Table 1 Research on the matching problem and its generalizations

(2-clique)

unweighted:

O(E)

1/2-approx. [16],[17],[19],

matching

$O\left(\sqrt{V}E\right)$[6]-[8],

1/2-approx. [22],

(1−ε)-approx. [18]

weighted:

O(E l o g V)

O(V(E+V l o g V)) [8]

1/2-approx. [23]

k-clique

NP-hard [14]

randomized

1/k-approx.

matching

in complete graphs

in our previous

for k=3 [20],[21]

paper [2]

H-matching

NP-hard [14]

—

unweighted:

max snp-complete [12]

- in unit-disk graphs [10],

- in planar graphs [11]