A **Vertex Cover** (VC) of a connected undirected (un)weighted graph **G** is a **subset of vertices V** of **G** such that **for every edge** in **G**, **at least one of its endpoints is in V**. A **Minimum Vertex Cover (MVC)** (**Minimum Weight Vertex Cover (MWVC)** for the weighted variant) of **G** is a VC that has the smallest cardinality (if unweighted) or total weight (if weighted) among all possible VCs. A graph can have multiple VCs but the cardinality/total weight of its MVC/MWVC is unique.

There is another problem called **Maximum Independent Set** (MIS) that attempts to find the **largest** subset of vertices in a (un)weighted graph **G** without any adjacent vertices in the subset. Interestingly, the **complement of an MVC of a graph is an MIS**.

At the end of every visualization, when an algorithm highlights an MVC solution of the currently displayed graph in orange color. For non-approximation solutions, the visualization will also highlight the MIS solution (which is V \ MVC) with light blue color.

MVC, MWVC, MIS, (and MWIS) are all NP-hard combinatorial optimization problems.

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There are two available modes: Unweighted (default) and Weighted. You can switch between the two modes by clicking the respective tab.

There are algorithms that work in both modes and there are algorithms that only work in a certain mode.

Pro-tip 1: Since you are not __logged-in__, you may be a first time visitor (or not an NUS student) who are not aware of the following keyboard shortcuts to navigate this e-Lecture mode: **[PageDown]**/**[PageUp]** to go to the next/previous slide, respectively, (and if the drop-down box is highlighted, you can also use **[→ or ↓/← or ↑]** to do the same),and **[Esc]** to toggle between this e-Lecture mode and exploration mode.

View the visualisation of the selected MVC algorithms here.

Originally, all vertices and edges in the input graph are colored with the standard black outline. As the visualization goes on, the color light blue will be used to denote covered edges and the color orange on edge will be used to show traversed edges.

At the end of the selected MVC algorithm, if it finds a **minimum** VC, it will highlight the MVC vertices with orange color and the non MVC vertices (a.k.a. the MIS vertices) with light blue color. Otherwise, if the found vertex cover is not proven to be the minimal one (e.g., the algorithm used is an approximation algorithm), it will highlight the vertices that belong to the found vertex cover with orange color without highlighting the MIS vertices.

Pro-tip 2: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution **or larger** (typical modern laptop resolution in 2021). We recommend using Google Chrome to access VisuAlgo. Go to full screen mode (**F11**) to enjoy this setup. However, you can use zoom-in (**Ctrl +**) or zoom-out (**Ctrl -**) to calibrate this.

There are two different ways to specify an input graph:

**Edit Graph**: You can edit the currently displayed undirected (weighted for MWVC mode) graph into any other undirected (weighted for MWVC mode) graph.**Example Graphs**: You can select from the list of example undirected (weighted for MWVC mode) graphs to get you started.

Pro-tip 3: Other than using the typical media UI at the bottom of the page, you can also control the animation playback using keyboard shortcuts (in Exploration Mode): **Spacebar** to play/pause/replay the animation, **←**/**→** to step the animation backwards/forwards, respectively, and **-**/**+** to decrease/increase the animation speed, respectively.

**Bruteforce**: It tries all possible 2^**V** subsets of vertices. In every iteration, it checks whether the currently selected subset of vertices is a valid vertex cover by iterating over all **E** edges in O(**E**) and checking whether all edges are covered by the vertices in the currently selected subset. This bruteforce algorithm keeps the smallest size of the valid vertex cover as the answer.

This bruteforce algorithm is available in both weighted and unweighted version.

Its time complexity is O(2^**V** × **E**), i.e., extremely slow. We are in the process to improve the visualization so that the 'boring' non-improving parts are cut-out and only the important 'candidate VC' subsets are highlighted.

Discussion: But there is an alternative O(2^**k** × **E**) parameterized solution if we are told that **k** is 'not-that-large'.

Unfortunately this part has not been digitized/visualized yet and is in the pipeline.

Please see CS4234 lecture note for the details.

**DP on Tree**: If the graph is a **tree**, the MVC problem can be formulated as a Dynamic Programming problem where the states are (position, take_current_vertex).

Then, it can be seen that:

DP(u, take) = cost[u] + sum(min(DP(v, take), DP(v, not_take))) ∀child v of u, and

DP(u, not take) = sum(DP(v, take)) ∀child v of u

This DP algorithm is available in both weighted and unweighted version.

Its time complexity is O(**V**), i.e., very fast, but only if the input graph is a tree.

**Greedy MVC on Tree**: Again, if the graph is an **unweighted** **tree**, it can be solved greedily by observing that if there is any MVC solution that takes a leaf vertex, we can obtain a "not worse" solution by taking the parent of that leaf vertex instead. After removing all covered vertices, we can apply the same observation and repeat it until every vertex is covered.

This greedy MVC algorithm is only available in unweighted mode.

Its time complexity is O(**V**), i.e., very fast, but only if the input graph is an unweighted tree.

**Kőnig's Theorem**: From Kőnig's Theorem, the size of MVC in an **unweighted bipartite** graph is equal to the cardinality of the maximum matching of the bipartite graph. In the case of **weighted bipartite** graph, we can see that this theorem also holds true, with a tweak in how we construct the graph. In this visualization, we use a reduction to max flow problem to get the value of the MVC.

This algorithm is available in both weighted and unweighted version.

Its time complexity is O(**V** × **E**) (for unweighted version; can be smaller with pre-processing) or O(**E**^2 × **V**)/O(**V**^2 × **E**) (for weighted version, depending on the max flow algorithm used).

There are several known approximation algorithms for MVC:

- For unweighted version, we have either the deterministic 2-approximation or probabilistic 2-approximation (in expectation),
- For weighted version we have the Bar-Yehuda and Even's 2-approximation algorithm.

Note that these algorithms only yield an "approximated" MVC, meaning that they are not a true **minimum** vertex cover, but a good enough one.

Unfortunately this part has not been digitized yet and is in the pipeline.

Please see CS4234 lecture note for the details.

You have reached the last slide. Return to 'Exploration Mode' to start exploring!

Note that if you notice any bug in this visualization or if you want to request for a new visualization feature, do not hesitate to drop an email to the project leader: Dr Steven Halim via his email address: stevenhalim at gmail dot com.

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Edit Graph

Example Graphs

Bruteforce

MVC on Tree

MVC on Bipartite Graph

Approximation

Parameterized MVC