CS5234 Combinatorial and Graph Algorithms

1. CS5234Combinatorial and Graph Algorithms Welcome!

2. Combinatorial & Graph Algorithms http://www.comp.nus.edu.sg/~cs5234/2013/ • Instructor: Seth Gilbert Office: COM2-323 Office hours: by appointment CS5234 Overview

3. Algorithms for Combinatorial Optimization

4. Algorithms What is an algorithm? • Set of instructions for solving a problem “First, wash the tomatoes.” “Second, peel and cut the carrots.” “Third, mix the olive oil and vinegar.” “Finally, combine everything in a bowl.” • Finite sequence of steps • Unambiguous • English, Chinese, pseudocode, Java, etc.

5. “If you need your software to run twice as fast, hire better programmers. But if you need your software to run more than twice as fast, use a better algorithm.” -- Software Lead at Microsoft

6. Goals: Algorithmic 1. Design (problem solving) 2. Analysis (rigorous, deep understanding) 3. Implementation (able to put it to use)

7. Goals: Algorithmic 1. Design (problem solving) 2. Analysis (rigorous, deep understanding) 3. Implementation(able to put it to use)

8. Algorithms for Combinatorial Optimization

9. Combinatorial Optimization Optimization: Find the minimum/maximum of…

10. Combinatorial Optimization Optimization: Find the minimum/maximum of: Continuous optimization: a function f Discrete optimization: a collection of items

11. Combinatorial Optimization Discrete Optimization: Find the “best” item in a large set of items. Combinatorial Optimization: Find the “best” item in a large set of items generated by a combinatorial process. Combinatorial: finite discrete structure Combinatorial process: counting, combining, enumerating

12. Combinatorial Optimization Discrete Optimization: Find the “best” item in a large set of items. Combinatorial Optimization: Find the “best” item in a large set of items generated by a combinatorial process. Graphs Matroids Similar structures…

13. Combinatorial Optimization Find the “best” item in a large set of items: Problem Set of items Size Difficulty Searching List of integersLinear Easy Shortest paths All paths in a graph Exponential Easy Minimum spanning tree All spanning trees Exponential Easy Steiner tree All steiner trees Exponential Hard Travelling salesman All possible tours Exponential Hard Matching All possible matchings Exponential Easy Bipartite vertex cover All possible covers Exponential Easy Vertex cover All possible covers Exponential Hard Maximum clique All possible subsets Exponential Very Hard

14. Combinatorial Optimization Find the “best” item in a large set of items: Problem Difficulty Maintain student records Easy Data compression Easy Program halting problem Impossible VLSI chip layout Hard Exam timetable scheduling Hard Job assignment problem Easy Computer deadlock problem Easy Finding patterns in a database Easy

15. Combinatorial Optimization Operations Research: How to make better decisions (e.g., maximize profit) Project planning / critical path analysis Facility location: where to open stores / plants Floorplanning: layout of factory or computer chips Supply chain management Berth assignment problem (BAP): port management Assignment problems (e.g., weapon target assignment) Routing / transportation problems: buses, subways, trucking. Airline ticket pricing

16. Combinatorial Optimization What do we do when problems are NP-hard? • Find exponential time solutions • Average performance • Approximate • Algorithm is efficient • Solution is sub-optimal • Provable guarantee: ratio of output to optimal

17. Combinatorial Optimization Five Representative Problems

18. b e h 1. Interval Scheduling Jobs don’t overlap Input:Set of jobs with start and finish times Output: Maximum cardinality subset of compatible jobs a b c d e f g h Time 0 1 2 3 4 5 6 7 8 9 10 11

19. 2. Weighted Interval Scheduling Input:Set of weighted jobs with start and finish times Output: Maximum weight subset of compatible jobs 23 12 20 26 13 20 11 16 Time 0 1 2 3 4 5 6 7 8 9 10 11

20. 3. Bipartite Matching Input:Bipartite graph Output: Maximum cardinality matching A 1 B 2 C 3 D 4 E 5

21. 1 4 5 6 4. Independent set subset of nodes such that no two joined by an edge Input:Graph Output: Maximum cardinality independent set 2 1 4 5 3 7 6

22. 5. Competitive facility location Input:Graph with weighted nodes Game: Two players alternate in selecting nodes. Cannot select a node if any of its neighbors have been selected. Goal: Select a maximum weight subset of nodes. 10 1 5 1 15 1 5 15 5 10 Second player can get 20, but not 25.

23. Five problems Variations on a theme: Independent Set Interval scheduling: Greedy O(n log n) Weighted Interval scheduling: Dynamic programming O(n log n) Bipartite matching: O(nk) max-flow algorithm Independent set: NP-complete Competitive facility location: PSPACE-complete

24. Combinatorial Optimization • General combinatorial problem • Given a finite, discrete set Sof objects • Minimize/maximize some function f (S) • Algorithmic Issues… • Representation of the set S • Efficient manipulation of the set S • Efficient algorithm to compute f (S)

25. Strategy Computability • Given a problem P, • Can it be solved? • If “Yes”, give an algorithm A for solving P, • Is algorithm A correct ? • How good is algorithm A ? • Can we find a better algorithm A’ ? • How do we define good? • How much time it takes. • How much space it uses. Verification Efficiency Time Complexity Space Complexity

26. Algorithms for Combinatorial Optimization Goals of this course: • Advanced design and analysis of algorithms: • Efficiency • Time: How long does it take? • Space: How much memory? How much disk? • Others: Energy, power, heat, parallelism, etc. • Scalability • Inputs are large : e.g., the internet. • Bigger problems consume more resources. • Solve important (fun!) problems in combinatorial optimization

27. Algorithms for Combinatorial Optimization Target students: • Beginning graduate students • Advanced (4th year) undergraduates • Anyone planning to do research in algorithmic design Prerequisites: • CS3230 (Analysis of Algorithms), or equivalent • Strong programming skills

28. Algorithms for Combinatorial Optimization You must already know these: • Data Structures (with analyses) • Stacks, Queues, Lists, • Binary search trees, balanced trees, • Heaps and priority queues • Algorithm Design Paradigms (with Analysis) • Standard sorting and searching algorithms • Graph algorithms: DFS, BFS, • Shortest Path Algorithms, MST Algorithms • Greedy Algorithms, Divide-and-Conquer

29. Algorithms for Combinatorial Optimization You must already know these: • Analysis of Algorithms • Expertise with Big-O, ,  notations • Summation of series, Master Theorem • Competent with Algorithmic Analysis: Quicksort, Heapsort, Divide-and-Conquer algorithms DFS, BFS, Shortest Path & MST algorithms

30. Algorithms for Combinatorial Optimization Assumed knowledge: • Data structures and algorithms • Good programming skills • CS3230 (Analysis of Algorithms) Not a course to learn algorithms: • If not, take CS3230 instead.

31. CS5234 Overview • Mid-term exam October 8 In class • Final exam November 19Reading Week Exams will be graded and returned.

32. CS5234 Overview • Grading 40% Problem sets 25% Mid-term exam 35% Final exam • Problem sets • 6-7 sets (about every 1-2 weeks) • A few will have programming components (C++).

33. CS5234 Overview • Problem sets released by tomorrow morning PS0: Covers background knowledge. Do not submit. PS1: Routine problems --- easy practice. Do not submit. Standard problems --- to be submitted. Advanced problems --- for a challenge/fun. Do not submit.

34. CS5234 Overview • Problem set grading Distributed grading scheme: Each of you will be responsible for grading one week during the semester. Grading supervised (and verified) by the TA.

35. CS5234 Overview • What to submit: Concise and precise answers: Solutions should be rigorous, containing all necessary detail, but no more. Algorithm descriptions consist of: 1. Summary of results/claims. 2. Description of algorithm in English. 3. Pseudocode, if helpful. 4. Worked example of algorithm. 5. Diagram / picture. 6. Proof of correctness and performance analysis.

36. CS5234 Overview • Policy on plagiarism: Do your work yourself: Your submission should be unique, unlike anything else submitted, on the web, etc. Discuss with other students: 1. Discuss general approach and techniques. 2. Do not take notes. 3. Spend 30 minutes on facebook (or equiv.). 4. Write up solution on your own. 5. List all collaborators. Do not search for solutions on the web: Use web to learn techniques and to review material from class.

37. CS5234 Overview • Policy on plagiarism: Penalized severely: First offense: minimum of one letter grade lost on final grade for class (or referral to SoC disciplinary committee). Second offense: F for the class and/or referral to SoC. Do not copy/compare solutions!

38. Textbooks Introduction to Algorithms • Cormen, Leiserson, Rivest, Stein • Recommended…

39. Textbooks Algorithm Design • Kleinberg and Tardos • Recommended…

40. CS5234 Overview • Topics (tentative, TBD) Introduction to combinatorial optimization Vertex cover, set cover, Steiner tree, TSP Flows and matching Maximum flow, bipartite matching Graph partitioningHeuristics, spectral bisection Linear programming LPs, duality, relaxations, rounding