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Heuristic & Approximation in Computational Social Networks

This lecture explores heuristic and approximation algorithms for optimization problems in computational social networks, including minimum set cover, maximum coverage, and maximum partition. It also discusses model-based community detection and various approximation algorithms for vertex cover, connected vertex cover, dominating set, and connected dominating set problems.

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Heuristic & Approximation in Computational Social Networks

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  1. Lecture 1-2 Heuristic & Approximation Weili Wu Ding-Zhu Du University of Texas at Dallas lidong.wu@utdallas.edu

  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 apolynomial-timeapproximation 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.

  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

  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

  37. References 1 2

  38. Not “Prerequisites”

  39. 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

  40. 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

  41. 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.

  42. THANK YOU!

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