1. Heuristic & Approximation
Lecture 1-2
Heuristic & Approximation
Weili Wu Ding-Zhu Du
University of Texas at Dallas
First, I want to thank you for you presence. ********In this presentation I will try to introduce The social network which is a theoretical structure to study relationships between individuals, groups, organizations, or even entire societies. 
It is related to a wide range of disciplines. These disciplines include, but are not limited to information science, biology, economics, geography, communication studies, and so on..
The study of social networks begins with the late eighteenth century, two sociologists (Émile [ei'mi:l] Durkheim and Ferdinand ['fɝdənænd] Fer迪南de Tönnies) foreshadowed the idea of social networks in their theories and research of social groups.
Nowadays, we study social networks using network analysis to identify social communities, pick influential person, and design good software.

2. An NSF Program

3. Algorithms in the Field (AitF)
Algorithms in the Field encourages closer collaboration between two groups of researchers: (i) theoretical computer science researchers, who focus on the design and analysis of provably efficient and provably accurate algorithms for various computational models; and (ii) applied researchers including a combination of systems and domain experts

4. (very broadly construed – including but not limited to researchers in computer architecture, programming languages and systems, computer networks, cyber-physical systems, cyber-human systems, machine learning, database and data analytics, etc.) who focus on the particular design constraints of applications and/or computing devices.

5. Each proposal must have at least one co-PI interested in theoretical computer science and one interested in any of the other areas typically supported by CISE. Proposals are expected to address the dissemination of the algorithmic contributions and resulting applications, tools, languages, compilers, libraries, architectures, systems, data, etc.

6. Performance Ratio

7. C-Approximation
c-approximation is a polynomial-time approximation satisfying:
1 < approx(input)/opt(input) < c for MIN or
c < approx(input)/opt(input) < 1 for MAX

8. Three Types of Problems
Many optimization problems in computational social networks belong to following three types:
Minimum set cover
Maximum coverage
Maximum partition

9. Min Set Cover

10. Min Set Cover
Red + Green

11. Greedy Algorithm

12. Observation

13. Theorem

14. Max Coverage

15. Max Coverage
Red + Green

16. Greedy Algorithm

17. Theorem

18. Lower Bound

19. Exercises

20. References

21. Max Community Partition

22. Question ? How to find a Community? The definition is ambiguous.
So, we can only do model-based detection.
This is an important research problem.
There exist many methods.
Show an example.

23. Model-Based Detection
Community Detection
Accurate or not?
Formulation (Model)
Solve formulated problem

24. Model-Based Physics The Real World Accurate or not? Newton Model
Solve physics problem

25. No Satisfied Community Model !

26. Question ? How to find a Community? A simplest way is
Connection-Based Detection
This is an important research problem.
There exist many methods.
Show an example.

27. An Example More connections inside each community.
Less connections between different communities.
There are several
ways to understand
this property.

28. A Connection-Based Condition
(Radicchi et al. 2004)
Each community has more connections inside
than connections to outside.

29. A Connection-Based Condition
Inside red >
outside blue
+ outside green
Each community has more connections inside
than connections to outside.

30. Max Community Partition
Theorem (Lu et al. 2013)

31. A Heuristic

32. Indicator
For example

33. Objective

34. Linear Constraints

35. Linear Constraints

36. Count 3 endpoints

39. References 1 2

40. Not “Prerequisites”

43. Exercises
Given a graph, a subset of vertices is called a vertex cover if every edge has an endpoint in the subset. The minimum vertex cover problem is a special case of the minimum set cover problem. Show that the minimum vertex cover problem has a polynomial-time 2-approximation.
A vertex cover is said to be connected if it induces a connected subgraph. Show that

44. the minimum connected vertex cover problem has a polynomial-time 3-approximation.
Given a graph, a subset of vertices is a dominating set if every vertex is either in it or adjacent to a vertex in it. Show that the minimum dominating set problem has a polynomial-time O(log n)-approximation.
A dominating set is connected if it induces a connected subgraph. Show that minimum connected dominating set problem has a polynomial-time O(log n)-approximation

45. Exercises*
5. Given a graph, a subset of vertices is called a vertex feedback set if every cycle contains a vertex in it. Show that the minimum vertex feedback set problem has a polynomial-time 2-approximation.

46. THANK YOU!