Graph Matching

1. Introduction

A Matching in a graph G = (V, E) is a subset M of E edges in G such that no two of which meet at a common vertex.


Maximum Cardinality Matching (MCM) problem is a Graph Matching problem where we seek a matching M that contains the largest possible number of edges. A possible variant is Perfect Matching where all V vertices are matched, i.e., the cardinality of M is V/2.


A Bipartite Graph is a graph whose vertices can be partitioned into two disjoint sets U and V such that every edge can only connect a vertex in U to a vertex in V.


Maximum Cardinality Bipartite Matching (MCBM) problem is the MCM problem in a Bipartite Graph, which is a lot easier than MCM problem in a General Graph.

1-1. Current Limitation: Unweighted Graphs

This visualization is currently limited to unweighted graphs only. Thus, we currently do not support Graph Matching problem variants involving weighted graphs...

1-2. Switching Modes

To switch between the unweighted MCBM (default, as it is much more popular) and unweighted MCM mode, click the respective header.


Here is an example of MCM mode. In MCM mode, one can draw a General, not necessarily Bipartite graphs. However, the graphs are unweighted (all edges have uniform weight 1).


The available algorithms are different in the two modes.

2. Visualisation

You can view the visualisation here!


For Bipartite Graph visualization, we will re-layout the vertices of the graph so that the two disjoint sets (U and V) are clearly visible as Left (U) and Right (V) sets. For General Graph, we do not relayout the vertices.


Initially, edges have grey color. Matched edges will have black color. Free/Matched edges along an augmenting path will have Orange/Light Blue colors, respectively.

3. Input Graph

There are three different sources for specifying an input graph:

  1. Draw Graph: You can draw any undirected unweighted graph as the input graph (note that in MCBM mode, the drawn input graph will be relayout into a nice Bipartite graph layout during algorithm animation),
  2. Modeling: A lot of graph problems can be reduced into an MCBM problem. In this visualization, we have the modeling examples for the famous Rook Attack problem (currently disabled) and standard MCBM problem (also valid in MCM mode).
  3. Examples: You can select from the list of our example graphs to get you started. The list of examples are slightly different in the two MCBM vs MCM modes.

4. MCBM Algorithms

There are several Max Cardinality Bipartite Matching (MCBM) algorithms in this visualization, plus one more in Max Flow visualization:

  1. O(VE) Augmenting Path Algorithm (without greedy pre-processing),
  2. O(√(V)E) Dinic's Max Flow Algorithm, see Max Flow visualization, select Modeling → Bipartite Matching → All 1, then use Dinic's algorithm.
  3. O(√(V)E) Hopcroft-Karp Algorithm,
  4. O(kE) Augmenting Path Algorithm++ (with randomized greedy pre-processing),

PS: Although possible, we will likely not use O(V3) Edmonds' Matching Algorithm if the input is guaranteed to be a Bipartite Graph.

4-1. O(VE) Augmenting Path Algorithm

Augmenting Path is a path that starts from a free (unmatched) vertex u in graph G (note that G does not necessarily has to be a bipartite graph), alternates through unmatched, match, ..., unmatched edges in G, until it ends at another free vertex v. If we flip the edge status along that augmenting path, we will increase the number of edges in the matching set M by 1 and eliminates this augmenting path.


In 1957, Claude Berge proposes the following lemma: A matching M in graph G is maximum iff there is no more augmenting path in G.


The Augmenting Path Algorithm is a simple O(V*(V+E)) = O(V2 + VE) = O(VE) implementation of that lemma (on Bipartite Graph): Find and then eliminate augmenting paths in Bipartite Graph G. Click Augmenting Path Algorithm Demo to visualize this algorithm on the currently displayed random Bipartite Graph.

4-2. Example C++ Code - Part 1

vi match, vis; // global variables

int Aug(int L) { // return 1 if ∃ an augmenting path from L
if (vis[L]) return 0; // return 0 otherwise
vis[L] = 1;
for (auto& v : AL[L]) {
int R = v.first;
if ((match[R] == -1) || Aug(match[R])) {
match[R] = L;
return 1; // found 1 matching
}
}
return 0; // no matching
}

4-3. Example C++ Code - Part 2

// in int main(), build the bipartite graph
// use directed edges from left set (of size VLeft) to right set
int MCBM = 0;
match.assign(V, -1);
for (int L = 0; L < VLeft; ++L) {
vis.assign(VLeft, 0);
MCBM += Aug(L); // find augmenting path starting from L
}
printf("Found %d matchings\n", MCBM);

Please see the full implementation at Competitive Programming book repository: mcbm.cpp|py|java|ml.

4-4. O(√(V)E) Dinic's Max Flow Algorithm

The MCBM problem can be modeled as a Max Flow problem. Go to Max Flow visualization page and see the flow graph modeling of MCBM problem (select Modeling → Bipartite Matching → all 1).


If we use one of the fastest Max Flow algorithm, i.e., Dinic's algorithm on this flow graph, we can find Max Flow = MCBM in O(√(V)E) time — analysis omitted for now. This allows us to solve MCBM problem with V ∈ [1000..1500] in a typical 1s allowed runtime in many programming competitions.

4-5. An Extreme Test Case

If we are given a Complete Bipartite Graph KN/2,N/2, i.e.,
V = N/2+N/2 = N and E = N/2×N/2 = N2/4 ≈ N2, then
the Augmenting Path Algorithm discussed earlier will run in O(VE) = O(N×N2) = O(N3).


This is only OK for V ∈ [400..500] in a typical 1s allowed runtime in many programming competitions.


Try executing the standard Augmenting Path Algorithm on this Extreme Test Case, which is an almost complete K5,5 Bipartite Graph.

4-6. O(√(V)E) Hopcroft-Karp Algorithm

The key idea of Hopcroft-Karp (HK) Algorithm (invented in 1973) is identical to Dinic's Max Flow Algorithm discussed earlier, i.e., prioritize shortest augmenting paths (in terms of number of edges used) first. That's it, augmenting paths with 1 edge are processed first before longer augmenting paths with 3 edges, 5 edges, 7 edges, etc (the length always increase by 2 due to the nature of augmenting path in a Bipartite Graph).


Hopcroft-Karp Algorithm has time complexity of O(√(V)E) — analysis omitted for now. This allows us to solve MCBM problem with V ∈ [1000..1500] in a typical 1s allowed runtime in many programming competitions — the similar range as with running Dinic's algorithm on Bipartite Matching flow graph.


Try HK Algorithm on the same Extreme Test Case earlier. You will notice that HK Algorithm can find the MCBM in a much faster time than the previous standard O(VE) Augmenting Path Algorithm.

4-7. O(kE) Augmenting Path Algorithm++

However, we can actually make the easy-to-code Augmenting Path Algorithm discussed earlier to avoid its worst case O(VE) behavior by doing O(V+E) randomized (to avoid adversary test case) greedy pre-processing before running the actual algorithm.


This O(V+E) additional pre-processing step is simple: For every vertex on the left set, match it with a randomly chosen unmatched neighbouring vertex on the right set. This way, we eliminates many trivial (one-edge) Augmenting Paths that consist of a free vertex u, an unmatched edge (u, v), and a free vertex v.


Try Augmenting Path Algorithm++ on the same Extreme Test Case earlier. Notice that the pre-processing step already eliminates many trivial 1-edge augmenting paths, making the actual Augmenting Path Algorithm only need to do little amount of additional work.

4-8. Another Hard Test Case

Quite often, on randomly generated Bipartite Graph, the randomized greedy pre-processing step has cleared most of the matchings.


However, we can construct test case like: Examples: Randomized Greedy Processing Killer to make randomization as ineffective as possible. For every group of 4 vertices, there are 2 matchings. Random greedy processing has 50% chance of making mistake per group. Try this Hard Test Case case to see for yourself.


The worst case time complexity is no longer O(VE) but now O(kE) where k is a small integer, much smaller than V, k can be as small as 0 and is at most V/2. In our empirical experiments, we estimate k to be "about √(V)" too. This version of Augmenting Path Algorithm++ also allows us to solve MCBM problem with V ∈ [1000..1500] in a typical 1s allowed runtime in many programming competitions.

5. MCM Algorithms

There are two Max Cardinality Matching (MCM) algorithms in this visualization:

  1. O(V^3) Edmonds's Matching algorithm (without greedy pre-processing),
  2. O(V^3) Edmonds's Matching algorithm (with greedy pre-processing),

5-1. O(V^3) Edmonds' Matching Algorithm

In General Graph, we may have Odd-Length cycle. Augmenting Path is not well defined in such graph, hence we cannot directly implement Claude Berge's lemma like what we did with Bipartite Graph.


Jack Edmonds call a path that starts from a free vertex u, alternates between free, matched, ..., free edges, and returns to the same free vertex u as Blossom. This situation is only possible if we have Odd-Length cycle, i.e., non-Bipartite Graph. Edmonds then proposed Blossom shrinking/contraction and expansion algorithm to solve this issue, details verbally.


This algorithm can be implemented in O(V^3).

5-2. O(V^3) Edmonds' Matching Algorithm++

As with the Augmenting Path Algorithm++ for the MCBM problem, we can also do randomized greedy pre-processing step to eliminate as many 'trivial matchings' as possible upfront. This reduces the amount of work of Edmonds' Matching Algorithm, thus resulting in a faster time complexity — analysis TBA.

6. Extras

We have not added visualizations for weighted variant of MCBM and MCM problems (future work).

6-1. Implementation

You are allowed to use/modify our implementation code for Augmenting Path Algorithm++:
mcbm.cpp
mcbm.java
mcbm.py
mcbm.ml