Topics in Advanced Algorithms 
Combinatorial Computing
CSCI761, Spring 2021
Instructor
bldg. 70B, room 3657,
(585) 4755193, spr@cs.rit.edu,
http://www.cs.rit.edu/~spr
office hours: TR 6:30pm7:30pm via
zoom, or email spr@cs.rit.edu anytime
Lectures
Tuesday/Thursday, 5:00pm6:15pm, room 701610
General Course Documents
College course document, and
common RIT policies and calendar.
Contents
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, tdesigns, 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.
Readings

Combinatorial Algorithms. Generation, Enumeration, and Search,
by Donald L. Kreher and Douglas R. Stinson, CRC Press, 1999.
 Handbook of Combinatorics by
R. Graham, M. Grotschel and L. Lovasz,
MIT Press, 1995 (complementary reference monograph).
 Introductory Combinatorics by
Kenneth Bogart, third edition,
HBJ Publ., 2000 (complementary combinatorial background).
Prerequisites:
CSCI665 or ((CSCI261 or CSCI264) and permission of the
instructor)
Evaluation
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. J4free graphs.
3/09.
Cliques and chromatic numbers are NPhard. #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
vertexarrowing and
edgearrowing.
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):

nauty
home by
Brendan McKay
at ANU in ACT of AUS.

g6format,
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.

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.

Overview of Computations in Ramsey Theory.
For homework #2, look at the tables I, II, III and IV on pages 4547 of a
very old paper posted at position #97 of the
list
(or get paper's
pdf directly,
or just the tables tabs88.pdf).

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).

Overview slides of
computational Ramsey theory is well complemented by
lecture 5 and
lecture 6 at MIT by
Jacob Fox.

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.

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.

Imre Leader's
problem of colored permutations.

Computing Cliques by Donald Kreher,
slides and chapter 4.6.3.

Exploring
Ramsey page at RITCS.

Ramsey arrowing, Folkman graphs and numbers:
An old 1pagelong 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
vertexarrowing and
edgearrowing.
The very special bicritical
graph, for Fv(3,3;4)=14.

Conference presentation on
Folkman problems.
Time for the G127 arrowing problem to be settled!

Several technical papers on Folkman problems coauthored 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.

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.

Diagonal Conjecture,
paper and
slides.

Shannon capacity of noisy channels modelled by graphs, and its relation
to Ramsey numbers,
paper,
slides
and a theorem.

Collatz 3n+1 conjecture.

On special Folkman graphs, the existence and search,
paper and
slides.
Other Online Resources