Minimum Vertex Cover (Bruteforce, Approximation, DP, Greedy)

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 VC 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 to a graph, it will also highlight its MIS (which is its complement) with light blue color.

Remarks: By default, we show e-Lecture Mode for first time (or non logged-in) visitor.
If you are an NUS student and a repeat visitor, please login.

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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.

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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 lightblue 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.

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There are two different sources for specifying an input graph:

Draw Graph: You can draw any connected (un)directed weighted graph as the input graph.
Example Graphs: You can select from the list of example connected undirected weighted 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.

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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 and checking whether there is any edge that is not 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., very slow.

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

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The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.

If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.

FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

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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, if the input graph is a tree.

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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, if the input graph is an unweighted tree.

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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).

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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.

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

About Team Terms of use Privacy Policy

About

VisuAlgo was conceptualised in 2011 by Dr Steven Halim as a tool to help his students better understand data structures and algorithms, by allowing them to learn the basics on their own and at their own pace.

VisuAlgo contains many advanced algorithms that are discussed in Dr Steven Halim's book ('Competitive Programming', co-authored with his brother Dr Felix Halim and his friend Dr Suhendry Effendy) and beyond. Today, a few of these advanced algorithms visualization/animation can only be found in VisuAlgo.

Though specifically designed for National University of Singapore (NUS) students taking various data structure and algorithm classes (e.g., CS1010/equivalent, CS2040/equivalent, CS3230, CS3233, and CS4234), as advocators of online learning, we hope that curious minds around the world will find these visualizations useful too.

VisuAlgo is not designed to work well on small touch screens (e.g., smartphones) from the outset due to the need to cater for many complex algorithm visualizations that require lots of pixels and click-and-drag gestures for interaction. The minimum screen resolution for a respectable user experience is 1024x768 and only the landing page is relatively mobile-friendly. However, we are currently experimenting with a mobile (lite) version of VisuAlgo to be ready by April 2022.

VisuAlgo is an ongoing project and more complex visualizations are still being developed.

The most exciting development is the automated question generator and verifier (the online quiz system) that allows students to test their knowledge of basic data structures and algorithms. The questions are randomly generated via some rules and students' answers are instantly and automatically graded upon submission to our grading server. This online quiz system, when it is adopted by more CS instructors worldwide, should technically eliminate manual basic data structure and algorithm questions from typical Computer Science examinations in many Universities. By setting a small (but non-zero) weightage on passing the online quiz, a CS instructor can (significantly) increase his/her students mastery on these basic questions as the students have virtually infinite number of training questions that can be verified instantly before they take the online quiz. The training mode currently contains questions for 12 visualization modules. We will soon add the remaining 12 visualization modules so that every visualization module in VisuAlgo have online quiz component.

We have translated VisuAlgo pages into three main languages: English, Chinese, and Indonesian. Currently, we have also written public notes about VisuAlgo in various languages:

id, kr, vn, th.

Team

Project Leader & Advisor (Jul 2011-present)
Dr Steven Halim, Senior Lecturer, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Senior Software Engineer, Google (Mountain View)

Undergraduate Student Researchers 1 (Jul 2011-Apr 2012)
Koh Zi Chun, Victor Loh Bo Huai

Final Year Project/UROP students 1 (Jul 2012-Dec 2013)
Phan Thi Quynh Trang, Peter Phandi, Albert Millardo Tjindradinata, Nguyen Hoang Duy

Final Year Project/UROP students 2 (Jun 2013-Apr 2014)
Rose Marie Tan Zhao Yun, Ivan Reinaldo

Undergraduate Student Researchers 2 (May 2014-Jul 2014)
Jonathan Irvin Gunawan, Nathan Azaria, Ian Leow Tze Wei, Nguyen Viet Dung, Nguyen Khac Tung, Steven Kester Yuwono, Cao Shengze, Mohan Jishnu

Final Year Project/UROP students 3 (Jun 2014-Apr 2015)
Erin Teo Yi Ling, Wang Zi

Final Year Project/UROP students 4 (Jun 2016-Dec 2017)
Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir

Final Year Project/UROP students 5 (Aug 2021-Dec 2022)
Liu Guangyuan, Manas Vegi, Sha Long, Vuong Hoang Long

Final Year Project/UROP students 6 (Aug 2022-Apr 2023)
Lim Dewen Aloysius, Ting Xiao

List of translators who have contributed ≥100 translations can be found at statistics page.

Acknowledgements
This project is made possible by the generous Teaching Enhancement Grant from NUS Centre for Development of Teaching and Learning (CDTL).

Terms of use

VisuAlgo is free of charge for Computer Science community on earth. If you like VisuAlgo, the only "payment" that we ask of you is for you to tell the existence of VisuAlgo to other Computer Science students/instructors that you know =) via Facebook/Twitter/Instagram/TikTok posts, course webpages, blog reviews, emails, etc.

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List of Publications

This work has been presented briefly at the CLI Workshop at the ICPC World Finals 2012 (Poland, Warsaw) and at the IOI Conference at IOI 2012 (Sirmione-Montichiari, Italy). You can click this link to read our 2012 paper about this system (it was not yet called VisuAlgo back in 2012) and this link for the short update in 2015 (to link VisuAlgo name with the previous project).

This work is done mostly by my past students.

Bug Reports or Request for New Features

VisuAlgo is not a finished project. Dr Steven Halim is still actively improving VisuAlgo. If you are using VisuAlgo and spot a bug in any of our visualization page/online quiz tool or if you want to request for new features, please contact Dr Steven Halim. His contact is the concatenation of his name and add gmail dot com.

Privacy Policy

Version 1.1 (Updated Fri, 14 Jan 2022).

Disclosure to all visitors: We currently use Google Analytics to get an overview understanding of our site visitors. We now give option for user to Accept or Reject this tracker.

Since Wed, 22 Dec 2021, only National University of Singapore (NUS) staffs/students and approved CS lecturers outside of NUS who have written a request to Steven can login to VisuAlgo, anyone else in the world will have to use VisuAlgo as an anonymous user that is not really trackable other than what are tracked by Google Analytics.

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For other CS lecturers worldwide who have written to Steven, a VisuAlgo account (your (non-NUS) email address, you can use any display name, and encrypted password) is needed to distinguish your online credential versus the rest of the world. Your account will be tracked similarly as a normal NUS student account above but it will have CS lecturer specific features, namely the ability to see the hidden slides that contain (interesting) answers to the questions presented in the preceding slides before the hidden slides. You can also access Hard setting of the VisuAlgo Online Quizzes. You can freely use the material to enhance your data structures and algorithm classes. Note that there can be other CS lecturer specific features in the future.

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