18.218 M.I.T. Spring 2016
18.204 Undergraduate Seminar in Discrete Mathematics
Class meets:
MWF 1 pm
Room 2136

Instructor:
Alexander Postnikov
apost at math
Room 2367
Course webpage:
http://math.mit.edu/~apost/courses/18.204/
Description:
Seminar in combinatorics, graph theory, and discrete mathematics in general.
Participants read and present papers from recent mathematics literature.
Instruction and practice in written and oral communication provided.
Course Level: Undergraduate
Syllabus
Ideas for topic of your first presentation:
Ideas for your second presentation
Schedule:
 W 02/03/2016. Introduction.
 F 02/05/2016. Alex Postnikov:
Random Walk on a Line, aka
Drunk Man Problem.
Related links:
Catalan numbers,
random walks.
 M 02/08/2016. Berj Chilingirian:
Arrow's Impossibility Theorem.
 W 02/10/2016. Liz Walker: Shuffling a Deck of Cards.
[Proofs for the Book, Ch. 28]
 F 02/12/2016. Anne Kelley: Zero Knowledge Proofs.
 Tuesday! 02/16/2016. (Monday schedule of classes on Tuesday)
Gabriella Baracchini: Linear Programming
 W 02/17/2016. Lynn Takeshita: Stable Marriage and Stable Roommate Theorems.
 F 02/19/2016. Xinyue Deng: Graph Search Algorithms.
 M 02/22/2016. Jon Lu: Simple Games with NotSoSimple Strategies.
 W 02/24/2016. Michael Wallace: The Probabilistic Method.
 F 02/26/2016. Open mike.
 M 02/29/2016. Jeremy Wohlwend: Facts About Pi.
 W 03/02/2016. Cali Gallardo: Latin Squares and Codes.
 F 03/04/2016. Margaret Carpenter: Knapsack Problem.
 M 03/07/2016. Gerrod Voigt: Diophantine Equations.
 W 03/09/2016. Alex Postnikov: Cayley's formula for
the number of trees and Prufer's codes.
 F 03/11/2016. Aaron Zwiebach: Diffie Hellman and RSA Encryption.
 M 03/14/2016. Greg Lubin: Pigeonhole principle and Ramsey numbers.
 W 03/16/2016. Alex Postnikov: Acyclic orientations of graphs.
 F 03/18/2016. Berj Chilingirian: Pink noise and sandpiles.
03/21/2016  03/25/2016. no classes  Spring vacation
 M 03/28/2016. Gerrod Voigt: Finding Hamiltonian cycles: Bondy's proof of Ore's algorithm.
Lynn Takeshita: Chromatic numbers of signed graphs.
 W 03/30/2016. Anne Kelley:
ErdosSzekeres theorem.
 F 04/01/2016. Jon Lu:
100 Prisoners Problem.
 M 04/04/2016. Xinyue Deng: Turan's Theorem
 W 04/06/2016.
Greg Lubin: Inductive Reasoning and Bounded Rationality
Margaret Carpenter: Cops vs Robber
 F 04/08/2016. Jeremy Wohlwend:
Elliptic Curve Cryptography
Aaron Zwiebach:
Percolation  A Study in Randomly Generated DAry Trees
 M 04/11/2016. Liz Walker: Bin Paking Problem
(papers:
[Korf] and
[MartelloToth])
Gabriella Baracchini: Bertrand's Ballot Problem.
 W 04/13/2016. Michael Wallace: A Probabilistic Proof
of a Formula for the Number of Standard Young Tableaux of a Given Shape.
 F 04/15/2016. Discussion about writing a paper.
M 04/18/2016. no class  Patriots Day
 W 04/20/2016. Berj Chilingirian:
The Sandpile LoadBalancing Problem for Latency Minimization.
 F 04/22/2016. Anne Kelley: Chip Firing Games,
Liz Walker: The Friendship Theorem.
 M 04/25/2016. Alex Postnikov: Abelian Sandpile Model and
Spanning Trees.
 W 04/27/2016. Lynn Takeshita:
Efficient Algorithms to Color Signed Graphs,
Jeremy Wohlwend: A Survey of Modern Public Key Cryptography.
 F 04/29/2016. Gerrot Voight:
Tree Decomposition, Treewidth, and NPHard Problems.
 M 05/02/2016. Xinyue Deng:
LenstraLenstraLovasz Lattice Basis Reduction Algorithm.
 W 05/04/2016. Margaret Carpenter:
The Single Jeep and Jeep Caravan Problems.
 F 05/06/2016. Aaron Zwiebach: Random Walks and Percolation: An Analysis of Current Research on Modeling Natural Processes,
Greg Lubin: Strategies for Pairwise Kidney Exchange.
 M 05/09/2016. Gabriella Baracchini:
Catalan Numbers and Possible Applications,
Jon Lu: Analysis of the ChowRobbins Game
 W 05/11/2016.
Michael Wallace: Housing Allocation with Multiple Tenants
Term papers:

Gabriella Baracchini:
Dyck Paths and UpDown Walks
(final version 5/12/2016)
This paper analyzes Dyck Paths and Alpha and Beta UpDown Walks, a subset of
Dyck Paths. It will show the natural implications of Catalan Numbers in counting
unique paths. Using Catalan Number knowledge and combinatorics we will count the
exact number of unique UpDown walks of length l as a function of l. A closed form
equation for the length of these paths is a natural extension of this problem and can
be found in the appendices.

Margaret Carpenter:
The Single Jeep and Jeep Caravan Problems
(final version 5/10/2016)
The Jeep problem was introduced in 1947 and has since spawned numerous
variations with reallife applications. The goal of the original
problem was to find the correct procedure for a single Jeep to reach a certain
distance using the minimum amount of fuel. The second part of this
paper introduces the idea of multiple Jeeps, called a Jeep Convoy[4] by
CG Phipps in 1947. The goal of the Jeep Convoy problem is to advance a
single jeep of the caravan as far as possible. Both Fine and Phipps found
solutions relating the Jeep problem to the Harmonic series, and proved
that given unlimited fuel and time a Jeep could cross any size desert. This
paper will cover the original problem introduce by NJ Fine[1] and will also
look at three variations of the Jeep Convoy problem.

Berj Chilingirian:
The Sandpile LoadBalancing Problem for Latency Minimization
(final version 5/12/2016)
I present the sandpile load balancer, an application of the abelian
sandpile model to dynamic load balancing. The sandpile load balancer exploits
the selforganized criticality of the abelian sandpile model to distribute web
traffic across a network of servers. I also propose the sandpile load balancer
latency minimization problem in which the goal is to minimize the latency of
packets across the network of servers. I show the number of possible strategies
for solving this problem and explore heuristicbased approaches via simulation.

Xinyue Deng:
An Introduction to LenstraLenstraLovasz Lattice Basis Reduction Algorithm
(final version 5/13/2016)
LenstraLenstraLovasz (LLL) Algorithm is an approximation algorithm of
the shortest vector problem, which runs in polynomial time and finds an
approximation within an exponential factor of the correct answer. It is a
practical method with enough accuracy in solving integer linear programming,
factorizing polynomials over integers and breaking cryptosystems. In
this paper, we introduce its background and implementation, analyze its
correctness and performance and discuss its applications.

Anne Kelley:
Chip Firing Games
(final version 5/11/2016)
Chip firing is a oneplayer game where piles start with an initial number of
chips and any vertex with at least two chips can send one chip to the piles on
either side of it. When all of the piles have no more than a single chip, the
game ends. In this paper we review fundamental theorems related to this game,
including the fact that termination and final configuration are independent of
the sequence of moves made and prove the number of moves required for
termination is bounded. We then extend the game to consider distinct chips,
where chips are represented by integers and firings result in a comparison of
two chips in a pile such that the smaller is sent left and larger is sent
right. We prove that for odd numbers of chips some final configurations are
sorted while others are unsorted and conjecture that for even numbers of chips
the final configuration is necessarily sorted.

Jon Lu:
Analysis of the ChowRobbins Game
(final version 5/10/2016)
Flip a coin repeatedly and stop whenever you want. Your payoff is the proportion
of heads and you wish to maximize this payoff in expectation. In this paper,
we will derive upper and lower bounds for the expected value of this game. We
will also examine stopping conditions for several different strategies and expected
payoffs for these strategies.

Greg Lubin:
Strategies for Pairwise Kidney Exchange
(final version 5/13/2016)
Because kidney donors are often not compatible with
their intended patients, exchanges can be arranged between one
incompatible patientdonor pair and another. Given a set of constraints,
there are certain strategies by which we can maximize the
efficiency of this process among a pool of patientdonor pairs while
incentivizing hospitals to submit truthful data. I will present a randomized
matching mechanism which has an approximation ratio
of 3/2 to the maximum cardinality matching, and is also expected
to motivate truthfulness. This is an improvement over previous
mechanisms, and is more efficient than any possible deterministic
truthful mechanism.

Lynn Takeshita:
Coloring Signed Graphs
(final version 5/12/2016)
This survey paper provides an introduction to signed graphs, focusing
on coloring. We shall introduce the concept of signed graphs, a proper
coloring, and basic properties, such as a balanced graph and switchings.
We will examine the chromatic number for six special signed graphs, upper
bound the chromatic number, and discuss practical applications of signed
graphs.

Gerrod Voigt:
Tree Decomposition, Treewidth, and NPHard Problems
(final version 5/12/2016)
This survey paper provides an introduction to the class of bounded
treewidth graphs, for which many NPhard problems can be solved effi
ciently. The notions of tree decomposition and treewidth are explained,
and subclasses of bounded treewidth graphs are analyzed. Furthermore,
recent findings about tree decomposition and treewidth are summarized.
Some (but not all) problems which are efficiently solvable on bounded
treewidth graphs are mentioned. In particular, solutions to two of them,
maximum weight independent set and coloring, are fully sketched in this
paper.

Liz Walker:
The Friendship Theorem
(final version 5/12/2016)
In this paper we explore the friendship theorem, which in graph theory is
stated as if we have a finite graph in which any two nodes have exactly one
common neighbor, then Then there is a node which is adjacent to all other
nodes. We provide a common proof of the friendship theorem, followed by two
extensions. The first extension relates to the number of common neighbors a
node must have. The second relaxes the friendship condition such that any two
nodes can have no common neighbor or one common neighbor.

Michael Wallace:
Housing Allocation: Existing Tenants and MultipleOccupancy
(final version 5/12/2016)
Housing allocation problems deal with assigning indivisible
objects (houses) to agents who have preferences over these objects. We
examine the housing allocation problem with existing tenants, then with
both new and existing tenants. In both cases, we present and evaluate
an algorithm for assigning houses. We then consider a version of the
housing allocation problem where agents are grouped into groups of size
k and houses can be occupied by k agents. We present an algorithm to
assign houses to agents in this case, and evaluate this algorithm.

Jeremy Wohlwend:
Elliptic Curve Cryptography: Pre and Post Quantum
(final version 5/12/2016)
Publickey cryptography has been at the center of online communication
and information transfer for decades. With computing power growing
at an exponential rate, some of the most widely used encryption schemes are
starting to show their limits. The RSA algorithm, which is still widely used
around the world, now requires very large keys to ensure security. Since these
systems may appear on low computing power devices such as mobile phones,
or chips, it has become essential to create protocols for which we can reach the
same level of security without spending considerable computing power setting
up the system in the first place. Elliptic curve cryptography (ECC) provides an
exciting alternative to RSA, and has shown to be a lot more efficient in terms
of key size. In this paper, we provide a description of how elliptic curves are
used in modern cryptography, as well as their current limitations and future
prospects. Because quantum computers pose a serious threat to the currently
in use publickey systems, we also describe the recent progress on super singular
elliptic curves isogenies, which may offer a quantum resistant cryptosystem
and a viable alternative for the future of elliptic curve based cryptography.

Aaron Zwiebach:
Random Walks and Percolation: An Analysis of Current Research on Modeling Natural Processes
(final version 5/12/2016)
In this paper we will analyze research that has been recently
done in the field of discrete mathematics, specifically relating to modeling
natural processes. We examine, summarize, and explain the proofs related to
the following natural processes: Percolation and Random Walks. We present
the main results of two papers and provide the proofs to demonstrate how
combinatorics, expected value, and generating functions, among other tools,
can be used to model such natural processes using discrete mathematics.
last updated: May 13, 2016