[SOLVED] algorithm COMP523 2019 Second CA Assignment Advanced Algorithmic Techniques

COMP523 2019 Second CA Assignment Advanced Algorithmic Techniques
Assessment Information
Assignment Number
2 (of 2)
Weighting
12.5%
Assignment Circulated
15 Nov 2019
Deadline
1 Dec 2019, 16:00
Submission Mode
Please submit your solutions electronically (PDF or DOC format) at the
electronic submission system of the Computer Science Department
which you can find at the following url. https://sam.csc.liv.ac.uk/COMP/Submissions.pl?strModule=COMP523.
Learning outcome assessed
3. Identify which of the studied design principles are used in a given algorithm taking account of the similarities and differences between the principles.
4. Apply the studied design principles to produce efficient algorithmic solutions to a given problem taking account of the strengths and weaknesses
of the applicable principles.
Purpose of assessment
Marking criteria
Design and analysis of algorithms for a given problem.
The aims are: to identify the complexity of a problem and design
good algorithms based on the different design principles covered in class.
Based on the marking descriptors of the Universitys Code of Practice on Assessment
Submission necessary in order to satisfy Module requirements?
No
Late Submission Penalty
Standard UoL Policy.
Plagiarism and Collusion
Please be aware of the University guidelines on plagiarism and collusion.

Task specification
Provide an answer to the following problem. You may consider any of the material that was presented in the lectures as known.
Problem 1 Necessary definitions
Consider the following measure of comparison between two algorithms A and B, based on their running time and approximation ratio:
Better running time: Algorithm A has a better running time than algorithm B if A is polynomial time and B is pseudo-polynomial time or exponential time, or if A is pseudo-polynomial time and B is exponential time.
Better performance: Algorithm A has a better performance than algorithm B if A has a smaller approximation ratio than B.
Remark: The comparisons above are based on the best possible analysis for the algorithms. For instance, a poly- nomial time algorithm is better than a pseudopolynomial time algorithm, only if the latter can not be proven to run in polynomial time. As a concrete example (from the lectures), you can either solve the max flow problem using the Edmonds-Karp algorithm, or using a polynomial-time algorithm for linear programming. If you use the Ford- Fulkerson analysis for Edmonds-Karp (which will give you a pseudopolynomial running time), it would be incorrect to claim that the LP-based algorithm is better than Edmonds-Karp, because the latter actually runs in polynomial time with a better analysis (and they both solve the problem exactly). Similarly, if you prove an approximation ratio for a polynomial-time algorithm A and an approximation ratio for a polynomial-time algorithm B, you can safely claim that A is better than B only if < and you can not getter a better approximation ratio for B with a better analysis.An algorithm A that has better running time and better performance than an algorithm B is better than B. An algo- rithm A is best, if there is no other algorithm that is better than it (assuming P = NP).From the definitions, it follows a polynomial-time algorithm that solves a problem P exactly is obviously best, but if the problem P is NP-hard, then a polynomial-time algorithm with the best possible approximation ratio > 1 is best, or a pseudopolynomial algorithm that solves it exactly (if it exists) is best. For example, for the 0/1-Knapsack problem, the pseudopolynomial algorithm based on dynamic programming is best, the FPTAS approximation algorithm is best, but the 2-approximation Greedy algorithm is not best, because there is an approximation algorithm which runs in polynomial time and achieves a better approximation ratio (the FPTAS).
The problem
Consider the following problem. A university lecturer aims to schedule a series of 1-hour Q&A sessions with n stu- dents (these are sessions for groups of students, not 1-on-1 meetings) and has set up a doodle poll where there are m available time slots. Every student has indicated which slots they could attend and it turns out that any student appears in at least 1 and at most k time slots in the doodle poll. The lecturer would like to minimise the number of sessions that they will have to do, making sure that they do at least one session for every student (i.e., every student will have a chance to attend some session).
Come up with three best algorithms for the problem.
Explain the limitations of your proposed solutions, as well as the limitations of the problem. Explain, whenever possible, why your proposed algorithms are best. If you are not sure whether they are best, what kind of improvements could you be looking for?
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