Schedule:

1. W 02/03/2016. Introduction.

2. F 02/05/2016. Alex Postnikov: Random Walk on a Line, aka Drunk Man Problem.
   Related links: Catalan numbers, random walks.

3. M 02/08/2016. Berj Chilingirian: Arrow's Impossibility Theorem.

4. W 02/10/2016. Liz Walker: Shuffling a Deck of Cards.
   [Proofs for the Book, Ch. 28]

5. F 02/12/2016. Anne Kelley: Zero Knowledge Proofs.

6. Tuesday! 02/16/2016. (Monday schedule of classes on Tuesday) Gabriella Baracchini: Linear Programming

7. W 02/17/2016. Lynn Takeshita: Stable Marriage and Stable Roommate Theorems.

8. F 02/19/2016. Xinyue Deng: Graph Search Algorithms.

9. M 02/22/2016. Jon Lu: Simple Games with Not-So-Simple Strategies.

10. W 02/24/2016. Michael Wallace: The Probabilistic Method.

11. F 02/26/2016. Open mike.

12. M 02/29/2016. Jeremy Wohlwend: Facts About Pi.

13. W 03/02/2016. Cali Gallardo: Latin Squares and Codes.

14. F 03/04/2016. Margaret Carpenter: Knapsack Problem.

15. M 03/07/2016. Gerrod Voigt: Diophantine Equations.

16. W 03/09/2016. Alex Postnikov: Cayley's formula for the number of trees and Prufer's codes.

17. F 03/11/2016. Aaron Zwiebach: Diffie Hellman and RSA Encryption.

18. M 03/14/2016. Greg Lubin: Pigeonhole principle and Ramsey numbers.

19. W 03/16/2016. Alex Postnikov: Acyclic orientations of graphs.

20. F 03/18/2016. Berj Chilingirian: Pink noise and sandpiles.

    03/21/2016 - 03/25/2016. no classes - Spring vacation

21. M 03/28/2016. Gerrod Voigt: Finding Hamiltonian cycles: Bondy's proof of Ore's algorithm.
            Lynn Takeshita: Chromatic numbers of signed graphs.

22. W 03/30/2016. Anne Kelley: Erdos-Szekeres theorem.

23. F 04/01/2016. Jon Lu: 100 Prisoners Problem.

24. M 04/04/2016. Xinyue Deng: Turan's Theorem

25. W 04/06/2016. Greg Lubin: Inductive Reasoning and Bounded Rationality
            Margaret Carpenter: Cops vs Robber

26. F 04/08/2016. Jeremy Wohlwend: Elliptic Curve Cryptography
            Aaron Zwiebach: Percolation - A Study in Randomly Generated D-Ary Trees

27. M 04/11/2016. Liz Walker: Bin Paking Problem (papers: [Korf] and [Martello-Toth])
            Gabriella Baracchini: Bertrand's Ballot Problem.

28. W 04/13/2016. Michael Wallace: A Probabilistic Proof of a Formula for the Number of Standard Young Tableaux of a Given Shape.

29. F 04/15/2016. Discussion about writing a paper.

    M 04/18/2016. no class - Patriots Day

30. W 04/20/2016. Berj Chilingirian: The Sandpile Load-Balancing Problem for Latency Minimization.

31. F 04/22/2016. Anne Kelley: Chip Firing Games,
            Liz Walker: The Friendship Theorem.

32. M 04/25/2016. Alex Postnikov: Abelian Sandpile Model and Spanning Trees.

33. W 04/27/2016. Lynn Takeshita: Efficient Algorithms to Color Signed Graphs,
            Jeremy Wohlwend: A Survey of Modern Public Key Cryptography.

34. F 04/29/2016. Gerrot Voight: Tree Decomposition, Treewidth, and NP-Hard Problems.

35. M 05/02/2016. Xinyue Deng: Lenstra-Lenstra-Lovasz Lattice Basis Reduction Algorithm.

36. W 05/04/2016. Margaret Carpenter: The Single Jeep and Jeep Caravan Problems.

37. 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.

38. M 05/09/2016. Gabriella Baracchini: Catalan Numbers and Possible Applications,
            Jon Lu: Analysis of the Chow-Robbins Game

39. W 05/11/2016. Michael Wallace: Housing Allocation with Multiple Tenants

• Gabriella Baracchini: Dyck Paths and Up-Down Walks (final version 5/12/2016)
  This paper analyzes Dyck Paths and Alpha and Beta Up-Down 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 Up-Down 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 real-life 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 Load-Balancing 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 self-organized 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 heuristic-based approaches via simulation.
• Xinyue Deng: An Introduction to Lenstra-Lenstra-Lovasz Lattice Basis Reduction Algorithm (final version 5/13/2016)
  Lenstra-Lenstra-Lovasz (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 one-player 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 Chow-Robbins 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 patient-donor 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 patient-donor 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 NP-Hard Problems (final version 5/12/2016)
  This survey paper provides an introduction to the class of bounded treewidth graphs, for which many NP-hard 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 Multiple-Occupancy (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)
  Public-key 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 public-key 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