Combinatorial Computing
VCSG-801
Fall 2011

Assignment 1, due Thursday, September 22, 2011

Part 1, no programming

1. Draw all nonisomorphic graphs on 4 vertices.
2. How many distinct 5 by 5 0-1 matrices are there representing a pentagon? Try to prove it in just a few lines of counting reasoning.
3. Draw all (nonisomorphic) graphs on 10 vertices and 5 edges (there are 26 of such graphs).
4. Prove that R(C_4,C_4) = 6. You may consider an intermediate lemma stating that every C_4-free graph has at most 7 edges.

Part 2, programming exercise

1. Build a file with 26 lines containing g6-formats of all nauty-canonically labeled graphs on 10 vertices and 5 edges.
2. With the help of nauty compute the Ramsey numbers R(C_4,K_n) for n = 3, 4 and 5, and build g6-formats of all nonisomorphic graphs witnessing the lower bound for corresponding numbers. For correctness compare your results to those in the paper at position 34 of the list.
3. Repeat the previous step for some n > 5. The case for n = 9 is an open research question.

Assignment 2, due Tuesday, October 4, 2011

Part 1, Dynamic programming and knapsack

1. Let the knapsack problem instance have the profits {3,5,7,9,12}, weights {1,2,3,4,5} and the capacity M=10.
2. What is the meaning of the entry P[3,7] ?
   Use the dynamic programming algorithm to produce the dynamic programming table P[j,m] for this instance.
3. Describe briefly how to enhance this algorithm so we can recover the actual choice of objects to be packed.
4. What is needed to decide if the optimal solution is unique ?

Part 2, Generating k-sets of an n-set

1. Give experimental running time comparison of this algorithm with the naive approach of the loop processing all n-strings with bit counting.
2. Prove that this algorithm correctly enumerates all k-sets of an n-set for n < WORDSIZE.
3. *(optional) Perform experiments with other algorihms for the same problem.

Midterm Exam, Thursday, October 13, 2011

Assignment 3, due Thursday, October 27, 2011

Part 1, for everybody

Implement coloring and clique finding algorithms as follows.

1. Greedy graph coloring as in Algorithm 4.16 page 137.
2. Maximum clique algorithm 4.19 page 139. Use at least two different bounding functions.

The algorithms of this assignment should be tested on three types of graphs:

• Small graphs of your choice.
• Erdos/Renyi-type random graphs build with the algorithm 4.21 page 140.
• Circular Ramsey-type graphs as discussed in class. Make your algorithm run first for small cases, then find the order of the maximum clique and independent set in larger cases. A number of test cases requiring clique algorithms to confirm lower bounds on Ramsey numbers can be found at Geoff Exoo's site (following the link Ramsey).

1. Trace in detail the behavior of each algorithm on one of the small graphs.
2. Compare your implementation in efficiency with the experiments done by the authors of the textbook, as reported in Tables 4.4 and 4.5 on page 140.
3. Hardcopy of the commented source code.

Part 2, for volunteers, open deadline

1. Evaluate the maximum clique finding algorithms developed by Patric Ostergard.
2. Find some new lower bounds for any of the cases listed in my survey paper Small Ramsey Numbers. Anything goes. You do need a fast clique testing algorithm, though in practice it may be targeted and tuned for very special parameters. Try non-classical cases (like K₆ versus K₇-e), try multicolor cases. When in doubt, discuss with me the details. GA's or simulated annealing are OK, team work is OK, any tools you can find on the web are OK, just give me some new lower bounds ... If your work on this part is good, you can be excused from completing Part 1.
3. Compute maximum clique order in Paley graphs up to 20000 vertices. Or, just try to be better than Shearer (up to 7000, with cpu times) and Exoo (up to 10000).

• Some comments on the experiments you performed.
• Hardcopy of the commented source code.

Assignment 4, due as soon as possible

Assignment 5, due Thursday, November 17, 2011

• Solve exercise 4.4a pages 145/146 (rational knapsack).
• Consider the general basic divisibility conditions for the existence of t-designs, and make them more specific for the cases of 2-(v,3,lambda) and 2-(v,4,lambda), i.e. state them for t=2, separately for k=3 and k=4, in terms of the number of blocks b, the repetitiveness number r, and the number of points v.
• Let lambda = 1. For each v <= 10 passing the above test for 2-(v,3,lambda) design compute the number of blocks and repetitiveness of potential designs with these parameters.
• Let lambda = 2. For each v <= 10 passing the above test for 2-(v,4,lambda) design compute the number of blocks and repetitiveness of potential designs with these parameters.
• STS(k) is a Steiner Triple System on k points, or equivalently it is the same as a 2-(v,3,1) design. Write a program which is able to construct some small Steiner Triple Systems. In particular, it must be able to generate at least one STS(13) and at least one STS(15). You can use any algorithm you wish. Each of the following should work for the task above: exhaustive search by backtracking, Stinson's algorithm, GAs, simulated annealing. For this assignment put main effort into the correctness. Performance would be of more importance for generating larger designs. Submit a commented hardcopy of your program and a printout of at least two STS's constructed by your program, on 13 and 15 points (as in sample STS(15).
• (for volunteers) Generate all nonisomorphic STS(15) - there are 80 such designs.

Final Exam, Thursday, 12:30-2:30pm, November 17, 2011, 70-3455

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