Summary of maximum match, minimum point coverage, minimum path coverage, and maximum independent set

(1) Maximum matching of binary graphs

Hungarian algorithm

(2) The minimum point coverage of the dichotomy graph

The minimum point coverage of a binary graph = The maximum match of two-minute graphs

Minimum point coverage: from the right side of all no matching points, according to the augmented road "alternately appear" requirements of DFS. At the end, there are no visited points on the right and the points to the left are the minimum point overrides.

Proof, see here.

(3) Minimum side coverage of the two-part graph

The least side overlay of the binary graph = The maximum match of the two-point graph

Prove:

First greedy Select a set of the largest matching edge put into the set, for the rest of the points that do not match, randomly select a link with the associated side into the collection, then the resulting set is the minimum edge overlay.

So there are: minimum side overlay = Maximum match + points -2* maximum match = points-Maximum match

(4) The maximum independent set of the dichotomy graph

Maximum independent set of binary graphs = number of points-two maximum matching of graphs

Prove:

We can think of this, first put all the points in the set, and then delete the fewest points and the associated side, so that all the edges are deleted, this is the minimum point coverage. So there are: maximum independent set = points-Minimum point coverage

(5) Minimum disjoint path coverage for a non-circular graph

We $v$ the points in the original image into two points $vx$ and $vy$, and for the side $a->b$ of the original, we even $ax->by$ in the new diagram. So least disjoint path overrides = original number of points-maximum match of new graph

Prove:

At first, each point is independent of a path, and in the dichotomy, the path is merged, and the number of each edge path is reduced by one. Because paths cannot intersect, there is no common point, which is exactly the definition of a match. So there are: least disjoint path overrides = original number of points-maximum match of new graph

Friendship question:

BZOJ1143[CTSC2008] Festival River

(6) A least-intersecting path covering a non-circular graph

First use Floyd to find the original image of the transmission closure, if a to B has a way, then add edge a->b. It then translates to the least disjoint path coverage problem.

(7) Mutual transformation of least disjoint path coverage and maximal independent set in a forward-free graph

With partial order sets, it is recommended to look at this blog first

Maximum independent set of a forward-free graph = least disjoint path overlay with a forward-free graph

Friendship question:

BZOJ3997[TJOI2015] Combinatorial mathematics

(8) The weighted maximum matching of the two-fractal graphs

KM algorithm.

Summary of maximum match, minimum point coverage, minimum path coverage, and maximum independent set