RIT Computer Science

Topics in Advanced Algorithms -
Combinatorial Computing
CSCI-761, Spring 2021


Stanisław Radziszowski

bldg. 70B, room 3657,
(585) 475-5193, spr@cs.rit.edu, http://www.cs.rit.edu/~spr
office hours: TR 6:30pm-7:30pm via zoom, or email spr@cs.rit.edu anytime


Tuesday/Thursday, 5:00pm-6:15pm, room 70-1610

General Course Documents

College course document, and common RIT policies and calendar.


This course will explore the possibilities and limitations of effective computations in combinatorics. The first half of the course will cover classical algorithms in combinatorial computing, together with the problems of generation, enumeration and manipulation of various types of combinatorial objects (graphs and finite set systems). The second part will concentrate on computational techniques for the search of different combinatorial configurations: Ramsey numbers, t-designs, Turan coverings, Folkman colorings and others. A number of topics suitable for further independent study, project or thesis development will be discussed.

Students will write a term paper, either theoretical based on literature or reporting student's own implementation or experiments with a chosen combinatorial problem. Depending on the size of the group, some or all students will give a presentation to the class.


Prerequisites: CSCI-665 or ((CSCI-261 or CSCI-264) and permission of the instructor)


Main Online Resources

  • Done So Far (with pointers to item numbers on the "To Do" list below, #n):

    1/26. Course logistics, my homepage, seating chart. General overview.
    1/28. Assignment 1 (on homeworks page), #1 (on To Do list).
    2/02. #2.
    2/04. #3, start #4.
    2/09. More on #4, start #5.
    2/11. Continue #5.
    2/16. Parts of #7. Permutation groups, cycle notation, generators.
    2/18. #6. Lecture 5 by Jacob Fox. We will be back to #7.
    2/25. Resolving doubts about assignment #3. Solutions to #1 and #2.
    3/02. #6. Lecture 6 by Jacob Fox. #7. Permuation groups.
    3/04. #7. Caley graphs. Paley graphs. J4-free graphs.
    3/09. Cliques and chromatic numbers are NP-hard. #9.
    3/11. #10. Computing cliques.
    3/16. #10. Greedy coloring and computing cliques.
    3/18. Midterm exam, online.
    3/23. Review of midterm exam, in class.
    3/25. Start #12.
    3/30. #12. What is in the Assignment #6.
    4/01. #12 and #13. Two theorems: on vertex-arrowing and edge-arrowing. A very special bicritical graph.
    4/06. #13 and #14. Graph G127, reducing Ramsey triangle arrowing to 3SAT.
    4/08. #15. Reconstruction Conjecture, part I.
    4/13. David Narváez, encoding Ramsey/Folkman problems, via zoom (OH link), slides.
    4/15. David Narváez, resolving the Keller's Conjecture, via zoom (OH link), slides.
    4/20. #15. Reconstruction Conjecture, part II.
    4/22. No class, recharge day.
    4/27. Reconstruction Conjecture, complexity.
    4/29. #16. Diagonal Conjecture.
    5/03. #17. Shannon capacity.
    5/04. #17. Shannon capacity, part II. #18. Collatz conjecture.
    5/13. Final exam posted, due 5/15 23:59.

  • To Do (with links to supporting materials):
    1. nauty home by Brendan McKay at ANU in ACT of AUS.
    2. g6-format, a very useful way to write graphs to files and pipes. You do not need to know all its details, but you need to know how to make your programs read and write graphs encoded in g6.
    3. Wolfram MathWorld page on graph isomorphism problem, GI, with further links to canonical labeling, automorphisms and such. Wolfram is also famous because of Mathematica and the book A New Kind of Science.
    4. Overview of Computations in Ramsey Theory. For homework #2, look at the tables I, II, III and IV on pages 45-47 of a very old paper posted at position #97 of the list (or get paper's pdf directly, or just the tables tabs88.pdf).
    5. There are three nonisomorphic (3,4;8)-Ramsey critical graphs for K3 versus K4, check out their drawings linked to cycle representation of their automorphism group generators. Two papers with Jan Goedgebeur, at positions #35 and #31 of the list, present recent developments of what is known about (3,k;n,e)-Ramsey graphs (reading these two papers is not required, but I encourage you to do so).
    6. Overview slides of computational Ramsey theory is well complemented by lecture 5 and lecture 6 at MIT by Jacob Fox.
    7. Basics of groups: the symmetric group, alternating group, sign, cycles and such. Then Cayley graphs of groups are explaining things even more, like the Cayley graphs of the group of automorphisms of pentagon for three different pairs of generators.
    8. Refreshing what you know about Pascal triangle of binomial coefficients and Catalan numbers (use also Wolfram pages, not only wiki) will help in better understanding of the upper bound on R(s,t). And/or, if you really want to know more about CS side of Newton binomial coefficients dive into the book A=B by Petkovsek, Wilf and Doron Zeilberger. A=B is free.
    9. Imre Leader's problem of colored permutations.
    10. Computing Cliques by Donald Kreher, slides and chapter 4.6.3.
    11. Exploring Ramsey page at RIT-CS.
    12. Ramsey arrowing, Folkman graphs and numbers: An old 1-page-long article by Ron Graham on the first cool Folkman number Fe(3,3;6)=8. Slides 21+ of Computations in Ramsey Theory. About 40 years later, a paper on the most wanted Folkman graph, with associated slides. Two theorems: on vertex-arrowing and edge-arrowing. The very special bicritical graph, for Fv(3,3;4)=14.
    13. Conference presentation on Folkman problems. Time for the G127 arrowing problem to be settled!
    14. Several technical papers on Folkman problems co-authored by the instructor are posted at publications. Christopher Wood wrote a comprehensive survey of the area. A more recent PhD thesis by Aleksandar Bikov covers extensively all computational af the area.
    15. Reconstruction Conjecture, presentation based mainly on the MS work by David Rivshin, first paper, second paper, third paper, slides A, slides B, and slides C.
    16. Diagonal Conjecture, paper and slides.
    17. Shannon capacity of noisy channels modelled by graphs, and its relation to Ramsey numbers, paper, slides and a theorem.
    18. Collatz 3n+1 conjecture.

    19. On special Folkman graphs, the existence and search, paper and slides.

    Other Online Resources