随机算法 (Fall 2011)/Graph Coloring and 组合数学 (Fall 2011): Difference between pages

From TCS Wiki
(Difference between pages)
Jump to navigation Jump to search
VisualWikitext
imported>WikiSysop
m (Protected "随机算法 (Fall 2011)/Graph Coloring" ([edit=sysop] (indefinite) [move=sysop] (indefinite)))
 
imported>Etone
No edit summary
 
Line 1: Line 1:
=Graph Colorings=
{| style="float: right; border: 1px solid #BBB; margin: .46em 0 0 .2em;"
A '''proper coloring''' of a graph <math>G(V,E)</math> is a mapping <math>f:V\rightarrow[q]</math> for some integer <math>q</math>, satisfying that <math>f(u)\neq f(v)</math> for all <math>uv\in E</math>.
|- style="font-size: 86%;"
| valign="top" |
{{Infobox
|name        = Infobox
|bodystyle    =  
|title        = <font size=3>'''组合数学  <br>
Combinatorics'''</font>
|titlestyle  =


We consider the problem of sampling a uniformly random proper coloring of a given graph. We will later see that this is useful for counting the number of proper colorings of a given graph, which is a fundamental combinatorial problem, having important applications in statistic physics.
|image        = [[File:LW-combinatorics.jpeg|border|100px]]
|imagestyle  =
|caption      =
|captionstyle =
|headerstyle  = background:#ccf;
|labelstyle  = background:#ddf;
|datastyle    =


Let's first consider the decision version of the problem. That is, given as input a graph <math>G(V,E)</math>, decide whether there exists a proper <math>q</math>-coloring of <math>G</math>. Denote by <math>\Delta</math> the maximum degree of <math>G</math>.
|header1 =Instructor
* If <math>q\ge \Delta+1</math>, there always exists a proper coloring. Moreover, the proper coloring can be found by a simple greedy algorithm.
|label1  =
* If <math>q=\Delta</math>, <math>G</math> has a proper coloring unless it contains a <math>(\Delta+1)</math>-clique or it is an odd cycle. ([http://en.wikipedia.org/wiki/Brooks'_theorem Brooks Theorem])
|data1  =
* If <math>q<\Delta</math>, the problem is NP-hard.
|header2 =
|label2  =
|data2  = 尹一通
|header3 =
|label3  = Email
|data3  = yitong.yin@gmail.com  yinyt@nju.edu.cn 
|header4 =
|label4= office
|data4= 计算机系 804
|header5 = Class
|label5  =
|data5  =
|header6 =
|label6  = Class meetings
|data6  = Thursday, 10am-12pm <br> 仙逸B-104
|header7 =
|label7  = Place
|data7  =
|header8 =
|label8  = Office hours
|data8  = Wednesday, 2-5pm <br>计算机系 804
|header9 = Textbook
|label9  =
|data9  =
|header10 =
|label10  =
|data10  = ''van Lint and Wilson,'' <br> A course in Combinatorics, 2nd Ed, <br> Cambridge Univ Press, 2001.


Sampling a random coloring is at least as hard as deciding its existence, so we don't expect to solve the sampling problem when <math>q<\Delta</math>. The decision problem for the case <math>q=\Delta</math> is also nontrivial. Thus people are interested only in the case when <math>q\ge \Delta+1</math>.
|belowstyle = background:#ddf;
|below =
}}
|}
 
This is the page for the class ''Combinatorics'' for the Fall 2011 semester. Students who take this class should check this page periodically for content updates and new announcements.
 
= Announcement =
* <font size=3 color=red>由于有事需要外出，9月14日星期三下午的office hour改在9月13日下午。</font>
* <font size=3 color=red>第一、二次课的slides已发布，见lecture notes部分。</font>
 
= Course info =
* '''Instructor ''': 尹一通
:*email: yitong.yin@gmail.com, yinyt@nju.edu.cn,
:*office: 804
* '''Teaching fellow''': TBA
:*email: TBA
* '''Class meeting''': Thursday 10am-12pm, 仙逸B-104.
* '''Office hour''': Wednesday 2-5pm, 计算机系 804.


The following is a natural Markov chain for sampling proper colorings.


{{Theorem|Markov Chain for Graph Coloring|
= Syllabus =
:Start with a proper coloring of <math>G(V,E)</math>. At each step:
 
# Pick a vertex <math>v\in V</math> and a color <math>c\in[q]</math> uniformly at random.
=== 先修课程 Prerequisites ===
# Change the color of <math>v</math> to <math>c</math> if the resulting coloring is proper; do nothing if otherwise.
* 离散数学（Discrete Mathematics）
}}
* 线性代数（Linear Algebra）
* 概率论（Probability Theory）
 
=== Course materials ===
* [[组合数学 (Fall 2011)/Course materials|教材和参考书清单]]
 
=== 成绩 Grades ===
* 课程成绩：本课程将会有六次作业和一次期末考试。最终成绩将由平时作业成绩和期末考试成绩综合得出。
* 迟交：如果有特殊的理由，无法按时完成作业，请提前联系授课老师，给出正当理由。否则迟交的作业将不被接受。
 
=== <font color=red> 学术诚信 Academic Integrity </font>===
学术诚信是所有从事学术活动的学生和学者最基本的职业道德底线，本课程将不遗余力的维护学术诚信规范，违反这一底线的行为将不会被容忍。


For a fixed graph <math>G(V,E)</math>, the state space of the above Markov chain is the set of all proper colorings of <math>G</math> with <math>q</math> colors.
作业完成的原则：署你名字的工作必须由你完成。允许讨论，但作业必须独立完成，并在作业中列出所有参与讨论的人。不允许其他任何形式的合作——尤其是与已经完成作业的同学“讨论”。


{{Theorem|Lemma|
本课程将对剽窃行为采取零容忍的态度。在完成作业过程中，对他人工作（出版物、互联网资料、其他人的作业等）直接的文本抄袭和对关键思想、关键元素的抄袭，按照 [http://www.acm.org/publications/policies/plagiarism_policy ACM Policy on Plagiarism]的解释，都将视为剽窃。剽窃者成绩将被取消。如果发现互相抄袭行为，<font color=red> 抄袭和被抄袭双方的成绩都将被取消</font>。因此请主动防止自己的作业被他人抄袭。
The followings hold for the above Markov chain.
# Aperiodic.
# The transition matrix is symmetric.
# Irreducible if <math>q\ge \Delta+2</math>.
}}


The followings are the two most important conjectures regarding the problem.
学术诚信影响学生个人的品行，也关乎整个教育系统的正常运转。为了一点分数而做出学术不端的行为，不仅使自己沦为一个欺骗者，也使他人的诚实努力失去意义。让我们一起努力维护一个诚信的环境。
{{Theorem|Conjecture|
#The simple Markov chain defined above has mixing time <math>O(n\ln n)</math> whenever <math>q\ge\Delta+2</math>.
# Random sampling of proper graph colorings can be done in polynomial time whenever <math>q\ge\Delta+1</math>.
}}


These two conjectures are still open. People approach them by relax the requirement for the number of colors <math>q</math>. Intuitively, the larger the <math>q</math> is, the more freedom we have, the less dependency are there between non-adjacent vertices.
= Assignments =


=Coupling: <math>q\ge 4\Delta+1</math>=
= Lecture Notes =
# [[组合数学 (Fall 2011)/Basic enumeration|Basic enumeration]]  | [http://lamda.nju.edu.cn/yinyt/notes/comb2011/comb1.pdf slides1] | [http://lamda.nju.edu.cn/yinyt/notes/comb2011/comb2-1.pdf slides2]
# [[组合数学 (Fall 2011)/Generating functions|Generating functions]] | [http://lamda.nju.edu.cn/yinyt/notes/comb2011/comb2-2.pdf slides1]
# [[组合数学 (Fall 2011)/Sieve methods|Sieve methods]]
# [[组合数学 (Fall 2011)/Pólya's theory of counting|Pólya's theory of counting]]
# [[组合数学 (Fall 2011)/Counting and existence|Counting and existence]]
# [[组合数学 (Fall 2011)/Discrete probability|Discrete probability]]
# [[组合数学 (Fall 2011)/The probabilistic method|The probabilistic method]]
# [[组合数学 (Fall 2011)/Extremal graph theory| Extremal graph theory]]
# [[组合数学 (Fall 2011)/Matching theory|Matching theory]]
# [[组合数学 (Fall 2011)/Flow and matching | Flow and matching]]
# [[组合数学 (Fall 2011)/Optimization|Optimization]]
# [[组合数学 (Fall 2011)/Matroid|Matroid]]
# [[组合数学 (Fall 2011)/Extremal set theory|Extremal set theory]]
# [[组合数学 (Fall 2011)/Ramsey theory|Ramsey theory]]
# [[组合数学 (Fall 2011)/The Szemeredi regularity lemma|The Szemeredi regularity lemma]]


=Path Coupling: <math>q\ge 2\Delta+1</math> =
= Concepts =
* [http://en.wikipedia.org/wiki/Binomial_coefficient Binomial coefficient]
* [http://en.wikipedia.org/wiki/Composition_(number_theory) Composition of a number]
* [http://en.wikipedia.org/wiki/Combination#Number_of_combinations_with_repetition Combinations with repetition], [http://en.wikipedia.org/wiki/Multiset_coefficient#Multiset_coefficients <math>k</math>-multisets on a set]
* [http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind Stirling number of the second kind]
* [http://en.wikipedia.org/wiki/Partition_(number_theory) Partition of a number]
* [http://en.wikipedia.org/wiki/Twelvefold_way The twelvefold way]
* [http://en.wikipedia.org/wiki/Partition_(number_theory)#Ferrers_diagram Ferrers diagram] (and the MathWorld [http://mathworld.wolfram.com/FerrersDiagram.html link])
* [http://en.wikipedia.org/wiki/Inclusion-exclusion_principle The principle of inclusion-exclusion] (and more generally the [http://en.wikipedia.org/wiki/Sieve_theory sieve method])
* [http://en.wikipedia.org/wiki/Derangement Derangement], and [http://en.wikipedia.org/wiki/M%C3%A9nage_problem Problème des ménages]
* [http://en.wikipedia.org/wiki/Generating_function Generating function] and [http://en.wikipedia.org/wiki/Formal_power_series formal power series]
* [http://en.wikipedia.org/wiki/Fibonacci_number Fibonacci number]
* [http://en.wikipedia.org/wiki/Binomial_series Newton's formula]
* [http://en.wikipedia.org/wiki/Catalan_number Catalan number]
* [http://en.wikipedia.org/wiki/Double_counting_(proof_technique) Double counting] and the [http://en.wikipedia.org/wiki/Handshaking_lemma handshaking lemma]
* [http://en.wikipedia.org/wiki/Cayley's_formula Cayley's formula]
* [http://en.wikipedia.org/wiki/Pigeonhole_principle Pigeonhole principle]
:* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem Erdős–Szekeres theorem]
:* [http://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem Dirichlet's approximation theorem]
* [http://en.wikipedia.org/wiki/Probabilistic_method The Probabilistic Method]
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model Erdős–Rényi model for random graphs]
* [http://en.wikipedia.org/wiki/Graph_property Graph property]
* Some graph parameters: [http://en.wikipedia.org/wiki/Girth_(graph_theory) girth <math>g(G)</math>], [http://mathworld.wolfram.com/ChromaticNumber.html chromatic number <math>\chi(G)</math>], [http://mathworld.wolfram.com/IndependenceNumber.html Independence number <math>\alpha(G)</math>], [http://mathworld.wolfram.com/CliqueNumber.html clique number <math>\omega(G)</math>]
* [http://en.wikipedia.org/wiki/Extremal_graph_theory Extremal graph theory]
* [http://en.wikipedia.org/wiki/Turan_theorem Turán's theorem], [http://en.wikipedia.org/wiki/Tur%C3%A1n_graph Turán graph]
* Two analytic inequalities:
:*[http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality Cauchy–Schwarz inequality]
:* the [http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means inequality of arithmetic and geometric means]
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Stone_theorem Erdős–Stone theorem] (fundamental theorem of extremal graph theory)
* [http://en.wikipedia.org/wiki/Dirac's_theorem Dirac's theorem]
* [http://en.wikipedia.org/wiki/Hall's_theorem Hall's theorem ] (the marriage theorem)
* [http://en.wikipedia.org/wiki/Birkhoff-Von_Neumann_theorem Birkhoff-Von Neumann theorem]
* [http://en.wikipedia.org/wiki/K%C3%B6nig's_theorem_(graph_theory) König-Egerváry theorem]
* [http://en.wikipedia.org/wiki/Dilworth's_theorem Dilworth's theorem]
* [http://en.wikipedia.org/wiki/Sperner_family Sperner system]
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem Erdős–Ko–Rado theorem]
* [http://en.wikipedia.org/wiki/VC_dimension VC dimension]
* [http://en.wikipedia.org/wiki/Kruskal%E2%80%93Katona_theorem Kruskal–Katona theorem]
* [http://en.wikipedia.org/wiki/Ramsey_theory Ramsey theory]
:*[http://en.wikipedia.org/wiki/Ramsey's_theorem Ramsey's theorem]
:*[http://en.wikipedia.org/wiki/Happy_Ending_problem Happy Ending problem]
:*[http://en.wikipedia.org/wiki/Van_der_Waerden's_theorem Van der Waerden's theorem]
:*[http://en.wikipedia.org/wiki/Hales-Jewett_theorem Hales–Jewett theorem]
* [http://en.wikipedia.org/wiki/Lov%C3%A1sz_local_lemma Lovász local lemma]
* [http://en.wikipedia.org/wiki/Combinatorial_optimization Combinatorial optimization]
:* [http://en.wikipedia.org/wiki/Optimization_(mathematics) optimization]
:* [http://en.wikipedia.org/wiki/Convex_combination convex combination], [http://en.wikipedia.org/wiki/Convex_set convex set], [http://en.wikipedia.org/wiki/Convex_function convex function]
:* [http://en.wikipedia.org/wiki/Local_optimum local optimum] (see also [http://en.wikipedia.org/wiki/Maxima_and_minima maxima and minima])
* [http://en.wikipedia.org/wiki/Linear_programming Linear programming]
:* [http://en.wikipedia.org/wiki/Linear_inequality linear constraint]
:* [http://en.wikipedia.org/wiki/Hyperplane hyperplane], [http://en.wikipedia.org/wiki/Half_space halfspace], [http://en.wikipedia.org/wiki/Polyhedron polyhedron], [http://en.wikipedia.org/wiki/Convex_polytope convex polytope]
:* [http://en.wikipedia.org/wiki/Simplex_algorithm the Simplex algorithm]
*  The  [http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem Max-Flow Min-Cut Theorem]
:* [http://en.wikipedia.org/wiki/Maximum_flow_problem Maximum flow]
:* [http://en.wikipedia.org/wiki/Minimum_cut minimum cut]
* [http://en.wikipedia.org/wiki/Unimodular_matrix Unimodularity]
* [http://en.wikipedia.org/wiki/Dual_linear_program Duality]
:* [http://en.wikipedia.org/wiki/Linear_programming#Duality LP Duality]
* [http://en.wikipedia.org/wiki/Matroid Matroid]
:* [http://en.wikipedia.org/wiki/Weighted_matroid weighted matroid] and [http://en.wikipedia.org/wiki/Greedy_algorithm greedy algorithm]
:* [http://en.wikipedia.org/wiki/Matroid_intersection Matroid intersection]
* [http://en.wikipedia.org/wiki/Laplacian_matrix Laplacian]
* [http://en.wikipedia.org/wiki/Algebraic_connectivity <math>\lambda_2</math> of a graph] and [http://en.wikipedia.org/wiki/Expander_graph#Cheeger_Inequalities Cheeger Inequalities]
* [http://en.wikipedia.org/wiki/Expander_graph Expander graph]
* [http://en.wikipedia.org/wiki/Szemer%C3%A9di_regularity_lemma Szemerédi regularity lemma]

Revision as of 15:56, 11 September 2011

组合数学
Combinatorics
Instructor
尹一通
Email yitong.yin@gmail.com yinyt@nju.edu.cn
office 计算机系 804
Class
Class meetings Thursday, 10am-12pm
仙逸B-104
Office hours Wednesday, 2-5pm
计算机系 804
Textbook
van Lint and Wilson,
A course in Combinatorics, 2nd Ed,
Cambridge Univ Press, 2001.
v · d · e

This is the page for the class Combinatorics for the Fall 2011 semester. Students who take this class should check this page periodically for content updates and new announcements.

Announcement

  • 由于有事需要外出，9月14日星期三下午的office hour改在9月13日下午。
  • 第一、二次课的slides已发布，见lecture notes部分。

Course info

  • Instructor : 尹一通
  • email: yitong.yin@gmail.com, yinyt@nju.edu.cn,
  • office: 804
  • Teaching fellow: TBA
  • email: TBA
  • Class meeting: Thursday 10am-12pm, 仙逸B-104.
  • Office hour: Wednesday 2-5pm, 计算机系 804.


Syllabus

先修课程 Prerequisites

  • 离散数学（Discrete Mathematics）
  • 线性代数（Linear Algebra）
  • 概率论（Probability Theory）

Course materials

成绩 Grades

  • 课程成绩：本课程将会有六次作业和一次期末考试。最终成绩将由平时作业成绩和期末考试成绩综合得出。
  • 迟交：如果有特殊的理由，无法按时完成作业，请提前联系授课老师，给出正当理由。否则迟交的作业将不被接受。

学术诚信 Academic Integrity

学术诚信是所有从事学术活动的学生和学者最基本的职业道德底线，本课程将不遗余力的维护学术诚信规范，违反这一底线的行为将不会被容忍。

作业完成的原则：署你名字的工作必须由你完成。允许讨论，但作业必须独立完成，并在作业中列出所有参与讨论的人。不允许其他任何形式的合作——尤其是与已经完成作业的同学“讨论”。

本课程将对剽窃行为采取零容忍的态度。在完成作业过程中，对他人工作（出版物、互联网资料、其他人的作业等）直接的文本抄袭和对关键思想、关键元素的抄袭，按照 ACM Policy on Plagiarism的解释，都将视为剽窃。剽窃者成绩将被取消。如果发现互相抄袭行为， 抄袭和被抄袭双方的成绩都将被取消。因此请主动防止自己的作业被他人抄袭。

学术诚信影响学生个人的品行，也关乎整个教育系统的正常运转。为了一点分数而做出学术不端的行为，不仅使自己沦为一个欺骗者，也使他人的诚实努力失去意义。让我们一起努力维护一个诚信的环境。

Assignments

Lecture Notes

  1. Basic enumeration | slides1 | slides2
  2. Generating functions | slides1
  3. Sieve methods
  4. Pólya's theory of counting
  5. Counting and existence
  6. Discrete probability
  7. The probabilistic method
  8. Extremal graph theory
  9. Matching theory
  10. Flow and matching
  11. Optimization
  12. Matroid
  13. Extremal set theory
  14. Ramsey theory
  15. The Szemeredi regularity lemma

Concepts

Retrieved from "https://tcs.nju.edu.cn/wiki/index.php?title=随机算法_(Fall_2011)/Graph_Coloring&oldid=4735"