Matching

A matching in a graph means the selection of a set of edges that are pairwise non-adjacent, i.e. they have no common incident vertices. A matching is maximal if it is not a proper subset of any other matching.

Usage

is_matching(graph, matching, types = NULL)

is_max_matching(graph, matching, types = NULL)

max_bipartite_match(
  graph,
  types = NULL,
  weights = NULL,
  eps = .Machine$double.eps
)

Arguments

graph: The input graph. It might be directed, but edge directions will be ignored.
matching: A potential matching. An integer vector that gives the pair in the matching for each vertex. For vertices without a pair, supply NA here.
types: Vertex types, if the graph is bipartite. By default they are taken from the ‘type’ vertex attribute, if present.
weights: Potential edge weights. If the graph has an edge attribute called ‘weight’, and this argument is NULL, then the edge attribute is used automatically. In weighted matching, the weights of the edges must match as much as possible.
eps: A small real number used in equality tests in the weighted bipartite matching algorithm. Two real numbers are considered equal in the algorithm if their difference is smaller than eps. This is required to avoid the accumulation of numerical errors. By default it is set to the smallest \(x\), such that \(1+x \ne 1\) holds. If you are running the algorithm with no weights, this argument is ignored.

Value

is_matching() and is_max_matching() return a logical scalar.

max_bipartite_match() returns a list with components:

matching_size: The size of the matching, i.e. the number of edges connecting the matched vertices.
matching_weight: The weights of the matching, if the graph was weighted. For unweighted graphs this is the same as the size of the matching.
matching: The matching itself. Numeric vertex id, or vertex names if the graph was named. Non-matched vertices are denoted by NA.

Details

is_matching() checks a matching vector and verifies whether its length matches the number of vertices in the given graph, its values are between zero (inclusive) and the number of vertices (inclusive), and whether there exists a corresponding edge in the graph for every matched vertex pair. For bipartite graphs, it also verifies whether the matched vertices are in different parts of the graph.

is_max_matching() checks whether a matching is maximal. A matching is maximal if and only if there exists no unmatched vertex in a graph such that one of its neighbors is also unmatched.

max_bipartite_match() calculates a maximum matching in a bipartite graph. A matching in a bipartite graph is a partial assignment of vertices of the first kind to vertices of the second kind such that each vertex of the first kind is matched to at most one vertex of the second kind and vice versa, and matched vertices must be connected by an edge in the graph. The size (or cardinality) of a matching is the number of edges. A matching is a maximum matching if there exists no other matching with larger cardinality. For weighted graphs, a maximum matching is a matching whose edges have the largest possible total weight among all possible matchings.

Maximum matchings in bipartite graphs are found by the push-relabel algorithm with greedy initialization and a global relabeling after every \(n/2\) steps where \(n\) is the number of vertices in the graph.

Author

Tamas Nepusz ntamas@gmail.com

Examples

g <- graph_from_literal(a - b - c - d - e - f)
m1 <- c("b", "a", "d", "c", "f", "e") # maximal matching
m2 <- c("b", "a", "d", "c", NA, NA) # non-maximal matching
m3 <- c("b", "c", "d", "c", NA, NA) # not a matching
is_matching(g, m1)
#> [1] TRUE
is_matching(g, m2)
#> [1] TRUE
is_matching(g, m3)
#> [1] FALSE
is_max_matching(g, m1)
#> [1] TRUE
is_max_matching(g, m2)
#> [1] FALSE
is_max_matching(g, m3)
#> [1] FALSE

V(g)$type <- rep(c(FALSE, TRUE), 3)
print_all(g, v = TRUE)
#> IGRAPH 2775395 UN-B 6 5 -- 
#> + attr: name (v/c), type (v/l)
#> + vertex attributes:
#> |     name  type
#> | [1]    a FALSE
#> | [2]    b  TRUE
#> | [3]    c FALSE
#> | [4]    d  TRUE
#> | [5]    e FALSE
#> | [6]    f  TRUE
#> + edges from 2775395 (vertex names):
#> [1] a--b b--c c--d d--e e--f
max_bipartite_match(g)
#> $matching_size
#> [1] 3
#> 
#> $matching_weight
#> [1] 3
#> 
#> $matching
#>   a   b   c   d   e   f 
#> "b" "a" "d" "c" "f" "e" 
#> 

g2 <- graph_from_literal(a - b - c - d - e - f - g)
V(g2)$type <- rep(c(FALSE, TRUE), length.out = vcount(g2))
print_all(g2, v = TRUE)
#> IGRAPH 58829d5 UN-B 7 6 -- 
#> + attr: name (v/c), type (v/l)
#> + vertex attributes:
#> |     name  type
#> | [1]    a FALSE
#> | [2]    b  TRUE
#> | [3]    c FALSE
#> | [4]    d  TRUE
#> | [5]    e FALSE
#> | [6]    f  TRUE
#> | [7]    g FALSE
#> + edges from 58829d5 (vertex names):
#> [1] a--b b--c c--d d--e e--f f--g
max_bipartite_match(g2)
#> $matching_size
#> [1] 3
#> 
#> $matching_weight
#> [1] 3
#> 
#> $matching
#>   a   b   c   d   e   f   g 
#> "b" "a" "d" "c" "f" "e"  NA 
#> 
#' @keywords graphs