7    VisuAlgo.net / /mvc Login Unweighted Minimum Vertex Cover Weighted MVC
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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) 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 value of MVC 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.

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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.

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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.

Another pro-tip: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution or larger (typical modern laptop resolution in 2017). 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:

  1. Draw Graph: You can draw any connected (un)directed weighted graph as the input graph.
  2. Example Graphs: You can select from the list of example connected undirected weighted graphs to get you started.
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Bruteforce: It tries all possible 2^V subset 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 check 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.

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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:

  1. For unweighted version, we have either the deterministic 2-approximation or probabilistic 2-approximation (in expectation),
  2. For weighted version whe 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.

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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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MVC on Tree

MVC on Bipartite Graph



General Graph

Linear Chain

Unweighted 2-approx Killer

Weighted 2-approx Killer



Bipartite Graph

CS4234 Sample

DP on Tree

Greedy MVC on Tree

Kőnig's Theorem

Deterministic 2-opt

Probabilistic 2-opt

关于 团队 使用条款


VisuAlgo在2011年由Steven Halim博士概念化,作为一个工具,帮助他的学生更好地理解数据结构和算法,让他们自己和自己的步伐学习基础。
VisuAlgo包含许多高级算法,这些算法在Steven Halim博士的书(“竞争规划”,与他的兄弟Felix Halim博士合作)和其他书中讨论。今天,一些高级算法的可视化/动画只能在VisuAlgo中找到。
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Dr Steven Halim, Senior Lecturer, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Software Engineer, Google (Mountain View)

本科生研究人员 1 (Jul 2011-Apr 2012)
Koh Zi Chun, Victor Loh Bo Huai

最后一年项目/ UROP学生 1 (Jul 2012-Dec 2013)
Phan Thi Quynh Trang, Peter Phandi, Albert Millardo Tjindradinata, Nguyen Hoang Duy

最后一年项目/ UROP学生 2 (Jun 2013-Apr 2014)
Rose Marie Tan Zhao Yun, Ivan Reinaldo

本科生研究人员 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

最后一年项目/ UROP学生 3 (Jun 2014-Apr 2015)
Erin Teo Yi Ling, Wang Zi

最后一年项目/ UROP学生 4 (Jun 2016-Dec 2017)
Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir

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



这项工作在2012年ACM ICPC世界总决赛(波兰,华沙)和IOI 2012年IOI大会(意大利Sirmione-Montichiari)的CLI讲习班上进行了简要介绍。您可以点击此链接阅读我们2012年关于这个系统的文章(它在2012年还没有被称为VisuAlgo)。
这项工作主要由我过去的学生完成。最近的最后报告是:Erin,Wang Zi,Rose,Ivan。
VisuAlgo不是一个完成的项目。 Steven Halim博士仍在积极改进VisuAlgo。如果您在使用VisuAlgo并在我们的可视化页面/在线测验工具中发现错误,或者如果您想要求添加新功能,请联系Dr Steven Halim博士。他的联系邮箱是他的名字加谷歌邮箱后缀:StevenHalim@gmail.com。