristics for Initial Matching Constructions. Each function can be asked to start
with either an empty matching (heur = 0), a greedy matching (heur = 1) or an (adapted)
fractional matching (heur = 2); by default, the fractional matching heuristic is used.
Graph Structure. All functions assume the underlying graph (type graph) to be connected,
simple, loopfree and undirected (i.e., no anti-parallel edges).
Edge Weight Restrictions. The algorithms use divisions. In order to avoid rounding
errors for the number type int, please make sure that all edge weights are multiples
of 4; the algorithm will automatically multiply all edge weights by 4 if this
condition is not met. (Then, however, the returned dual solution is valid only with
respect to the modified weight function.) Moreover, in the maximum-weight (non-perfect)
matching case all edge weights are assumed to be non-negative.
Arithmetic Demand. The arithmetic demand for integer edge weights is as follows.
Let C denote the maximal absolute value of any edge weight and let n be the number of
nodes of the graph.
In the perfect weighted matching case we have for a potential pot[u] of a node u and for
a potential z B of a blossom B:
−nC/2 ≤ pot[u] ≤ (n + 1)C/2 and − nC ≤ z B ≤ nC.
In the non-perfect matching case we have for a potential pot[u] of a node u and for a
potential z B of a blossom B:
0 ≤ pot[u] ≤ C and 0 ≤ z B ≤ C.
The function CHECK WEIGHTS may be used to test whether the edge weights are
feasible or not. It is automatically called at the beginning of each of the algorithms
provided in this chapter.
Single Tree vs. Multiple Tree Approach: All functions can either run a single tree
approach or a multiple tree approach. In the single tree approach, one alternating tree is
grown from a free node at a time. In the multiple tree approach, multiple alternating trees
are grown simultaneously from all free nodes. On large instances, the multiple tree approach
is significantly faster and therefore is used by default. If #define SST APPROACH
is defined before the template file is included all functions will run the single tree approach.